Download Fergany, Nader.; (1970)On the macro-dynamic stochastic treatment of the size and age structure of a human population."

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ON THE MACRO-DYNAMIC STOCHASTIC TREATMENT OF THE SIZE
AND AGE STRUCTURE OF A HUMAN POPULATION
By
Nader Abd-Elmaksoud Fergany
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 666
February 1970
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NADER ABD-ELMAKSOUD FERGANY. On the Macro-Dynamic Stochastic Treatment
of the Size and Age Structure of a Human Population. (Under the
direction of H. B. WELLS.)
Macro-dynamic stochastic formulations of the size and age structuIe
of a human population are considered.
No restrictive assumptions are
placed on the physical process of population change and demographic variables are assumed to have the simple stochastic structure of a constant
depending on time plus a stationary random error component.
Probabilistic characterizations are obtained for the population
size at a point in time under discrete and continuous probability structures, and the probability distribution of the number of individuals in
two different age groups at a certain point in time is described by its
lower moments in general and in the special case of an underlying multi-
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variate normal distribution.
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of a sample of independent observations are biased and that it is possible
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The theory developed is illustrated using
data for the population of Egypt.
The problem of estimation of a certain function of joint moments
of a multivariate distribution using a sample of correlated observations
is discussed briefly.
It is shown that the usual estimators in the case
to arrive at unbiased estimators in the case of m-dependent processes.
The complexity of the variances of estimators considered prevented a
more conclusive study based on the mean square error criterion.
A Monte
Carlo study of a special case in small samples indicates that the degree
to which an estimator uses the sample information is the major factor in
determining how well the estimator performs in terms of mean square error.
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ON THE MACRO-DYNAMIC STOCr~STIC
TREATMENT OF THE SIZE A~~ AGE
STRUCTURE OF A Hill-1AN POPULATION
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by
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Nader Abd-Elmaksoud Fergany
A thesis submitted to the faculty of
the University of North Carolina at
Chapel Hill in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy in the Department
of Biostatistics
Chapel Hill
1970
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BIOGRAPHY
The author was born August 20, 1944, in Gieza, U.A.R.
He
graduated from the Saidia high school in Gieza in 1959.
He received the Bachelor of Science degree with a major in
applied statistics from the Faculty of Economics, Cairo University
in 1963.
He then joined the Faculty as an instructor.
Since
October, 1965 he has been on leave from his position pursuing
graduate studies in statistics at the University of North Carolina.
The author is married to Moushira Khattab of Cairo, U.A.R.
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ACKNDtVLEDGME1TTS
The author expresses his appreciation to members of his advisory
committee, Professors A. Hawley, D. Quade, M. Sheps, H. Van Der Vaart
and H. B. Wells.
Professor Wells endured the arduous task of advising the author
for more than three years and all members of the Committee gave valuable
assistance and counsel.
The financial support of the Population Council and the Carolina
Population Center
~s
gratefully acknowledged.
Mrs. Gay Goss typed the dissertation beautifully and efficiently.
Finally, it gives the author
~reat
pleasure to acknowledge the
significant contributions of the many wonderful people who were associated with
the Faculty of Economics at Cairo University during his years
of study and work there to his education and edification.
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TABLE OF CONTENTS
Page
LIST OF TABLES .
vi
LIST OF FIGURES.
vii
Chapter
I.
INTRODUCTION AND REVIEW OF LITERATURS. .
1
1.1.
1.2.
1
2
2
8
1.3.
II.
POPULATION SIZE . •
2.2.
2.3.
2.4.
III.
Theoretical Formulation, Discrete Probability
Structure. . . • • • • . • . • • •
A Discrete Error Probability Law . .
Theoretical Formulation, Continuous
Probability Structure. •
An Example
.•••
AGE STRUCTURE. •
3.13.2.
3.4.
IV.
Introduction. • . . . • • • • .
..• .
Review of Macro-Analytic ~fude1s in Demography.
1.2.1. Deterministic Models • • • • • • • •
1.2.2. Stochastic Models • • . • • • • • • • •
Objectives of the Present Study and Approach
Adopted. •
• . . .
Introduction
Statistical Formulation.
3.2.1. Introduction ••
3.2.2. The Multivariate Normal Case •
A Discrete Approximation to the Human
Survivorship Function.
An Example . . • • • • •
SOME ASPECTS OF THE ESTIMATION PROBLEM •
4.1.
Estimation of a Certain Function of Joint
Moments of a Multivariate Distribution Using
the Same Function of the Joint Moments of a
Sample of Dependent Observations • • •
13
15
15
20
27
30
37
37
42
42
45
46
48
52
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Page
Chapter
4.2.
A Monte Carlo Study of Some of the Properties
of Five Estimators for the Variance of the
Product of the Two Components of a Bivariate
Normal Time Series in Samll Samples . •
4.2.1. Introduction
4.2.2. Procedure ••
4.2.3. Conclusions.
.
.
V.
....
60
60
63
66
SUMMARY AND SUGGESTIONS FOR FURTHER RESEARCH .
69
5.l.
5.2.
69
71
SlJlTIIO.ary • • • • • • • • • • • • •
Suggestions for Further Research .
Appendices
A.
DATA AND INTERMEDIATE CALCULATIONS FOR SECTION (3.4) .
73
B.
COMPUTER PROGRAM . .
78
C.
NUMERICAL RESULTS OF THE MONTE CARLO STUDY
87
LIST OF REFERENCES • • • • • • • • • • • . .
92
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LIST OF TABLES
2.2.1.
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2.4.1.
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3.4.1.
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A-2.
A-3.
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A-4.
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A-5.
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Page
Table
C-l.
Comparison of the Discrete Error and Syrr~etric
Binomial Distributions for k = 1,2,3 . • . .
26
Estimated Probability Distribution of the Population
of Egypt in Thousands on 9/30/1960 . . • • • •
35
Estimated Male Births in Thousands and Survival
Probabilities. • • • •
50
Birth Rate per 1000 Population Corrected for
Under-Registration, Egypt (1940-1959).
73
Estimated Mid-Year Population in Thousands,
Egypt (~940-1959). • . • •
• ••••
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Reported Death Rate per 1000
(0-4), Egypt (1940-1959) .
75
}~les
in the Age Group
. .•.•.
Average Ratio of Single-Year Death Rates to the Death
Rate of the Age Group (0-4), Egypt, 1937, 1947 and
1960 Life Tables .
• • • • •
••.•• • • .
76
Ratio of Crude Death Rates Corrected for UnderRegistration to Reported Rates, Egypt (1940-1959).
77
Relative Mean Square Error of the Five Estimators ••
87
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LIST OF FIGURES
Figure
2.4.1.
Page
Annual Rate of Natural Increase,
Egypt, 1947-1959 . . . . • • . . • • • . • . . . • . . .
34
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CHAPTER I
II
INTRODUCTION A1'n REVIEH OF LITERP.TURE
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1.1.
IntroG'Jctic.d
The use of
math~natics
as a
mediu~
of expression in a certain
branch of human knO\vlcdgc. is considered by some to
of an eXact f.:ci.ence. g::owing out of that brallc.:h.
b(~
a sign of the davm
Vie,vcd in that light,
the incre.:'sed use of mathematical models in dCinography in recent year" -
as means of both enr:tching the theory and deriving sound and ,\\lorkable
methodologies - is a definite manifestation of the advent of an era of
sc:ienU.E:ie 8cphisticLdon in th:L:; arCH.
That consideration of stochastic varie.tJons in a Tllodel improves
its
resem'l:llanct:~
to reality and thus increases its utility is indicate!'!
by Mindel C. Sheps [27J in the special case of the study of patterns of
human reproduction:
"\\hile
wan or expf:c:tcd v2.1ucs given by proh2hiJ··
j1
istic model.s agree with those of deterministic
n~dcls
based on suffi-
ciently simile.T assumptions, stochastic models offer insight into the
effects of chance and into the variations that occur natu'rally in the
biologicaJ system."
However, we must not lose sieht of the fact that
the role played by pure chance variations in the problems of human
population c1Jsn.ge tends to be small since the number of individuals
considered is usually very large.
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Unfortun.1.tely, each of the existing m;1cro-analytic stochastic
models of population change suffers one or more of the following
comings:
~;hort­
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2. couplicGl t e.d
of needed
p<'4r<,.mc,t~:rs
li,O th'l1T,,'.~ U.cal
d~,.t:iv~1tions,
[rom ac:tual data \vhich
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both
ji:1P(11.tWlt
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chal1engil1g.
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1.2. .P..c'yjE::z..s~LJ1f.9:Q=AT:"Q.:1:.Y.t1_c:-11o d f;:2.::s jE__ Q~'Jg<?Zl.:~~.rhy
O~·
continuo'ls time and age
variabl(~~;.
The cle.ssic of discrete time deterudI1istic fornl'..llntions of t:kc
problem is that of
b~;cat1se
briefly
]~es15.e
[l9] and
it .serves as the
v.T€~
shall
st~1.rti!lg
):C.ViE··....'
it!::
ba~;ic [.,t·rl:cL.;re
pDint of alT'lost all ege-
specific discrete time stochastic treatments of the problei 1 of populn1
tion change.
Leslie considered only the female popt1letiOll and assu,n2d th;::L Dge-·
specific rates are independent of time.
He also made thz follcMing
definitions:
l1
xt
- the number of femaler; alive
hl th(~
age group (x, x+l]
at t.ime t)
P
x
"" the probability that a female in the ;:-.go group (x, x+ 1]
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at time t will be alive in the age group (x+l, x+2]
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at time t + 1, 1
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Px
~
0,
F = the number of daughters born in the interval (t, t+1]
x
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per female alive aged (x, x+l] at time t, who will be
alive in the age group (0,1] at time t + 1, F
> 0,
x-
and
F
M=
o
o
o
...
m-1
F
m
o
o
o
o
P
o
m-1
where (m, m+l] is the last age group attainable in the population.
It then follows that
~+1 =~
where
Et
• • • (1.2.1.1)
is an (m+l) column vector giving the female age distribution
at time t, or
n
where t
=0
t
= Mt --0
n
• • • (1.2.1.2)
stands for some initial time period.
Leslie then notes that if (k, k+l] is the last age group in which
reproduction occurs, F
x
= 0 for x > k and M may be partitioned symmet-
rically at this point as
whence
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+
[-f(;'::cl J ~j
t
=
M
t
But, since C
=0
for t ~ m - k, he concluded that it is the submatrix
A which is of primary importance and gave his mathematical treatment
in terms of it.
The prinicpal continuous time deterministic formulation is due
to Alfred J. Lotka [21], [22], [23].
Since it will serve as the basis
for our stochastic treatment, some aspects of Lotka's theory that are
relevant to this treatment will be reviewed here.
Let x denote age, x
~
0,
t denote time,
N(t) be the number of individuals in the population at time t,
c(x,t) be the proportional age distribution function at time t;
a
i.~., J
2
c(x,t) dx
= the
proportion of individuals in the age group
00
J
c(x,t) dx
= 1,
o
B(t) be the number of births in the population at time t and
bet)
= B(t)/N(t)
be the corresponding birth rate,
D(t) be the number of deaths in the population at time t and
d(t) = D(t)/N(t) be the corresponding death rate, and
p(x,t) be the probability that an individual born at time
(t-x) will survive to be of age x at time t.
Considering a closed population,
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and using the terminology
1A closed population is one that increases only through birth and
decreases only through death.
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presented above, Lotka derived the fo11mV'ing relations
co
N(t) =
J
B(t-x) p(x,t) dx
• • •(1.2.1.3)
o
B(t-x) p(x,t)/N(t)
c(x, t)
• • .(1. 2.1. 4)
He then remarked that if the survivorship function p(x,t) and
the proportional age distribution function c(x,t) are mathematically
independent of time, both the birth and death rates of the population
will be constant and so will be the rate of change in the size of the
population.
If the birth and growth rates are denoted by band r
respectively, it follows that
N(t) = N(tt»
e
rt
.B(t) -- B(ta) e
rt
D(t)
e
rt
• (1.2.1.5)
where to is some initial time point.
It also follows, from relation
(1.2.1.4) above, that
c(x)
b p(x)
erx
• • • (1. 2.1. 6)
We observe that all functional forms of the age structure presented above depend fundamentally on the survivorship function p(x,t).2
In the following discussion, we shall be concerned with one generation real or synthetic - survivorship function, or a p(x,t) that is constant
2It should be noted that p(x,t) is rather complex. It is not a
"generation" survivorship function in the sense of a pure death process
describing the mortality experience of a cohort of births; nor is it a
"period" survivorship function describing the age-specific mortality
experience of a certain population at some point in time applied to a
synthetic cohort of births. Actually, p(x,t)V x at tDne t describes
segments of the mortality experience of as many birth cohorts as there
are ag~ points at time t, with the mortality experience of younger ages
represented more often than that of older ages.
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over time; hence the argument
will not be displayed.
t
In addition to the prominent role the survivorship function plays
in det.ermining the age structure, it determines a life table completely.
For example, the life expectancy at age x, e
x
say, is given by the
expression
ClO
ex = (
J p(z)
• . . (1.2.1. 7)
dz)/p(x)
x
Consider a pure death process in which a cohort of N(O) births
goes through life until the last member of the cohort dies.
Let N(x)
denote the size of the cohort at age x (in such a process, age and time
since the start of the process coincide, of course) then the survivorship function is given by p(x)
= ~~~~
If the instantaneous death rate at age x,
~
~(x),
is a constant,
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say, then
-d
a~x)
= -
~ p (x) which gives the well-kno\im negative exponen-
tial relation
p(x)
on using the initial condition p (0) = 1. [7]
• • • (1.2.1.8)
The last relation defines
what might be called a constant-attrition survivorship function.
However, a survivorship function that describes human mortality
conditions cannot be expected to follow the constant-attrition survivorship function, for the risk of human mortality is a function of age.
The most general approach to a human survivorship function is to consider the probability of death as a function of age,
~(x),
in which case
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we have
E.-Pd;Z-(x)
condition p(O)
=
"
' il.la
' 1
= - II (x) P (x) glv1ng
- on ut1'1"1Z1 ng t Ile 1n
1,
x
p(x)
=
expo (-
I ll(~)
ds)
• • • (1.2.1.9)
o
It remains, of course, to determine the mathematical form of ll(x).
The earliest attempts to find a general mathematical expression
that
i·~·,
descri~es
the human survivorship function aimed' at it directly;
considered p(x) or N(x) rather than ll(x).
De Moivre conjectured
in 1725 that p(x) decreases in an arithmetic progression.
Gompertz introduced his famous "Laws of Mortality." [8]
which was later modified by Makeham in 1860 to ll(x)
In 1825,
ll(x):: 13yX
= a + Byx. [7]
However, this law was not applicable at all for ages before tr.alurity.
Later developments - surveyed by Frechet [13] - started with ll(x) or
a variation thereof.
Two remarks are in order here.
First, most later attempts did
not aim at a law of mortality that is based on a general theory regarding the underlying biological process but rather at fitting mathematical expressions to observed age specific probabilities of death.
This resulted in expressions that were too complicated formally and/or
too specific to yield the analytic advantages that would result from
a general mathematical expression.
Second, although the reasons are
clear - namely, the complexity of a general expression that describes
human survivorship for all ages - the shift of the attention of
researchers in this area from p(x) to ll(x) and from starting with an
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und~rl yi Db theory to fit Un:", appropri ate fune tional [orw:; to observed
lo,;~;
p(x) values is nnftn-tuDc.te and represents
[l
in the dt'gl'('(' of
generali ty of the re.sultinG exprcs,d ons.
1'or not much faith
('.pn
~IC
placed i.n an empi.d cal fitting that is not backoG by an acc(1"ta1110
theory; and - clearly .- if
).1
(x) represent" I-he probability of deatll in
an interval 6x starting at age x, then Jl ();) is a monotonicall y nondecreasing function of t..x and the fitted form - although it may not
chDuge basically - will depend on the age units used in the fitting
process.
Nore recently, typical patterns of mortality have
be~~ll
through the presentation of serie's of "model" life tables.
.
1. 2.2. Stochastic Nodels.
c):ist for
clD~;sical
debc,ihed
[8]
Although mathCrfi3 tical fonaulnti
birth and dc::ath processu; tk; t
ghTt'~
the population as a ftlnctton of ti.me, the syt';1:eJ11s of
or,~.:
the size of
equati(lm~
resul t--
ing \olhen birth and de.ath rates are allowed to he ge.neral functions of
time are not easy to solve and practical applications an. c:umberse.·l'1c.
[3L
[12]
The first age-specific stochastic model of population change is
ascribed to Bartlett. [4]
age groups.
Bartlett used discrete time and discrete
He considered females only, assuTIied fertility and mortalit.y
rates to he constant over time -and wi.thin each age group} and derived
asymptotic forms for the linear and quadratic moments of thc num1H'r of
females in an age group when the age groups and time intervals \-.'ere
both made small.
Three years later, Kendall
[17] discussed an age-specific con-
tinuous time stochastic model of population changr~.
Kf";nc1all formulated
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the problem using N(x,t) to describe the state of the population at
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time t in the sense that the Stieltjes integral'
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the number of individuals in the age group (x ,x ].
l 2
2
dN(x,t) is
He admitted that
distribution problems are extremely difficult but found it possible to
arrive at some of the moments.
Under the following assumptions:
1. one sex
2. the subpopulations generated by two co-existing individuals
develop
in complete independence of each other
3. the birth rate varies with age but not with time
4. the death rate does not vary,
either with age
or with
time, Kendall developed equations for
E (dN (x, t) ) ,
Var (dN(x,t», and
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Cov (dN(x,t), dN(y,t».
The equations are difficult to
solve, except in some unrealistic special cases.
For example, in his
paper, Kendall solved the case in which both the birth and death rates
are independent of age and time.
Simpler methods exist for solving
this problem now. [12]
More recently, age specific stochastic models of population
change were formulated in terms of discrete time and corresponding discrete age grouping.
Two formulations are worth mentioning here.
The
first, due to Pollard [24], revived interest in this type of model and
the second, due to Sykes [28], contains one of the most general treatments of the problem available.
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Pollard considered females only, restricted his age distribution
to prercoprodl1ctive and reproductive ages
Cllld
DssUlUed thnt
1. fertility rates and survival probabilities in Leslie's
formulation represent binomial probabilities that are independent of
time,
2. a certain individual's death or procreation is independent
of other individuals' death or procreation.
Then using Leslie's notation, he derived an expression for
e
the vector of expected number of females in the (k+l)
-t
age groups (0,1], (1,2], •.. (k,k+l] at time t and
c(t) the vector that has as its elements the variances and
,
covar1ances
0
f t h e num b er
0
f f ema 1 es i n t1e
1 1. th and'J th age groups at
time t, c (t) say, l'1ste d in a dictionary order according to the subij
scripts i and j
(for i f: j,
ci~)
and
c;~)
are both listed).
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This expression is
(~n)) - (-:-
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A 9
=(~A-::A)n~O- :(0)-+- ;nj=l
• • • (1.2.2,1)
where B is a (k+l)2 x (k+l)2 matrix whose elements are functions of
F
x
and P , x
x
= 0,
1, 2, ••• , k, and A
matrix A by itself.
moments of '!!'t'
e.
A is the direct product of the
These results were generalized to higher order
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It is in order here to point to the tremendous computational
effort involved in applying (1.2.2.1) for a moderate number of age
groups, if ,,-,e decide that the simplifying assumption on which the above
results are basod are tolerable.
Sykes also restricted attention to (k+l) age groups at or below
the limiting age of reproduction and defined stochastic versions of
Leslie's model by assuming respectively that:
1. the deterministic model is subject to additive random
errors,
2. the elements of the transition matrix A are binomial
probabilities,
3. the transition matrices are random variables.
If A
t
is the' Leslie transition matrix for the time interval
(t,t+l], Sykes shows that in all three cases considered
n ) = ( 1T
E(=t
t-l
A.)
i=O].
Eo
= fl
-t
• . • (1.2.2.2)
We give below some of hi.s further results together with the assumptions
required for them to hold in the three above mentloned cases.
• •• (1.2.2.3)
(1)
where {£ } is a sequence of random vectors with
-t
E(_€t)
=0
and Cov (£
-s , -£t)
= r s,t
s, t
0, 1, 2, ...
whence
Var (n
)
-t
=
L
t-l
j=O
( 1T
t-l
k=t-i
• • . (1. 2 •2.4)·
12
Further, if {~1:} is a normal stochastic process,
as a multivariate normal with mean
Qt
_
~ -~)
,
-1
Vt
}l-t
<It-t; -
3
t.hen !It is distributed
and variance Vt.
~)
Hence
2
is distributed as Xk+l
· • . (1.2.2.5)
which may be used to place confidence intervals on
(2) Assume that individuals in the i
th
~.
age group give birth cmd
die with probabilities tbi and (1 - tSi) respectively at time t and that:
all events result from mutually independent binomial trials.
This is
siulilar to Pollard's formulation except that Sykes allo'lt!s the transi-·
tion matrix to depend on time.
The variance matrix of
~
is given by
the relation
• • • (1. 2 . 2 .6)
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where
= (t a .. (1 - ta..»
~J
~J
(3)
· • . (1.2.2.7)
where At is a known Leslie matr~x and' {6
t
matrix random
variabl~s
with E (6 )
t
}
is a sequence of independent
= 0 and Var (6 t ) = St.
case, Sykes gave the variance matrix of
~
3i.~., for any finite subset (t l , t z'
In this
as
... ,
sequence of nonnegative integers, the vector ~'
has the n(k+l) - variate normal distribution.
t ) of indices in the
n
=
,
~
1
E'
-t ,
2
£'
••• , --1:
)
n
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vt =
t-1
L
t=O
t-.Q,-l
( 'IT
i=O
t-t-l
Ai) Pt (
'IT
i==O
A.) ,
1.
. • • (1.2.2.8)
where
P
t
is a (k+l) x (k+l) matrix whose ijth element is given by
and
Again, we can easily see that, even if we accept the various
restrictive assumptions of the above formulations, it is very difficult
to put their results to reasonably detailed practical applications.
As a case in point, in the example Sykes gave in his paper, only three
age groups were considered and the transition matrix did not vary with
time.
1.3. Objectives of the Present Study and Approach Adopted
The purpose of this study is to investigate some aspects of the
mode of change in the size of a human population and its age structure
under simple and realistic stochastic set-ups with the hope of producing
practical methodologies for the demographer who wishes to incorporate
stochastic variations in his calculations.
The line of development of
the stochastic formulations employed, which is a stochastic extention
of some of Alfred Lotka's work [23], is not directly related to any of
the existing macro-analytic stochastic models and the resulting formu1ations do not suffer from most of the shortcomings of the previous ones
summarized in (1.1).
There is some conceptual difficulty in a stochastic treatment of
14
lnnnan population change in
3
certain society since in reali.ty we have
only one such society and it i.s difficult to imagine hm,T that society
could be subject to chance variations.
HOHever,
observed pattern of population change as a
from a
tlniv.~rse
of possible patterns.
VIC
s~uple
may consider the
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of one observation
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This is the approach adopted
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here.
The unit of analysis considered in this treatment is a human
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population possessing certain characteristics and subject to vital
change continually affecting the number and characteristi.cs of its
members.
This vital change is effected through two
proces~'es,
il1cremen-
tation by the birth of new individuals and imnligration, and decreruentation by the death of living individuals and emigration.
Such a popula-
tion may be described by three types of random variables expressing
forces of incrementation, decrementation, and the resulting population
as a function of time.
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Thi.s way of describing the dynamics of hUlr.an
population change is most natural and could be considered as an attempt
to theorize the underlying concepts of population change in classical
(numerical) demographic analysis.
In the present formulation, demographic variables are assumed to
have the stochastic structure of a constant depending on time plus a
stationary random error component.
Although the exposition of the
present formulation is set forth entirely in terms of human populations,
the principles involved are applicable to all other types of populations, and so are the results.
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CHAPTER II
POPULATION SIZE
2.1.
Theoretical Formulation, Discrete Probability
Consider the probability space·
en,?,
p) where
St~~cture
n =. {w l 'w 2 ' ... }
is a set of a countable number of elements,~ is the power set of
and P is a probability measure on
Elements of
n
:P.
n are identified here with conglomerations of
relative socio-cultural conditi.ons - in the broadest sense of the word
whose effect on the factors affecting change in demographic variables
is to introduce stochastic variations as a component of their total
variation.
Although the theory presented here does not depend on the
nature of the probability
as normal-type
l
m~sure
P, stochastic variations are treated
discrete errors around an expected value determined by
overall socio-cultural cllange expressed in terms of time trends in
demographic variables.
n are
Accordingly, elements of
having a one-to-one correspondence with the class
~f
conceived of as
integers - or a
symmetric subset of it - such that they could be arranged as
n = L .. ,w_2 ,
w_ , w ' WI' w ' ••• } where Wi' i
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2
=
0, ±l, ±2,
••• ,
denotes an i-unit deviation. from the expected value; and the probability measure P is conceived of as assigning probabilities to elements
li.~., symmetric around the expected value which is also the
mode of the distribution.
16
of
jJ in
the fashion of an error probability distribution about some
expected value.
It is probably more realistic to think of relative sociocultural conditions as forming a continuum, but the case where st is
countable has a simplified probability structure and for this reason is
presented first.
A continuous probability structure version will be
considered in (2.3) below.
st -+J:lwhere 0 < T < t and']{is the set of non-
Let Nd("r,e)
=
negative integers such that Nd(T,W)
n
d
(T) is the size of the popu1aW
tion at time T corresponding to wEst, which is a discrete function of
time.
For the sake of mathematical convenience, we shall approximate
Nd(T,e) by the differentiable function of time N(T,e).
st -+ A, where 0 < T < t and A is a countable set of
Let R (T, e) :
points on a finite real
inte~"Va1
such that R (T,W)
=
r (T) is the rate
W
of change in the size of the population of time T corresponding to
WEst defined by
. (2.1.1)
-1
such that R
(T,e) exists.
From (2.1.1), it follows that
nw(t)
=
nw(O) expo
~ f
t
• . . (2.1.2)
o
where t
=0
stands for some initial time point.
In order to consider the probability structure while allowing
the relative socia-cultural conditions to change with time we define
'W (e)
: (O,t] -+ st
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11
such that~(T)
= WT
existing at time
functions W.
'l" E:
is the set of relative socia-cultural conditions
(O,t].
Let-;t (T) be the set of all possible such
We may now think of R(T,·) T
T
into a space of functions.:J (T, .) T
R(T,W J
1
= rW
(O,t] as mappingdt (T)
E:
(0, t] whose typical element
E:
(L) represents the functional dependence of the rate of
L
change in the population size on time and the path of relative sociocultural conditions prevailing in the time interval (O,t].
also be redefined in a similar manner.
Note that r
W
T
N(T,·) may
(T) and n
W
(T)
L
are "sample functions" in the ordinary sense of stochastic processes i f
we take our underlying sample space to be
Q.
7f
O<T<t
Now, equation (2.1.2) may be written as
n
* (t)
=
(0) exp.
n
(1.)0
(J
t
o
r
W
(L) dL)
T
¥wL E: 1: (T)
. . . (2.1.3)
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for relati.on (2.1.1) holds for any path W
L
replacing W at T.
(Note that r
w
s"£
(L) L E: (O,t] with III
T
(L) is a function of T only).
1"
Clearly, the sample space to consider in discussing the probabili.ty structure of r 1" (1") and n (1") is ~ (T).
W
w
However, any probability
1"
measure on &
(ct
(1"»
- the 0- field on i.. (L) - is of limited practical
interest in the problem of human population change of one society since
we would not be able to estimate it.
All the data normally available
represent only one element i~ (f (T).
A more useful probability structure results from considering
equation (2.1.2) in conjunction with the probability measure P
when it is assumed to renmin stationary over time.
~n jQ
Given the initial
18
= 0,
population size at t
the foun of the time-trend in
R(T,") - its
expected value as a function of time - and the probability law P, the
probability distribution of N(t,") could be determi.ned - or approxi-mated - making use of remarks (1) and (2) below.
Remarks:
n
(1) Let X
Al
y
where
n,
~
AI' and A
2
Al
~
X(w) = x,
A :::,;) y (x) ::: y
2
-+
1
are countable sets elements such that P(X- ) and
P(X- l y-l) are defined.
In other words,
~
~-l
X :
(n,
y
(Al ,6!I' P X ) ~ (A2'~' P X
P) ->- (AI' \..:;t-l' P X
L:::V,
.
-1
),
-1
-1
Y )
1
where P is a probability measure on the cr-field~, (p X- ) is the
probability measure on ~l induced by X, and (p X-I y-l) is the probability measure on ~2 induced by Y.
The probability law of the random variableY is determined completely by the probability law of X, or by P on ~, in a straightfoward fashion.
For, clearly,
::: p' {w}
(2) A discrete probability distribution may be regarded as an
approximation to a continuous (or a finer discrete) one with the probability mass points of the approximation located at 'some well-defined
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19
position within intervals of the continuous distrihution
(CJi
o
g1:0U}'S
of
mass points for the finer discrete one) and carrying the probability
measure associated with the corresponding intcl"vdls (or group of mass
points).
(3) In practice, annual rates of natural increase would be the
most reasonable estimates of rates of change in the population size
for they are usually available on a yearly basis and honce W'ould provide a sample of reasonable size over a relatively short period of
time.
But the rate of natural increase is not the same as the rate
of change defined by (2.1.J.).
For w
~
n,
let n(t+l)
= net)
r
e .
(Note that the subscri.pt
u)
is suppressed); i.£., the rate of population change is asswned to be
.
constant throughout the time unit (t, t+l].
Assuming a closed popula-
tion, the rate of natural increase for the same time period is defined
by
r'
- net)
= n(t+l)
n(t;J-~)
.(2.1.4)
=
r/2 ]
[net) e r - n(t)]/[n(t) e
=
e
r/2
-r/2
- e
= 2{ (r)
2
= r+
1
+ 3!
3
(1:)
{l
r
2 3: ('2)
2
3
5
1
+ 5: (L)
2
1
+5T (r)
2
+ ••• }
5
+ ... }
i..~.
r' > r
for
r > 0
and
r' < r
for
r < 0
• . • (2.1.5)
20
and
The absolute difference
bet,~een
rand r' is very small in the
customary range of values for these variab1es t but when we consider the
app1icati.on of these rates to relatively large
popu~ations
for a large
number of intervals such that r is constant within each interval but
may vary from one to another, we might do well to make the necessary
adjustment.
Tables of hyperbolic fUllctions would be of use in numer-
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ical conversions.· [14]
If the population is open to migration t the above4nentioned
relations hold with the only difference that r' represents the balance
of natural increase and net migration.
2.2. A Discrete Error Probability Law
From the properties of the probability measure P described at
the beginning of (2.1), it seems natural that the probability distribution of a variable like
I
R(T,·) at a certain point in time would take
the form of an error probability distribution around an expected value
determined by the time trend in
R(T,·).
An appropriately-labeled normal distribution or a binomial distribution with probability parameter
~
would serve this purpose.
How-
ever, the data we usually encounter in demographic estimation are not
commensurate with high degrees of accuracy in the sense of probability
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21
mass concentration around the mean, which makes an alternative to the
normal di.stribution in this regard desirable.
The same observation
applies to the synnnetric binorlinl distribution.
The purpose of this section is, taking the norm;)l distribution
as a standard, to devise a simple discrete error pro1nbility di.stribution that does not display a high degree of accuracy in the sense
described above.
It is well
k.110\vn
that the sum of independent con-
tinuous uniform random variables approaches the normal distribution as
the number of components increases, hence it was reasonable to expect
that the sum of independent discrete uniform random vari.ables would
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approach the form of a discrete error distribution as the number of
components increases.
The probability mass function (p.m.f.) of the
sum of three sucre random variables was judged to achieve a balance
between simplicity and closeness to the desired fonn.
This p.m.f. has,
in addition, the advantage of simple calculations of probabilities and
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the disadvantage of a restricted number of mass points.
Consider the probability distribution
x
= -k,
-k + 1, ..• , 0, ••. , k - 1, k
• • . (2.2.1)
Lemma (2.2.1)
It follows from the relation (2.2.1) that the probability
generating function (p.g.f.) of X is
G (5)
X
= E(sx)
=
-k
2k
('t' sx)
2k+l
'-'
s
x=o
• • • (2.2.2)
22
clearly,
° and
E(X) -
2
Vsr(X) =, k 3+k
may be calculated directly or as (;~ (1) •
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Let Xl' X be independently distributed according to (2.2.1),
2
pr{y :: y} ::
.::.J..yJ + 2k + 1
y :: -2k,-2k+1, •.. ,O, ... ,2k-l,2k
(2k + 1)2
. (2.2.3)
. The proof is immediate from first principles.
This gives the p.m.f.
of a discrete triangular distribution and may be used as the building
block for the next step ..
Theorem (2.2.1)
Let Xl' X , X be independently distributed according to (2.2.1),
2
3
m
1
=-
Iz I + 3k + 1,
ID
2
=-
Iz I +
Pr{Z
and
k, then
= z} = ~(z)/(2k+l)3 where
, Iz I ::
~(z)
=
ID
1 (m1 + 1)/2 - 3m2 (m2 + 1)/2,
3k
2
+ 3k + 1
Izl ::
z
k, k+l, ""
1, 2, ""
=°
3k
k-l
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.23
Proof
Let
~ (z) = number of \,'ays in \·:h:!_ch xl + xl + x 3 = z, from first
principles - using Lemma (2.2.2), or expanding the p.g.f. for Z,
2k
( L
4k
(x+l) sX +
x=o
2k
(4k-x+l) sx)( L sx)
x=2k+l
x=O
L
we get the following table
~(z)
Izl
+
+ (k+2) + (k+l)
+ (k+l) +
+ (lk+l) +
+ (k+3) + (k+2)
+ (2k+l) +
+ (k+4) +
+
(k+l) + (k+2) +
o
k
I
2
(k-l) +
k
+
(lk+l)
k-2
3
+
4
+
+ (2k+l) +
k-I
2
+
3
+
+
k
1
+
2
+
+ (2k-l) +
1
+ 2
+
k+l
(2k)
(2k)
-I-
(2k)
+ (lk+l)
(2k-l) +
I
3k-l
+ (2k-l)
+ (2k+l) +
(2k)
(k+3)
+
(2k)
2
I
3k
Clearly, for Z
=0
~(O) =
2
(2k+l) + (2k ) + 2
= 3k2 +
3k + 1
k(~+ll
24
Iz I =
and for
k, k+l, ""
3k
$(2) == m
(m +l)!2
1
l
Finally, for
Izi
1,2, " ' , k-l, after some algebraic monip01a-
=
. Further,
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tion we get
==
I
m (m +1)/2 - 3 m (m +1)!2
1
2
2
l
3k
3k
L:
~(Z) == (3k2+3k+1) + 2 L: i3k-z+.l)?k-Z+~.l
z=-3k
z=l
k-1
- 2 L: 1 (k-z) (k-z+l)
z==1 2
_I
3 2 3
+ 12k + 6k + 1 ;: (2k+l)
== 8k
i.e., (2.2.4) is a proper p.m.f.
Q.E.D.
We note the following:
(1) For Z == 0, m (m +1)/2 - 3 m (m + 1)/2 == 3k
2
2
1
1
2
+ 3k + 1
and hence Pr{Z == z} may be thought of as composed of three second
degree polynomials in Z as it is the case with the continuous uniform
variates.
(2) It is clear, from the symmetry of the p.m.f. around the
mean, that all the odd moments of Z vanish.
Using the p.m. f. or the
p.g.f. to evaluate the even moments directly is straightforward but
tedious.
However, it is easy to arrive at the variance as
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3
Var (z) ~ L Var (X.) by virtue of independence
~
. 1
~=
(3) The constant k determines both the number of probability
mass points, (6k+1), and the variance of the distribution
(4) lVar(Z)
.!...~., k < 0z <
k +
=
0 = ;k2+k
Z
~ since
k
+ k < (k +
t) 2
which means that all the
probability mass of the distribution of Z is concentrated in the range
. (5) The probability mass concentrated in the interval (-O'z ,oz)
cannot exceed
=
P
k-l
(2k+1)-3 {(3k2+3k+l) + 2 L [(3k-z+l) (3k-z+2) - 3(k-z) (k-z+1)]/2
~l
c
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2
+
k+l
(3k-z+l) (3k-z+2)/2}
L
z=k
...
l6k 3 + l4k
2
3 (2k+1)
- 2k - 3
3
< 2/3
• (2.2.5)
which means that the p.m.f. for Z allows for more uncertainty around
the mean than the normal distribution.
Alternatively, we may compute the kurtosis of Z
2
+ 13k
= 13k5k(k+1)
- 1
which lies in the interval
[2.5, 2.6) compared to the normal kurtosis of 3.0.
20
(6) T<lh1e
2.2.1
comp<1rc~
the proh,.l1d lily
llI:1SS,'~>
[1ssocL·,t.c·J
with (6k+l) mass poh1Ls for th(~ SY"1~'l;;c·tric tinonial ,mel tl iscret c en.o)"
distributior,s for k == 1, 2, 3 \·Jit.h the (Gk-Il) maSs points nu~,lbunc.!
1,2, ••• , 6k+1 in order, sbO\.Jlng that the:
di~;cr8te
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error disl:-il),llio;1
a11o\Js for more uncertainty tban the synunctri.c binomi;Jl.
TABLE 2.2.1
COMPARISON OF TIlE DISC1\ETE ElumR A!\D SYH:fETRIC Ell';OHIAL
DISTlaBUTIO:iS HJR 1<. :. 1, 2, 3
k
Mass Poi.nt
Prob3 bi}o!-_!:'y_l~:J ~'!___ ._.. __._
Q:Ls..£E.~.~e En:..or
1
2
3
13 i n..llli!.:1.0J_.CJ.__
1 or 7
.037
.0] 6
2 or 6
.111
.094
3 or 5
.222
.234
4
.259
1 or 13
.008
.0002
2 or 12
.024
.0029
3 or 11
.048
.0161
4 or 10
.080
.0537
5 or 9
.120
.1208
6 or 8
.144
.1934
7
.152
.2256
1 or 19
.0029
.000 004
2 or 18
.0087
.000 069
3 or 17
.0175
.000 584
4 or 16
.0292
.003 113
5 or 15
.0437
.011 673
6 or 14
.0612
.032 684
7 or 13
.0816
.070 816
8 or 12
.0962
.121 399
9 or 11
.1050
.166 924
10
.1079
.185 471
.313
---_._-
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Theor..c~i~?l.1'.QP!lt!lation, Continuous _troba~:iJ.it.Y....§!.rucJ:ur,=­
2.3.
Consider the probability space
line,
13
(:R,
15 ,
P) where ~ is the rea.l
is the Borel field on ~ , and P is an a.bsolutely continuous
probability measure on
J3 .
Although the theory presented here does not depend on the nature
of the probability measure P, we think of points on ~ as having the
same physical meaning as elements of
n
in (2.1) above with x
~
E:
expressing an x-unit deviation from the expected value of the random
variable X defined by P on
13
with probability density function
(p.d.f.) p(x); c;ind of P as an error-type absolutely conti.nuous probability measure, the prime example being the normal probability
measure on the Borel line.
For the purposes of this section we consider R(T,X) L
x
E:
1("
E:
(O,t],
to he a measurable stochastic process, and require that any
two distribution functions F(r
(x» and F(r (x» corresponding to
L
2
1
R(T ,X) and R(T ,X) L ' L E (O,t] differ only in location.
2
1
2
1
L
It is well known that if A
parameter set and if
£
(O,t] is a Lebesgue measurable
JE /R(L,x)/d T <
00
then - since the value of
A
an absolutely convergent iterated integral is independent of the order
of integration E{
JR(T,X) d~} = J E{R(T,X)} dT,
A
Hence, if we define V(t,x)
_
[10J
t A t
=
I rx(T)
dT, and if
o
we have
J IE(R(T,X»ldT < 00,
0
t
E(V(t,x»
=
J E(R(T,x»dT
o
. • • (2.3.1)
28
Also,
rO(T) =
~
where, we may note) E(R(T)X»
~(T)
say.
Y·le hav~,
= E(V(t,x)
Var (V(t,x»
- E(V(t,x»)2
t
f (rx(T)
= E(
o
= E( 'IjJ
(x) t)2
where 'IjJ (x) is the difference between r ('r) and r (T) "Thich is
x
O
constant
fOl:
all
Further, 'IjJ (x) is symmetric around zero
(0, t].
T €:
and has the simple form 'IjJ (x)
Var (V(t,'»
=
=>
!Y-X,
a. > O.
Hence
Cut cr )2 where 0 2 is the
x
x
• • • (2.3.2)
variance of the random variable X with p.d.f. p(x).
Now, let aCT) and b(T) have continuous derivatives in (O,t],
Pr{a(t) < V(t,x) ~ bet)}
t
=>
Pr{
J
t
a'(T)dT
~ J rx(T)
o
dT
~
0
= pr{ a' (T) -<
r (T) < b' (T)
x
-
t
f b'(T)
dT}
0
¥ T €:
(0 , t] }
since, because of the parallelism of R(T,X ) and R(T,X 2 ) xl' x 2
l
T
€:
€
(O,t] resulting from the properties of R(T,X) given earlier,
aCt) ~ V(t,x) ~ bet) holds if and only if a'(T) ~ rx(T) ~ b'(T)
¥
T €:
(O,t],
Further, if we choose
a'(T) = ~(T) - a
a, b > 0, then
and
b'(T) = ~(T) + b where a, b
€
~
,
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then
~t =
I
lG,
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29
aCt)
~
EeV(t,x»
- at Bnd bet)
and, becnuse of the fact th2L F(r
T
1
~
(x»
E(V(t,x»
b~
i
diffllr only" i.n
and F(r. (x»
1:
2
location, we find that
b (1")
a'lT
Pda I Cr) -<
:.--:
l'
X
(1:) _.
< b ' (1")}
b '(1")
f
:=
dF(r " (x»
1
b
a' (-r)
= pd-a ..$. x ~ b}
f
-a
""
p(x) dx
does not: depend on '[.
To establish tp.nt 'jr
t
;:: a'lT
b
"le deULc the pr-ob':lbility space
(R(.1e,t), RC"B,t), R(P»by the follow::llg one·-to···one conespoucbncc
relations.
x E:
B
E.:::B
X"",>
{R(l,X)
-=>' {R(T,X) J.j x
R(P)(R(B,t»)
VT
£ E,
E;
V-
= Pl-{R(l,X) V
=
(O,t]
E.
R(J(. ,t)
T £ (O;t]}
x
£.
E,
V-
R(B ,t) and
£.
1" £(O,t.]}
PCB)
=
J p(x)
dx
B
Finally it should be clear that the transformation V(t,x) preserves
probability structure in the sense that
Pr{V(t,x) : x £ B}
or, more specifically,
~
Pr{R('T,x) : x
€
B, T
£.
(O,t]}
30
TI
= Pr{a(t)
t
< V(t,x) < b(t)} =
-
.(2.3.3)
TI
a b
-
For example, if p(x) is the standard normal p.d.f., then R(T,·) is
2 2
distributed as normal (rO(T), a). It then follows that V(t,·) is
2 2
distributed according to normal (E(V(t,·», a t ), and, finally, N(t,·)
is distributed lognorma11y with density
1
f(n(t»
=
---=-- expo
{_~[
n (t) v'27i-at
In n(t) -
~
(t) - In n(O) 2
v
] }
at
and
• • . (2.3.4)
variance
median
=n
2
(0) expo (2 ~ (t)
= n(O)
v
expo
2 2
2 2
+ a t )[ea t
- 1]
(~v(t»
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~
where
v
(t) = E(V(t,·).
Also, f is skewed to the right and has positive kurtosis compared
to the normal distribution. [1],[16],[18]
2.4. An Example
Data on the population of Egypt were used to illustrate some
aspects of the previous theory.
The 1947 census gave the country's population as of the 31
2
st
of
We should actually use a standard normal variate truncated to a
finite interval, ~.z., (-4,4) or (-5,5), if R(T,·) is to be considered
finite.
..
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31
March as 18967 thousands.
This figure was cOMdclerec1 t.o hove suffered
frorn an overall over-enumeration stimulated by a rationing census "'hich
was taken in 19Q5, and a sizeable under-enumeration of children less
than ten years of age.
A correction for over-enumeration carried out
by El-Badry [llJ resulted in an estimated population of 17907 thousands.
Another correction for under-enumeration of children less than ten
years of age - using the standard demographic technique of surviving
estimated births in the ten years preceding the census date - inflated
that estimate to 19619 thousands.
This last figure.was used here for
illustration purposes.
Egypt's is virtually a closed population - 1..£., external migration is a negligible component of population change - hence the annual
rate of natural increase is a proper measure of population change.
The
series of rates of natural increase in the period (1947-1959) was taken
to estimate the rate of population change defined by (2.1.1).
The
conversion of rates of natural increase to rate of change in accordance
wi.th remark (2) at the end of section (2.1) above was not deemed necessary since the difference did not exceed 1 x 10- 6 in any year (it is
implied in this comparison, of course, that the rates of change remain
constant for a whole year).
E1-Badry [1l], gave the £0110\-1ing series
of rates of natural increase, adjusted for under-regi.stration of births
and deaths, per 1000 population.
Year
1947
1948
1949
1950
1951
1952
1953
Rate
22.5
21.3
20.4
23.8
24.6
27.0
22.5
Year
1954
1955
1956
1957
1958
1959
Rate
24.4
23.8
23.7
19.1
23.3
25.0
32
A polynomial in time was used to estimate the time trend in the
rates of natural increase.
The least squares method was employed to
estimate the coefficients of this polynomial.
An assrnnption needed for
the optimum properties of least squares to hold, namely the independence
of the stochastic elements of R(T,") over time, would not, in general,
be met by the data of the type considered here.
are, in general, serially correlated.
These random variables
In this case, least squares
esti~·
mators are still unbiased but not guaranteed to be minimum variance.
It was decided to approximate the time trend with a quadratic function of time.
The estimated least squares function was
E(~» =
with T
=0
.021531 + .0006466 T - .00004501 T
at the 1947 census date;
l.~.,
2
1946.25
• • • (2.4.1)
The mean accounted for 99.3% of the total sum of squares.
Although the linear and quadratic terms contributed only .025 and .079
respectively to the square of the multiple correlation coefficient, it
was decided to keep them in the prediction equation.
Higher degree
polynomials were excluded from consideration because the resulting time
patterns were judged to be inappropriate in terms of projecting the trend
eVen to the near future; the cubic and quartic coefficients contributed
.093 and .355 to R2 respectively.
Although the choice of the second
degree polynomial is rather subjective, it was deemed to both improve
the predictive value of the prediction equation and provide a more
interesting example than the caSe of a constant trend.
We consider next the probability structure of the rate of growth
of the population in the time period under consideration.
Under the discrete probability structure, a constant multiple of
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33
a random variable that has a probability distribution of the type
described in section (2.2) above HaS superposed on the variation of the
growth rates that was not explained by the quadratic function of time.
A 19-·interval probability laY],
2-_•.£., k
= 3, \-7as used, this only detcr-
mines the degree of detail desired in the probability distribution.
The sample variance about the regression line,
u
i
.
1S
TI
2
13
2
1: u. where
1.
i=l
.
t h e 1. til res1'd ua 1 , was tak en to est1.mate
t h e vari.ance o.f the sto-
chastic component of R(T,·)
=a2
(3
2
+ 3)
= l2a 2 ;
and the step size for
the translated random variable R(T,·) was estimated by equating
TI
2
= 45956
-10
2
A
x 10
to 12a as a
=
.000619.
The fitted regression curve was taken as the center of a grid of
19 such parallel curves each .000619 units above or belm.,]' the next in
either direction (if there is one) with associated probability masses
determined according to the probability law considered.
It should be
noted that it is implied here that the random mechanism introducing the
stochastic variation around' the mean does not change with time, although
the mean itself does and that this probability structure amounts to
approximating the structure resulting from considering (p, r (T)) over
W
~he
time period (O,t] by continuous grouping of elements of
n
into 19
groups in a nlanner consistent with the structural characteristics of P.
If we are willing to project the time pattern of growth rates and
the estimated probability structure into the future, it could be used
to determine an approximate probability distribution for future population estimates making use of remark (1) in (2.1).
On the other hand,
we could stipulate future changes in the functional form of the time
·028
.026
.024
.022
.020
.018
.016
1947
1949
1951
1953
1955
1957
1959
V=I-1\R
'l... \
FIGURE (2·4· I)
ANNUAL RATE OF NATURAL INCREASE
EGYPT- (1947-1959)
•
•
w
.::--
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35
pattern of
gro~.,th
rates and retain the estimated probability structure.
As an examplc-! of the first case, the population of Egypt at the time of
the 1960 census was estimated using the 1947 adjusted census figure and
the information provided by the rates of natural increase.
of the fact that the 1960 census is referred to the 30
th
Making use
of Septenilicr,
equation (2.4.1), and the above theory and discussion, the following
results were obtained.
TABLE 2.4.1
ESTIMATED PROBABILITY DISTRIBUTION OF THE POPULATION OF EGYPT
IN THOUSANDS ON 9/30/1960
Interval
Prob~bilic!Y..
24774 - 24981
.002 915
24982 - 2519b
.008 746
25191 - 25402
.017 493
25403 - 25615
.029 155
25616 - 25830
.043 732
. 25831 - 26047
.061 224
26048
26265
.081 633
26266 .- 26486
.096 210
26487 - 26708
.104 956
26709 - 26932
.107 872
26933 - 27158
.104 956
27159
27386
.096 210
27387 - 27616
.081 633
27617 - 27847
.061 224
27848 - 28081
.043 732
28082 - 28317
.029 155
28318 - 28554
.017 493
28555 - 28794
.008 746
28795 - 29037
.002 915
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36
vlith
= 26832 thousands and
mean
standD.rd deviation == 777 thousands
Under the continuous probability fonnulation, the following
results were obtained, in the nonna1 case
~
v
(13.5)
E(N(l3.5,·»
= 0.312
676
- 26832 thousands
Also, properties of the normal p. d. f. may be used to make appl.·oxi.mate probability statements about V(t,·) and, consequently, N(t,·).
In our example,
P~{25312 ~
N(13.5,·)
~ 28419} =
0.95
The 1960 census count was 25984 thousands.
When corrected for
under-enumeration of children less than ten years of age, the population
figure rose to an estimated 26896 thousands.
-.
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CHAPTER III
AGE STRUCTURE
3.1.
l
Introduction
Let N(t) and ret) have the same definitions as N(t,-) and R(t,-)
respectively in (2.1.1),
B(t), bet) and c(x,t) have the same definitions as in Lotka's
formulation in (1.2.1),
N(x,t)
= c(x,t)
N(t),
m(x,t) be the birth rate for individuals of age x at the time
epoch t, and
ex t)
p,
= number
of individuals of age x at time t in the population
number of births at time (t-x) in the population
This generalized formulation of survivorship incorporates the
balance of forces of migration affecting the population in addition to
mortality conditions.
In this case, p(x,t) may, conceivably, exceed 1,
and "generalized" death rates computed in such a situation may become
negative.
However, such oddities do not introduce much logical diffi-
culty and it is in this generalized sense that death and survivorship
will be considered here.
Hyrenius.
The roots of this formulation are ascribed to
[29 J
lAll variables considered in this chapter are random variables.
The explicit dependence of these functions on elements of the underlying
sample space is suppressed in the interest of clarity because it does
not serve a significant purpose.
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38
Using
th(~
ahove notation, tIle folloHj rLg relations cbscritle thl' age
structure of the population at time t:
N(x, t) = B(t--x) p(x, t)
..
. (3.1.1)
..
• (3.l.2)
-.
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ClO
c(x,t) = B(t-x) P (x, t) /
J
B(t-x) P (x, t) dx
t
0
'" b (t--x) P (x, t) expo (-
J
r(t) dt)
t-x
making usc of the relation N(t)
= N(O)
t
exp (
J
reT) dT).
o
Equati.on (3.1.1) is fundamentally simpler, conceptually easier and
more a1l1enable to statistical manipulation than (3.1.2) \lhich is a generalization of the familiar fomulation of Latka and probably more
advantageous from t!he demographic analysis point of vieH since for its
purposes it is both easier and more meaningful to speak in terms of
rates of occurrence rather than the number of occurrences.
Of course,
if the total age span possible at time t is considered, then c(x,t) is
only a constant multiple of N(x,t).
The last observation points out the major source of difficulty
in determining the complete age structure of a population at a point
in time under the present formulation if all the demographic variables
are allowed to be general functions of time.
If w is the highest
t
attainable age at time t, then to determine the complete age structure
at this point in time, the history of birth and survival in the population under consideration must be available for the time period
({t-w ), t].
t
This ambitious requirement - as well as mathematical con-
venience - has led students of demography pursuing this formulation to
consider special cases in which some of the demographic variables are
el
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39
mathematically independent of time.
If p(x,t) and bet) arc mathematically independent of time in such
a way that ret) docs not depend on time in the interval «t-b-a),(t-a)]
a
~
0, b > 0, then c(x,t) must also be mathematically independent of
time for a < x < (a+b) and we have
c (x, t) = c(x)
=
bp (x) e
-rx
a
which is Alfred Latka's celebrated equation.
shown to hold at t
-+
00
~
x :5.. (a+b)
The same form may be
if p(x,t) and m(x,t:) do not deper..d on time. [21]
A slightly modified version of this formulation requires knowledge
of the age distrib'4tion function at a point in time prior to t,
N(x,t-a),a> 0 say, and the course of birth and survival in the popu1ation in the intervening time period «t-a), t] only in order to determine the complete age structure at time
*
p (x, a, t)
=
t.
For this formulation, define
number of individuals of age x at time t in the~~pulati~
number of individuals of age (x-a) at time (t-a) in the population
x > a > 0
and note that p * (x,a,t)
and
p * (x,x,t)
= p(x,t)/p(x-a;t-a)
= p(x,t).
Then, we can write
N(x, t) = N(x-a, t-a) p * (x,a, t)
= B(t-x)
p(x,t)
x > a
x < a
}
.(3.1.3)
The last relation is exactly the same as in the previous formu1ation and is written in the usual demographic practice as
40
00
=(
J m(s,
t-x) N(s, t-x)
d~)
p(x,t).
o
This is the model of component proje.ctions which is clearly qui.te
practical in a deterministic set-up.
This is so because it does not
require hard-to-get historical data.
But, on the other hand, si.nce
estimates of the age structure are available only through rather
infrequent surveys and censuses, the estimation of the underlyi.ng stochastic mechanism - in a stochastic treatment of the age structure is severely hindered.
Consequently, the distribution of N(x,t) is
usually given conditional on the initial N(x,t-a).
Data on birth and
survival are available much more frequently.
In actual applications, irrespective of the formulation used, we
are usually interested in age groups rather than age points;
i.~.,
in
the age distribution function as a set - (rather than a point) - function of age.
Hence we consider
c
bNc (t)
f
= J
= f
=
N(x,t) dx
b
c
b
B(t-x) p(x,t) dx
c.:::,a
c
N(x-a, t-a)p*(x,a,t) dx b > a
b
.(3.1.4)
Since all the functions involved in the above relations are
always non-negative, then - by the second mean value theorem - we have
c
bNc (t) = K1
f
B(t-x) dx
x > a > c > b
b
=~
B
(t-c) (t-b)
• • . (3.1.5)
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41
where
p(x,t) < Y
min
~l
-
b < x < c·
max
<
- b
p(x,t) and
< x <
is the
B
(t--c) (t-b)
C
number of births in the time interval «t-c), (t-b)]; or
bNc(t)
~
c
f .N(x-a,
K2
t-a) dx
b
~
K
~
K
2
• . . (3.1.6)
N(t-·a)
(b-a) (c-a)
2
where
min
b < x <
p * (x,a,t)
C
~
max
p * (x,a,t) and
N (t-a)
b < x < c
(b-a) (c-a)
is the number of individuals in the age group «b-a), (c-·a)] in the
population at time. (t-a).
K and K are usually approximated by some sort of average value.
l
2
For example, if survivorship conditi.ons are constant in the time pe.riod
«t-a),t] for birth cohorts of the time period «t-a-c), (t-a-b)] and
survivorship probabilities from age (x-a) at time (t-a) to age x at
time t are weighted by the cohort size at time (t-a), we have
c
K
2
~
~
f
b
c
P* (x,a) p(x-a) dx/
f
b
=
p(x) dx/
I
p(x-a) dx
b
L
(c-b) b
/
L
(c-b) (b-a)
x+n
where
L
n x
~
f
x
p(x-a) dx
b
c
c
I
p(x) dx
which is the regular demographic practice.
..
. (3.1. 7)
42
3.2.
Statistical Formulation
3.2.1. Introduction
In this section we discuss the naturc of the prok1hili ty c1istdl.ution of the age structure at a point in time.
For a fixed age point and a fixed timepolnt, xO' to respectively
- under both thc origi.nal Lotkian formulation and the component proj ections set-up - the problem of determining the probability dis tri.lmtion of N(x ' to) is one of either finding the distribution of the pro-'
O
duct of two random variables that cannot, in general, be assumed sto-chastically independent, or fi.nding the probabil:t ty distri.bution of a
constant multiple of a random variable.
2
Further, the probability distribution of N(x, to) is the joint
distribution of N(x ' to) for all xO' and, in general, the marginal
O
distributions - for fixed age and time points - are not expected to be
those of independent random variables.
Also, implementing the approximations discussed in (3.1), it is
clear that the stochastic treatment of the age distribution function ?s
a set-function of age is essentially the same as its stochastic treatment as a point function of age.
We shall consider the joint distribution of the number of individuals in two different age groups in the population at some point in
time.
Demographic variables will be assumed to have the simple stochas-
tic structure of a constant depending on time (a deterministic component)
2Note that in the case of using m(x,to-x ) to arrive at B(to-X O)'
O
the probability distribution of B(tO-x ) is a transformation of that
O
of m(x,to-x ) since i t is usually not possible to treat N(x,tO-x O) as
O
a random variable.
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43
representing the expected value of the variable plus a stationary - in
time - stochastic error component.
This restricted form will not be
required in the theoretical development but is suggested as the basis
for estimation in practical application.
The joint distribution referred to in the preceding paragraph is
a function of the joint distri.bution of a number of demographic variables that d.epends on the underlying formulation (Lotkian, component
projections or a combination of both).
This number of component vari-
ables ranges from two to four depending en the situation.
In the most
general case, Lotkian formulation, namely,
N(x1,t)
= B(t-xl )
p(x1,t)
N(x ,t) :;: B(t-x } p(x ,t)
2
2
2
we have an initial joint distribution of four random variables expressing
demographic phenomena, two for number of births and two for survival
probabilities.
These four random variables are not - in general -
stochastically independent.
This set of four components is divided into
two subsets of two random variables each,. (one for the number of births
and the other for the corresponding survival probability),- and the finaL
probability distribution is the joint distribution of the procfucts of.'
the two random variables in each of the two subsets ~
All. ather: cases.
may be deduced from this by considering degenerate. random variables •.
The distribution of the product
of two stochastically dependent
random variables is usually difficult to obtain:t let alone- the hivariate
distribution of the two two-variate products of four stochasti.c:aIly
dependent random variables.
.-
Because o:f this difficul.ty',. we shall. be'".
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44
satisfied with describing the last distribution by its lower moments;
!.~.,
the means, variances and covariance of its two components.
covariance is probably of the least practical importance.
The
These moments
-.
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are functions of the joint moments of the original four-variate distribution.
An interesting and important statistical problem arises in the
estimation of the joint moments of the original four-variate distribution.
The usual methods of estimation are not strictly applicable
here since the basic requirement of a sample of n independent obscrvations is not met in time series data.
However, we shall use the same
n
functions of sample moments (1
r
4
r.
Xji) r j
4 or
n
n i=l j=l
= 0,
1, 2 to estimate
j
functions of population moments E( n x. ) and the usual unbiased
j=l Ji
estimators of covariances in the normal case.
In general, let
Then
i
=
1, 2
• • • (3.2.1.1)
In actual applications, we may set out to estimate these quantities without specifying a parent distribution.
Alternatively, we may
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45
stipulate an underlying probability distribution as a basis for the
stochastic variation in the demographic variables, and use its pro··'
perties to arrive at expressions for the required litOnlents and then
estimate these, hoping that the latter procedure, being more specific,
may prove to be easier.
One such parent distribution, the multivariate
normal,3 is considered here as providing the random error componE:nt
in the variation of the demographic variables.
3.2.2.
The Multivariate Normal Case
If Xl and X are distributed identically and independently as the
2
standard normal variate, then the distribution function of their product is
; KO(Z) where KO(Z) is the Bessel functior; of the second kind
of a purely imaginary argument of zero order.
Further, if Xl and X are
2
not stochastically independent, the distribution function of their product is available in a closed form. [9]
Unfortunately, this form is
too complicated to be of much analytical use.
Limited tabulations of
the distribution function are ·available in the case of independence. [2]
Consider the four dimensional normal variable X' = (Xl X X X )
2 3 4
with mean vector ~' - (~l ~2 ~3 ~4) and covariance matrix V
i, j
= 1,
variables.
2, 3 ,4 •. Let !.'
=
(t
l
t
2
t
3
=
(Oij)
t ) be a vector of mathematical
4
Then the moment generating function of X is given by
~x(t)
== expo (1!.'!.
+ ~ !.' V !.).
Making use of the properties of ~ X <.!), we get
3 In an exact treatment, the multinormal distribution would have
to be truncated to a finite rectangular region in the four dimensional
space •
46
E (XiX. )
J
Var (X.X. )
1 J
.. + ll.1 llj'
= o1.J
2
2
+
= O..
1J +Oii llj
2
20 .. lli llj + o .. p.
1
1J
JJ
• . . 0.2.2.1)
+
0 .. 0 ..
11
JJ
i,j
and
which are sufficient to determine the lower moments of the distribution
3.3. A Discrete Approximation to the Human
Survivorship Function
An approximation to the survivorship function p(x) based on a
finite number of age groups is motivated by the following consideration.
In practice we can only observe a finite approximation to p(x) and
for most practical purposes we do not need more than that.
One such
approximation may be arrived at as follows:
Divide the possible age span (O,w] into n intervals of the form
(ai' a i +l ] ,
i = 1, 2, ••• , n where a l = 0 and a n+l = w such that ll(x)
can be considered approximately constant within each of the n age
groups.
Let the probability of death per age unit within the i
th
age
interval be approximated by qi say.
Clearly, the assumptions needed for a constant - attrition survivorship function are approximately satisfied within each of the n
age intervals considered and we have the original p(x) function approximated by n negative exponentials for
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47
n
1J (x)
where
,e
I
(a i ,a i +1]
n
= 1
i=l
a.1. < x
~
qi 1 (ai,a
Hl ]
• • • (3.3.1)
a i +1
= 0 otherwise is the indicator function of the
It then follows - from (1.2.1.8) - that
p(x)
~
m-1
exp.{-( ~ q1..(a + - a.) + (x - a )q )}
i 1
i=l
1.
m m
p (x)
n
m-1
=
1T
where am < x
~
exp.{- qi.Q,.}
i=l
= km e
e
-q (x-a )
m
m
1.
-~(x-am)
a m+1 , .Q,i = a i +1 - ai' and
• . . (3.3.2)
~
is the probability of
th
survival from birth to the beginning of the m age group.
Ie
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1
~
1J (x) =
Note that the conditional probability of death in the i th age
interval given survival to the beginning of the interval is equal to
. • • (3.3.3)
which bears a strong resemblance to the Reed and Merrell formula
resulting from the direct assumption that In p(x)
=a +
Sx within the
age interval. [25]
Further, it turns out that qi is equal to the central death
rate
4
of the i
th
age group, d. say, which is the measure of age-specific
1.
mortality usually available, since
4Relating the number of deaths in a certain age group over a
certain period of time in a population to the number of person-years
lived in this population by individuals in those age groups and time
period.
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48
-q.9J.
k. - k
1.
k.
1.
The
approxim~.tion
e
i
1
--
q.
1
-,
1. 1.
(1 - e
-a.£.
'1. 1.
)
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(3.3.2) may be used to calculate life expect··
ancies rather easily, for
.
e(a. )
:;:
00
1
- .
• k·
1.
1.
p (x) dx
n
f
a.
1
n
1
= -
k.
1
L
j=i
k.
J
(j+l
a
e
-q (x-a.}
j
J
dx
j
. • • (3.3.5)
3.4. An Example
In this section we apply some aspects of the previous theory to
estimate the number of males in the t\170 age groups (0) and (1-4) ..
using the conventional census age grouping - in Egypt on January 1,
1965.
Males in these two age groups are the survivors of male births
in the period January 1, 1960 - December 31, 1964.
Yearly male births \\rere estimated using a series of crude birth
rates corrected for under-registration by El-Badry [11], a sex ratio
at birth of 105 males per 100 females and a series of estimated mid-year
population figures - using census figures adjusted for under-enumeration
of children less than ten years of age and assuming exponential grm\Tth
between census counts.
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49
Approximate single-year central death rates were derived from thE'
reported' death rate for the age group (0-4) using divisors computed from
complete life tables for the years 1937, 1947 and 1960. [6]
were multiplied by correction factors computed by relating crude death
rates adjusted for under-registration to reported crude death rates. [1]]
Finally, the resulting rates were transformed to survival probabilities
using the method described in (3.3) and separation factors of 0.75, 0.7,
0.65, 0.6 and 0.55.
Appendix (A) includes the original data and some of
the intermediate calculations.
Table 3.3.1 shows the resulting births
and survival probabilities estimates.
The next step is to estimate the time trend in the four demographic
variables under consideration and then use deviations from the trend to
estimate the moments of the underlying distribution.
We shall confine
ourselves to polynomials and use least squares to get unbiased estimates
of their coefficients.
For Xl' X and X , we fitted straight lines.
2
4
The square of the
multiple correlation coefficient was 0.845, 0.907 and 0.991 for Xl' X
2
and X respectively, and the inclusion of a quadratic term in the
4
2
regression equation did not contribute to R more than .01 in any of
the three cases.
The fitted regression lines were
----
E(X )
l
= 409.86 +
8.9I5t
-
= 0.8585 +
0.00386t
E(X )
2
E(X ) = 0.5939 + 0.01057t
4
with t
,.
I
These rates
.-
=0
at mid-year 1940;
i.~.,
1939.5.
• • • (3.4.l)
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TABLE 3.4.1
ESTE1ATED
Calendar
year
~1ALE
BIR'nIS IN n!OUSANDS 1',T,;]) SUlzVl\'j\t rnor,ABIL UIES
Male births
in thous~mds
in year t
Probability
of survival
from birth
to be in
the age
group (0)
Hale llirths
in thou[;,'mds
in the fOl1l"year period
starti.ng with
year t.
at the
start of
_.
bjrth in a fouryear period
E:tiJrting \dth
year
t to be in
thQ age group
(1-./1) at the
start: of year
(t+l)
(t+5)
__._--- -----t
Pl.-oh::lhih.Ly of
survivul fro~l
X,q
Xl
1940
432
.861
1656
.597
1
426
.8611
1656
.605
2
391
.856
1702
.617
3
407
.876
1764
.635
4
432
.877
1852
.638
5
472
.871
1899
.641
6
453
.889
1907
.654
7
495
.897
1964
.661
8
Il79
.870
2002
.672
9
480
.894
2071
.685
1950
510
.901
2125
.697
1
533
.896
2159
.708
2
548
.906
2185
.723
3
534
.905
2165
.738
4
544
.918
2142
5
559
.912
2157
6
528
.932
7
511
.921
8
559
.927
9
598
.930
••
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.,
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51
t
II
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t
For X , it is important to realize that yearly births determine
3
period births completely, and since we have expressed Xl as
=
1
I
i = 0, 1, ••. , 19
£.
1
It follows that
j+3
j = 0, 1, ••• , 15
L
i=j
= (4a + 6S) + (48)
j
+
j+3
L
E:. )
1
i=j
=
where E(e:~)
J
= 0,
E(X )
3
Ie
.-
O.
where E(E:.)
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a + Si +
a' + 8' j +
e:j
therefore (3.4.1) implies
= 1692.93 +
• • • (3.4.2)
35.66t
If we assume that the observed time trends will continue for a
period of five years (which we do not have to do if we want to stipulate
different trends), and that the underlying stochastic structure about
these trends is stationary, and using the expected values of the four
demographic random variables on 1/1/1965 predicted from the estimated
regression lines in conjunction with the deviations from the lines, we
get the following results for the number of males in Egypt as of
January 1, 1965.
Age
Group
Method
0
distribution free
593.3
21.5
normal
593.3
22.1
distribution -120.2
free
distribution free
1939.5
36.3
normal -132.6
1939.5
37.6
1-4
normal
(*) in thousands.
Mean(*)
Standard (*)
Deviation
Covariance(*)
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CHAPTER IV
SOME ASPECTS OF THE ESTIMATION PROBLBf
4.1. Estimation of a Certain Function of Joint Moments of a Multivariate Distribution Using the Same Function of Joint ~1oments of a
Sample ~f Dependent Observations
Consider the k-variate random variable Y' = (Y
c.d.f. Fy (')'
1
Y ••• Y ) with
2
k
Suppose we have a sample of n observations from this
parent distribution,
I 1 , Y2'
Y where
... , -n
i = 1, 2, ... , n
such that the n observations are not independent.
k
r
r
t
Let X denote the product (TI
Yo.),
in chapters 2 and 3, we used
i
t= 1 Nl.
n
k r
n
t
1
r
1
and
say, where r = 0, 1, 2,
=
m = r (TI Yti) n r Xi'
t
r
n
i=l t=l
i=l
...
t = 1, 2,
... , k as an estimator for
used (m - m mt ) to estimate
r
s
(~r
-
~
k r
r
t
= E( TI YR;") = E(X ).
r
i
t=l il-
~s ~t)'
We also
In this chapter, we inves-
tigate some of the properties of these estimators.
In the course of such an investigation, we may want to consider
i
. of _Z' = (Y'
y' ..• -n
y')
t he kn-variate distribution of the samp 1 e,. _.~.,
-1 -2
with c.d.f. F {')'
Z
Unfortunately, however, we end up with a sample
of one observation only from FZ (')'
It is easy to show that m is an unbiased estimator for
r
~
r
since
53
E(m )
r
1
r
= -
n
:;
n
J
i=l
1
r
--
n
J
x
xII
n
Y
1.
kn
fxli... xf
i=l
dF
X:
X: d
1.
ki
J...
Xu
r
x
I
F
Y
kn
t
I
except
1
r
=n
n
1
E(X:)
r
n
1.
i=l
n
i=l
...
xli
x ki
lJ r = lJ r •
Also, the calculation of the covariance of two stich estimators
m and m is straightforward.
r
s
1
Cov(m ,m ) = E{[r
s
n
=
r
n
i=l
We have
r
(X. - E(X:»]
1.
l2 {n[E(X~+S)
1.
[1:.n
n
L::
i=l
1.
1.
- E(X~) E(X~)]
n
1
r
s
+ [r
E(X~A~) - n(n-l) E(X ) E(X )]}
i;j =1
J
n
r ,X.)
S
+ 1:.2 r
s
= -1 Cov (X.
Cov (X.r ,X.)
n
1.
1.
n i;j=l
1.
J
1
r s
1
r s
= - Cov (Xi ,Xi) + (1 - _..) Cov (X. ,X.)
n
n
1.
J
1
n
rs
1
+ (1 - -) .. -a
n 1.J rs
say
l
.
.(4.1.1)
where
Cov (Xi,X j ) = ij 0rs denotes the arithmetic mean of the n(n-l)
covariances.
r
s
k
'IT
,Q,=l
r,Q,+s,Q,
~i
f
I
I
(X~ - E(X~» ]}
n
=-<1
,
.1
,
-,
1
I
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t
,I,
t.
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54
If the n observations are independent, the second term of the
right hand side of (4.1.1) vanishes, and if any two observations have
the same dependenc.e struc.ture, hence the same distribution, then i'O-
J rs
is equal to the common covariance, 'jO
1
rs
As a special case of (4.1.1) when r
say.
1
= sl'
1 = 1, 2, •.. , k we
get
Var (m )
r
= 1:.n
=
1:.
n
Var (X:) + (1 1
0"
rr
+
(1 -
1)
n
Cov (x:,X:)
1
J
1)
.' (1
n 1J rr
• • • (4.1.2)
This is a restatement of the fact that if a set of n random variables
, are distributed such that they all have the same variance, thp.n their
.
. 2
2
average internal covariance must not be less than CO~l) where 0 is
n-
.
It follows that, in the limit, 'jO
their conW-lon variance.
1
rr
is non-
negative.
T~
situation gets more complicated >"hen ,ve consider (m 'm ) as
s t
an estimator of (~s'~t)'
To start with, the estimator is - in general -
biased for
E(m 'm ) =
s
t
11
11
t-'st-'t
+ {l
0"
+
n st
(1 -
1:.)
n
jj
0st }
• . • (4.1.3)
as may be shown either directly or as a corollary of (4.1.1) in conjunction with the relation E(XY)· = Cov (X,Y) + E(X) E(Y).
of fact, (m 'm ) cannot even be asymptotically unbiased for
s t
unless ijO"st
As a matter
(~s~t)
o.
Further, the bias term in (4.1.3) is not estimable by the usual
estimators of sample covariances.
It is easy to see that
55
-8
,.,here X = m
s
Hence
'I
e-I
I
I
n
E( 1:
i=l
I
. . • (4.1.4)
and
I
"
n
--~
(X~
= E {L
i=l
n
=~=1
L
- lls -
+~
{ast
n
S
X
-
lls)
n
(X~
- llt -
t
X
- llt) }
I
I
n
(L a
+ L
'j,a)
j=l st· jfj'=lJ
st
e,
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I
I
Clearly, the bias term in (4.1.3) cannot be' expressed as a func-
We also observe that
n
E(
t
r
ifj=l
X~ X ) = n(n-1) <. . cr
~
~J s
j
t
+ II llt)
,
• , ,(/+,1.6)
s
and
n
E(
r
S
t
X X )
i=l i i
= n(a st +
~ II )
I
, , ,(4,1.7)
s t
.1
"
I
I
I.
I
1
I
I)
I
t
,'\_
S "f
lience
c:ons:ider
\';C. inDy
n
--~--- L
n(n-l)
n
1
E
n 1"'1
esti~ators
o~o
~rJ::=
1
X~
~
X~~ X~1 as
X7J
and
\18 11
no;
8
TIl
3
s
m
t
as
bia::~('.d
11 ) .
for 01~ s"'t
An a1tenlative approach to "-That He have been considcri Ilg so f;n-
stipulates certain dcpendenc'3 patterns among the. n 0bser:voUons
A pa:rticuli3rly appealing pattern \·ias considered by P. K.
sample.
[26J
or
the
Serlo
Sen extended the U"statistics method of csti.l:l;:::t:ion of functio:1al,,;
of distribution func.tions [l~] to the case of m-dependent 8tEJtiC'!~;:;yy
stochastic processes.
Let {X } be a seqUf'.nce of vector~·v<,llJed );'i'mc1om v<1ri ...bles) Sen
Il
called such a sequence I;1-dependent if the randOI'l vector (X.l ,X 21 ••• ,Xl)
is stoch2stica11y inde:pendent of the random vector (X , X '1' ... )
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I.
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S
S-r
whenever s-r > m and it is sai.d to be stationary if the iuint: cH:·trihc,tion of Xi' •.. , X + is independent of i for 8.11 r.
i r
Let f(X
, X
o.~
0.1
,
.... , x )
~
be a
statistic syul;netric in its nybu·
If E(£(X
0.
,
... ,
g,
1
we call f
<it
kernel for g.
Defiue h
and the
= E(f(X(Xl ,
sj'!"~£t:etric
n > m (k-·J.)
+
... , X )\0.0+1 - a. > m for j = l,2, ... ,k-l)
(ik
J
. J
estimator based on a sample of n observat ions)
k,
... ,
E f(X
S
0.1
, ""
where the s",:,mntion extends ove.r all possible (
X )
ak
• . . (4.. 1.8)
n -om (l~ ~ 1) )
sets of
k
I
57
(a ,
l
•
t
•
,
a k ) satisfying
The estimate U(X ,
l
(Ct j +l
... ,
-
Ct
j
) > m.
X11, ) is unbiased for h.
Sen sho\ved that
.t
I
i t is also consistent and, under the conditions of existence of the
second and third order moments of f(X , ... "
I
X ), that U(X ," ... ,
n
l
is asynptotically normally distributed with mean h and variance k
X
Z
n
I
)
z;
I
,I
z; •
and gave a consistent estimator for
Under the m-dependence set-up, we may formally generalize c
get an unbiased estimator for (]..I s]..I)
t·'
1
to
Consider
I
t
11,
=
1
L
X~ Xt
(n-m)(n-m-l) i~j=l 1 j
• • • (4.1.9)
Ii-jl >m
Clearly mel i.s unbiased for (]..Is]..lt) and Oe l = el'
We may also want to
consider the estimator
• • • (4.1.10)
Recapi.tulating, we have five estimators that we may use to
estimate (11 l.l ).
s t
Let
c1
=
c'
I
=
(n-m) (n-m-l)
n
Z
en-m) (n-m-I)
11,(11,-1)
'
C
z=
m(Zn-m-l)
2
n·
m(Zn-m-1)
, c'Z = n(n-l)
are related by the following relations.
1
, c 3 = -11,
then these estimators
t
••I
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I
t
t
I
eI
I
I
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I.
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I
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I
58
c'
m
m
where ijOst is the average of the (n-m) (n-m-l)
s
covariances of (X., X~) such that Ii J.
J
jl
> m
(m)
+ ij 0 st
...s ....t
-
11
11
(m)
where ijOst
E(e )
;:::
1
E(e
I
I
e
(m) 1
and, in general,
e
I
I
I
I.
I
+ c'2
• • • (II .1.1J )
I
I
I
f
I
I
e
1 m 1
z) ;: :
covariances of (X si, X:) such that Ii J
m
- (m)
c'
+
c'
+
,
0
llsllt
1 J.'jO S.t
2 J.j s t
11
11
....s .... t
= "11
....s .... t
E(e )
3
is the average of the m(2n-m-l) • • . (4.1.12)
j/
< m
-
+ 0 st
m
+ c1
,,0 t
J.J s
+
C
z
+ c3
0
st
m
under m-dependence
c Z> c
2'
C
3
,,0
~
s
t ;::: 0 and asymptotically c ; ci ~ land
l
n~
--... 0, and we can distinguish two subsets of five estimators
n-+oo
accordingly.
On one hand, we have a fundamental estimator, namely mel'
on which two other estimators, e
l
and e , are based.
3
estimators are asymptotically equivalent.
m-dependence, e
l
and e
3
These three
Since mel is unbiased under
are asymptotically unbiased under m·-dependence
59
and have the same variance as mel.
and e
2
On the other hand,
~"e
have (m) ('1
which have the undesirable property that they are ahm)'s biased;
i .. e. for all sample sizes, unless - of course -
o(m): 0 or a
ij st
st
: 0
respectively which is very unlikely •.
Clearly \>Je should be more interested in small sample results
rather than asymptotic results because the length of time series of
demographic data j.s usually restricted.
Unfortunately the nature of
the bias terms of the five estimators does not allow preci.se compari'sons.
Only general tendencies can be indicated.
For the first subset of the five estimators, mel seems to be the
choice estimator for m-dependent processes and n > m(k - 1) + k,
otherwise mel
= O.
from the sample.
However, we need to either know m or estbuate it
Even if m-dependence does not describe the dependence
structure of the process under consideration adequately, mel may still
be preferred to (m)e
l
or e
l
if there is reason to expect that
or
respectively, as is usually the case with time series.
Between the two
groups of estimators, mel may-be preferred to (m)e l for the same reasons
that is may be preferred to el' and it would probably be less biased
m
than e
Z since it is reasonable to expect that lastl >
within the second subset of estimators, (m)e
(Ul)
if
I.J.j crs t I
I
I
I
I
I
I
I
.,
I
m
I.·cr
J.J s t I
J
.1
l
I.1J,0S t I.
Finally,
would be better than e Z
< last I which is likely to be the case.
However, we must remember that in estimation from small samples,
the relative merit of an estimator is determined to a great extent by
how much use it makes of the sample information, and the five estimators
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under consideration here use various portions of the sample.
example
For
e , e and e make increasingly more use of the sample informal
m l
3
tion, in that order.
Also, the amount of relative use of the sample
information for mel and (m)e l depends on the relation of m to n.
A still more conclusive comparison among the five estimators con-sidercd would be based on a more decisive criterion than unbiascdness
only, like the mean square error of an estimator. All indications arc,
however, that such a criterion would not be feasible analytically extrapolating from the experience with unbiasedness.
As a matter of fact the
variance ternlS needed for a study based on M.S.E. are too general and
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too complicated to be useful analytically.
Obviously, the problem gets to be considerably more complex when
we consider estimators for
t
-
~
s
~t)
based on m and the five estir
) discussed above.
In this section we use the Monte Carlo method of statistical
experimentation to estimate the mean square error of five estimators
for a special case of the estimation problem discussed in (4.1).
4.2.1. Introduction
Let
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r
4.2. A Monte Carlo Study of Some of the Properties of Five Estimators
for the Variance of the Product of the Two Components of a BivariateNormal Time Series in Small Samples
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(~ ~
mators for
(~
2.
{~}
.!..~.
be a (p x 1) vector-valued normal time series
with
, for any subset of indices (t l , t 2 , ••• , t n ) the vectors
X
-t
.-
2
) have the np-multivariate normal distribution.
n
(,1
and
l:h
;::
Cov(X
x +1)
-t , -t
1
(1 OJ .)
1
J
i,j ;:: 1, 2,
where
La
;::
1
P 12
P 2I
1
... ,
p,
Vt
PIp
...
P 2p
1
• • • (4.2.1)
Further, let
l:
such that
ea (m+1) 9
h
a.
ea h
;::
L
ct
a
>
a
• • • (4.2.1.2)
This defines a special case of an m-dependent
multi-normal time series.
Then a subsequence of n consecutive elements of {X
t
}
variate normal with
0'
(1 x np)
and
X
-t+1
Var
X
....t+2
X
-t+n
;::
...
...
L
n-l
L
;::
n-2
l:a
l:
(np x np)
has the np-'
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I.
I
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I_
I:
=
(n-l) a
I
-a
e
e
-a.
e
I
e
(n-Z)a.
_(n-l)a.
e
~
L for all t.
O
(n-2)a.
e
I
. . . (4.2.1.3)
In the most general problem considered in Chapters II and III, we
had p
=4
for the estimation of
components of
~,
note that this parameter is independent of t.
This
function of moments represents the covariance of the number of individuals in two different age groups.
Unfortunately, a complete investiga-
tion of this case using Monte Carlo methods would be both cumbersome
and of questionable utility silnply because of the large number of correlation patterns that we have to consider among the four components of
I
~.
ance will not be of much practical importance.
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Furthermore, as indicated earlier, it is likely that this covari-
A special case that is of considerable practical importance refers
to the problem of estimation of the variance of the number of individuals
in a certain age group.
In this case, p = 2 and the parameter is
This case will be considered here.
Equation (3.2.2.1) gives
.-
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in our special case, using the conditions included in (4.2.1.1), this
reduces to
2
• . • (4.2.1.4)
V = P12 + 1
1
n
2
2 2
Let eO = -- L (X X ) be the unbiased esti.mator for E(X X ),
2i
li
n i=l
1 2
and mel' (m)e l , e , e 2 and e be defined from the corresponding formulae
1
3
in (4.1) by putting
t
= si = 1,
i
i
= 1,
2, and X.~
= X- .X2 ..
-~J..J.
The fol-
lowing estimators for V were considered
where
A
51
= eO
- mel
52
- (m)e l
53
= eO
= eO
54
= eO
55
=
• • • (4.2.1.5)
- el
- e3
A2
A A2
A
A
A
A A2
A A
°12 + °Il P2 + 2°12 Pl P2 + °22 PI + °11°22
We note that (eO - e )
2
= 0,
estimator for V although eO and e
which does not make it a useful
2
may be - under certain conditions -
2 2
2
reasonable estimates of E(XIX ) and E (X X ) respectively
2
1 2
4.2.2. Procedure
The following values of n, m and P12
n
= 5,
= P,
say, were considered
10, 15
m= 0, 1, 3, 5, 10, 15
P = -.75, (.25) .. 75 .3
.-
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denotes the usual unbiased estimator of the parameter in the
case of a sample of independent observations.
3p
.1
=±
1 were not included to keep L positive definite.
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6!;
for (:;lch cc,.:·.ir13Li.on of n, m nnd p, 1000 si"rpL':'i
dent nonn".] deviates
\>/eYC
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£
;~n
in(kp~'l'-
genC'l'.'ltcd using the }Ji1:1 E:uln.ot'tinc R?NliL: La
gcnerat c untform ra'ldoLl numbcn; [30J and J. Be)]' s wn.1ificn.t ion of the
r.ox··Hullcx routjhe to transfonn them to il'.dcpc;Hc1ent
nnn~:.11 dl~vi~'.t(.s. [:i]
This siOl'nple of 2n ohservations pas tran,;fo:-nwd to a Edu;;J C' rondo:;, obs.rvatioa frori>. a 2n'-variate T!ol-ul,:11 \'lith weaD ,Q ilnd
detc:rniDcd by n,
The follm,ing
(1) l.et
J"l,
t\o70
v~n'L1fiC(' n~c,t:r.:ix
::
p and the cov3d.ance structure c1.:,sc:db8U ill (li.2.J).
remQrks "Tere use.rl· in the trans fone"i. ie"i.
Y.--.... N(O, 1 )
(1'5(1)
~,. (Pxp)
and
z "" C Y then
(pxl) (pxp) (px'I)
Z-~/ N(Q, C C')
(2) Choleski's Fact.ori:<:ation
If
L
is a symmetric positive definite matrix, thcn
(f>xp)
1: = T T' ".There
I
C
T
is a lOHet tIHtngulcr In;,tr:ix.
(pxy) )
elements c,f Tare detennined' by the follol-dng relations.
t, .
J.J
t
--
i < j
°
n =-~ Vall
= ail/t n
t'l
1.
too
J.1
=Va
ii
-
i-I 2
t ik
k=l
= 1, 2, 3,
i
L
j--l
-. (00 0 J.J
L
k=l
t, k t Ok) It.
J.
J
0
JJ
i:!-j,i,.i
... ,
P
T11,;
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Obviously. the one observation from the 2n-variate normal may be
considered as a sample of n correlated observations from a bivariate
normal distribution determined by P , whose correlation structure is
determined by m, which is what we actually need.
For each of the 1000
samples. the five estimators Sl' S2' 53' 54 and 55 were computed and
then their mean and variance over the 1000 resulting values were used
to estimate their mean square error (M.S.E.).
In computing the estimators use was made of the following forms.
= 1,
i
2, ... , n, then
mel = (X 'E1 Y)/ «n-m)(n-m-l»
(m)e1
e
8
1
3
=
cr.'
=
<X' E2 Y)/n(n-l)
=
(X' (E1 + E2)Y) !n(n-l)
=
(I' J Y)/n
E2 Y)/(n(2n-m-l»
n>m+2
n <m
+
E2
ij =
2
ij
=
1~
• • • (4.2.2.1)
li-jl > m
li-j I
~m
0
i=j
0
li-j I > m
1
li-j I ~ m, i " j
I
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I
I
_,
2
where J is an (nxn) matrix of l' s,
El
.1
We may point out here that it would have been possible - but
vezy:arduous - to avoid the sampling experiment performed to arrive at
numerical values for the M.S.E. of the first four estimators for the
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combinations of n,
that can
b~~
TIl
and p considE.Ted here using the g8Dcr.ll
laborious} y obtained.
foYl-=-;~;
This proc.edurc would have the
advantage of the absence of sampling errors from the H.S.E. estimates.
However, such a procedure \-1aS cor,sidered prohibitively tedious for the
fifth estimator.
Consequently, in order to enhance comparability and
save time and effort,
vIC decidf~d
•
on the Monte Carlo sinm1ation pro'-
cedure fOT the five estimators.
4.2.3.
Conclusiol~.§.
The variance -Has invariably the major component of M. S. E. of the
five estil;:ators.
study.
Table
C.l
sU1T'lnarizes the results of the Monte Carlo
It gives the Relative Mean Square Error, defined as the ratio
of the N.S.E. to the parameter value being estimated and henceforth
denoted by R.N.S.E.
This will be our criterion for comparing the fi'l'::
estimators.
The main conclusion to
be-dra'~
out of the results of the study
is that for small sample sizes (up to 15 observations) making maxhlun
use of the sample information is mOTe important than avoiding corrda··
tion structures by basing estimators on the subsets of uncorrelated
observations in the sample that form kernels for the parameter being
estimated.
This is indicated by the facts that 54 had, un i f O"!:mly,
thc~
lowest R.M.S.E. of all estimators and 52 had lower R.M.S.E. than
Sl when m was relatively large compared to n.
Naturally, the performance of the five estimators, in terms of
R.H.5.E. --, improved as the sample size increased; also, the differences
among the estimators decreased considerably.
\\Cas 14.6, 3.8 and 2.7 for n
=
The range of the R.H.S.:E.
5, 10 and 15 respectively.
In particular,
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67
the magnittide of the relative superiority of 54 with respect to the
rest of the estimators declined as n i.ncreased.
Fut"thar, the uni.form
ordering pattern of the five estimators that held for n
~
5
ov~r
all
.-I
values of the correlation coefficient and time lag - with the exception
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of 8 and 8 2 exchanging relative positions as m increased - was replaced
1
by a criss-crossing pattern that was much in effect for n = 10, 15.
The fifth estimator, based on normal theory, had - consistently _.
the highest R.M.S.E. - and with a wide margin - except for m : 15 when
Sl came close to it, for n ::: 5.
Evidently .this wns clue to the large
number of component parameters that had to be estimated in the c:ourse
of computing S5 and the non-linear form in which they were used.
How-
ever, it showed the largest proportional decline in R.M.S.E. as n
increased.
Further, it was not uniformly the worst estimator starting
with n ::: 10.
for n
= 15,
Actually, it competed with 8
moderate correlation
~nd
4
for the lowest R.M.S.E.
intermediate time lags.
l.Je ll!ay
even speculate that for larger samples, 55 will probably compare more
favourably or have the lowest R.M.S.E. as the disadvantages of a large
number of component parameters and non-linear forms get overwhelmed by
sample size as the present results indicate.
For m ::: 0,
e
(m) l
==
0 and for m relatively small compared to n,
(m)e l makes relatively little use of the sample information; hence,
(m)e
l
had a relatively high R.M.S.E. as may be expected.
for m + 2 > n
mel
==
Similarly,
0 and for m relatively large compared to n mel
makes relatively little use of the sample information.
Clearly, 8
3
tended to be dominated by Sl or S2 depending on the re1ati.ve magnitude
of m to n.
For n = 5, with S5 being uniformly worse than all other
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68
C'stiElates, S2 and Sl exchanged the position of the scCOnd-\'7on:t esUr":l"
tor as m ix-.creased; and for n = 10, 15 they also exchang8d their l"L'h!-tive posit:ion - \olhich \-'as the \.".orst for n == 15.
The R.N.S.E. of Sl . . :Jld
52 tended to increase dY'w,tical1y in response to inc1."easps in
Ip I.
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-CI-li'..PTER V'
SUMM..A.RY AND SUGGESTIONS FOR FURTHER RESEARCH
5.1.
SUllimaq::.
Stochastic mone1s of human population change arc valuable
theoretical constructs and useful tools of practical applications.
However, available stochastic formulations are of limited use because
of over-simplified assumptions, complicated mathematics and difficulty
of applications.
In the present study, simple macro-analytic stochastic formulations of the problem of human population change over time are considered on two levels of detail, namely, total population size and age
structure of the population.
In the development of these formulations,
no conditions were imposed on the physical mechanism producing the
process of population change and the resulting methodologies are simpler
than what is presently available.
Demographic variables are assumed to have the simple stochastic
structure of a constant depending on time plus a stationary random error
component;
~.~.
the stochastic mechanism producing random variations
as a componant of the total variation in the population is assumed to
remain stationary over time.
In Chapter II, a basic probability space is defined whose elements
are identified with relative conglomerations of socio-cultural conditions that are responsible for the random component in the change of
70
the population.
A discrete error-type probability law that allows for more
uncertainty - in the sense of less probability mass concentration around
the mean - relative to the normal and symmetric binomial
is discussed.
distributio~s
Such a distribution would be of value for types of data
that are not commensurate with high degrees of accuracy as is the case
with demographic data for developing countries for exmnple.
Using characteristics of the above-mentioned probability space,
probabilistic characterizations of the distribution of the population
at a certain point in time are arrived at under both discrete and continuous probability structures.
The theory discussed is illustrated by
data for the population of Egypt.
The
proble~
of the probability distribution of the age distribu-
tion function is considered in Chapter III.
A generalized formulation
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of survivorship from birth incorporating net nigration is used to present the age distribution function formulations of Lotka and the model
of component projections.
The probability distribution of the number of individuals in two
different age groups at a certain point in time is described by its
lower moments,
~.~.,
the means,variances and covariance, in general and
in the special case of an underlying multivariate normal distribution.
A numerical example is given using Egyptian data.
Because demographic data needed for the application of stochastic
models of population change are usually in the form of time series,
classical methods of estimation are not strictly applicable for the
samples available are not - in general - composed of independent
observations.
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An investigation of some aspects of this estimation problem is
carried out in Chapter IV for the case of a certain function of joint
moments of a multivariate distribution.
It is found thnt estimators
usually considered are biased and that it is possible to define unbiased
estimators for m-dependent processes by basing estimators on subsamplcs
of independent observations that can be formed from the original sample.
Because the variances of estimators considered are too general and too
complex to be of much analytical use, a special case was investigated
using the Mante Carlo method of statistical experimentation to study
the mean square error of five estimators for the variance of the number
of individuals in a certain age group at a certain point in time,
assuming an underlying normal distribution, in small samples.
The main conclusions to be drawn out of the study are that maximum use of sample information outweighs all other considerations in
comparing the performance of estimators in terms of M.S.E., and that
an estimator based on normal theory does not compare favorably with
other estimators that do not assume an underlying normal distribution,
presumably because it depends on a large number of parameters in a nonlinear fashion.
5.2. Suggestions for Further Research
Two main directions for further useful research that are indicated
by this study are:
(1) investigation of probabilistic characterizations for the
distribution of the age distribution function that describe the distribution more fully than its moments,
(2) further examination of the problem of parameter estimation
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72
from time serJ.es data in general, and the estimation of functions of
joint moments of mult.iv..n:iate distributions from serially correlated
observations in particular.
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APPENDICES
A. DATA AND INTER}lliDIATE CALCULATIONS FOR SECTION (3.4)
TABLE A.l
BIRTH
P~\TE
PER 1000 POPULATION CORRECTED FOR UNDER-REGISTRATION
EGYPT (1940-1959)
==- --
Year
1940
1941
1942
1943
1944
Birth rate
46.2
45.1
41.0
42.2
44.3
Year
1946
1947
1948
1949
1950
1951
Birth rate
45.5
49.1
46.3
45.3
47.1
48.0
Year
1952
1953
1954
1955
1956
1957
Birth rate
48.2
45.9
45.7
45.9
42.3
40.0
Year
1958
1959
Birth rate
42.8
44.8
Source:
1945
E1-Badry. "Trends in the Components of Population GrO\vth in
the Arab Countries of the Middle East." [11J
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TABLE A.2
ESTIHATED HID-YEAR POPULATION IN THOUSANDS, EGYPT
(19 l I0-1959) 'I:
Year
Population
Year
Population
Year
Population
Y.ear
Population
1%0
1941
1942
1943
18236
18434
18635·
18838
190h3
19250
1946
1947
1948
1949
1950
1951
19460
19734
20201
20678
21167
21668
1952
1953
1954
1955
1956
1957
22180
22704
23241
23791
24353
24929
1958
1959
25518
26121
1945
*Using
census figures corrected for under-enumeration of children less
than ten years of age and assuming exponential population growth bet\veen
censuses.
Source:
Compiled from "Population Trends in the U.A.R."
[6J
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TABLE A.3
REPORTED DEATH RATE PER 1000 }L~LES IN THE AGE GROUP (0-4)
EGYPT (1940-1959)
1940
1941
1942
1943
117.4
112.1
116.5
100.1
99.5
106.9
Year
1946
1947
1949
1950
1951
Death rate
93.1
83.6
108.3
88.3
81.6
86.2
Year
1952
1953
1954
1955
1956
1957
Death rate
78.7
79.4
68.5
70.2
59.0
66.6
Year
1958
1959
Death rate
61.6.
57.0
Year
Death rate
Source:
"Population Trends in the U.A.R. lI
[6]
1945
76
TABLE A.4
\
AVERAGE RATIO OF SINGLE-YEAR nEATH JtATE TO THE DEATH RATE
OF THE AGE GROUP (0-4), EGYPT,
1937, 1.9/17 AND 1960
I~IFE TABT.ES
Year of life
Ratio
1
2
3
4
5
2.100
1.300
0.725
0.325
0.200
----_._-----_._---------_._-----------Source:
Compiled from IIPopu1ation Trends in the U.A.R." [6J
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TABl.E A.5
RATIO OF CRUDE DEATH RATES CORRECTED FOR UNDER-REGISTP-ATION
TO REPORTED RATES, EGYPT
(194(J-1959)
-=====::.:.:--=
Year
1%0
1942
1943
1944
1945
Ratio
1.221
1.276
1.267
1.258
1.235
Year
1946
1947
1948
1949
1950
1951
Ratio
1.204
1.243
1.225
1.209
1.220
1.212
Year
1952
1953
1954
1955
1956
1957
Ratio
1.198
1.200
1.197
1.256
1.141
1.174
Year
1958
1959
1.175
;1..222
Ratio
Source:
1941
Compiled from E1-Badry's paper. [11]
I
B. CO}fPUTER l'ROGRAN
The FORTHAN IV computei' program used in the Honte Cnr10 study
describc'J in
(l~.
2) is reproduced be1oH:
DI~mNSJON
X(50,50),T(50,50),S(50,50),SD(50),R(7),M(8),A(8),B(25)
1,Yl(50),Y2(50),Y(25),E1(25,25),E2(25,25),ES(5),ESM(5),ESV(5),
2EStlliS(5),ESR}lSE(5),T1(2500),E11(625),E21(625)
C BUILDING lAG AND CORRELATION ARRAYS
1'1(1)=0
M(2)=l
1'1(3)=3
DO 2
3=1~,
6
2 M(J)=(J-3)*S
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DO 3 J=1,6
3 A(J)=15./(M(J)+1.)
R(1)=-.75
DO 4 1=2,7
4 R(I)=R(I-l)+.25
DO 1 N=lO,30,lO
N2=N/2
WRITE(3,9)N2
9 FORMAT(lHl,13HSAMPLE SIZE =,5X,I5)
DO 1 J=1,6
WRITE(3,19)H(J)
19
FOR}~T(/////1HO,4HLAG=,14X,I5)
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79
C BUILDING COPJlliLATION bECAY FACTORS ARRAY
DO 5 K:1,25
5 B(K)=O.O
MJ=M(J)
IF(MJ.EQ.O)GO TO 91
DO 6 K=l,MJ
B(K)=~~(-A(J)*K)
IF(ABS(B(K».LT •• IE-I0)B(K)=O.O
6 CONTINUE
91 MJ2=M(J)+2
C BUILDING
Y~TRICES
REQUIRED FOR COMPUTING ESTIMATORS
IF(N2.LT.MJ2)GO TO 15
DO 17 K=I,N2
DO 17 L=1,N2
IF(IABS(K-L)-MJ)117,117,18
18 El(K,L)=l.
E2(K,L)=O.O
GO TO J.7
117 E1(K,L)=O.O
E2(K,L)=1.
17 CONTINUE
121 DO 49 K=1,N2
49 E2(K,K)=O.O
GO TO 16
15 DO 20 K=1,N2
DO 20 L=1,N2
so
--I
El(K,L)=O.O
IF(K-L)21,200,21
21 E2(K,L)=1GO TO 20
200 E2(K,L)==0.0
20 CONTINUE
16 DO 1 1=1,7
WRITE(3,29)R(I)
29 FOR}~T(/!/lHO,24HCORRELATION
C BUILDING VARIANCE MATRIX
DO 7 K=l,N
7 X(K,K)=l0
DO 8 K=2,N
8 X(K,K-l)=R(I)
DO 59 K=l,N,2
K2=K+2
DO 59 L=K2,N,2
LB=(L-K)!2
X(L,K)=B(LB)
59 X(L+l,K)=X(L,K)*R(I)
DO 10 K=2,N,2
K2=K+2
DO 10 L=K2,N,2
LB=(L-k)!2
X(L,K)=B(LB)
10 X(L-l,K)=X(L,K)*R(I)
I
COEFFICIENT~,F10.6)
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81
DO 81 K""l,N
DO 81 1,=l,K
IF(ABS(X(K,L».LT •• 1E-10)X(K,L)=0.0
81 CONTINUE
DO 101 K=l,N
DO 101 L::::1,K
101 X(L,K)=X(K,L)
C GENERATING HULTINORK<\L OBSERVATIONS
CALL FCTRIZ(X,N,T)
W=O.O
IX=I*}LJ2*N+1
DO 26 L=I,5
ESH(L)=O.O
26 ESV(L)=O.O
NS=1000
DO 11 NN=I,NS
DO 12 L=l,N
CALL
NOP~EV(IX,W,Z)
IF(ABS(Z).LT •• IE-10)Z=O.O
12 Yl(L)=Z
CALL PACKR(T,50,50,Tl,N,N)
CALL MULT(Tl,N,N,Yl,N,1,Y2)
DO 13 L=1,N,2
Ll=L+l
L2=L1/2
13 Y(L2)=Y2(L)*Y2(Ll)
82
C CONPUTATIUN OF ESTU1ATES
ESOl=O.O
DO 14 L=1,N2
14 ES01=FSOlfY(L)*Y(L)
ESO~ES01/K2
CALL PACKLS(El,25,25,Ell,N2,N2)
CALL ATRBA(Y,N2,1,Ell,N2,N2,ES(1»
CALL PACKLS(E2,25,25,E21,N2,N2)
CALL ATRBA(Y,N2,l,E21,N2,N2,ES(2»
ES(3)=ES(1)+ES(2)
ES (4)=ESOl+ES (3)
IF(N2.LT.MJ2)GO TO 22
ES(1)=ES(1)/«N2-M(J»*(N2-M(J)-1»
ES(2)=ES(2)!(N2*(2*N2-M(J)-1»
GO TO 23
22 ES(2)=ES(2)/(N2*(N2-1»
23 ES(3)=ES(3)/(N2*(N2-1»
ES(4)=ES(4)/(N)*N2)
DO 24 L=1,4
24 ES(L)=ESO-ES(L)
Yill=O.O
YM2=O.O
Vl=O.O
V2=O.O
V12=O.O
DO 25 L=1,N,2
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83
Ll=L+l
YMl::YNl+Y2(L)/N2
YH2==YH2+y2 (Ll) /N2
Vl~VI4Y2(L)*Y2(L)/(N2-1)
V2=V2+Y2(Ll)*Y2(Ll)/(N2-1)
25 VI2=V12+Y2(L)*Y2(Ll)/(N2-1)
Vl=VI-N2*Yl-U*nu/ (NZ-l)
V2=V2-N2*YH2*YH2/ (N2--1)
VI2=VI2-N2*~11*YM2/(N2-1)
ES(5)=VI2*VI2+Vl*YM2*~12+V2*YMl*YMl+2*VI2*YM1*nI2+Vl*V2
DO 11 L=I,5
ESM(L)=ESM(L)+ES(L)/NS
11 ESV(I.)=ESV(L)+ES(L)*ES(L)/NS
P=R(I)*R(I)+L
DO 27 L=I,5
ESV(L)=ESV(L)-ESM(L)*ESM(L)
ESMSE(L)=ESV(L)+(ESM(L)-P) **2
27 ESRMSE(L)=ESMSE(L)/P
1 WRITE(3,39)P,(L,ESM(L),ESV(L),ESMSE(L),ESRMSE(L),L=I,5)
39 FORMAT (lHO, 16HPARAMETER VALUE=,4X,FI2.6,///3X,9HESTlMATOR,12X,
14HMEAN,12X,8HVARIANCE,12X,6HM.S.E.,14X,8HR.M.S.E.//(6X,I5,6x,4
2(FI4.6,5X»)
STOP
END
84
SUBROUTINE FCTRIZ(X,N,T)
DIMEKSION X(50,50),T(SO,50),S(50,SO),SD(50)
DO 14 K=I,N
DO ll. V=K,N
J.4 T(K,L)=O.O
T(I,l)""SQRT(X(I,l»
IF(N-l)ll,12,2
2 DO 3 I:;:2,N
3 T(I,l)=X(I,I)/T(!,l)
Nl=N-l
IF(Nl--1)12,J.,4
4 DO 5 K=2,Nl
Kl=K+l
DO 5 I==Kl,N
5 S(I,K)=T(I,I)*T(K,I)
DO 6 J=2,N
6 SD(J)=T(J ,1)*T(J ,1) .
DO 10 K=2,N1
T(K,K)=SQRT(X(K,K)-SD(K»
Kl=K+l
DO 9 I=Kl,N
9 T(I,K)=(X(I,K)-S(I,K»/T(K,K)
IF(NI-K)l2,1,7
7 DO 8 J=K1,N
8 SD(J)=SD(J)+T(J,K)*T(J,K)
DO 10 L=Kl,Nl
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85
,.
L1=L+1
DO 10 M=L1,N
10 S(M,L)=S(M,L)+T(M,K)*T(L,K)
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1 SD(N)=SD(N)+T(N,N1)*T(N,N1)
T(N,N)=SQRT(X(N,N)-SD(N»
RETURN
11 WRITE(3,13)
13 FORMAT(1HO,33H VARIANCE MATRIX DIMENSION ERROR)
12 RETURN
END
SUBROUTINE NORDEV(IX,W,Z)
IF(W)1,1,2
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2 Z=2.*W*V*U
W=O.O
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RETURN
1 CALL RANDU(IX,IY,V)
IX=IY
CALL RANDU(IX,IY,U)
IX=IY
U=2.*U-1.
W=V*V+U*U
IF(1.-W)1,3,3
3 CALL RANDU(IX,IY,X)
IX=IY
,.
I
W=SQRT(-2.*ALOG(X»/W
.-
I
86
Z= (V*V-U'~U)
,'w
RETURN
LND
The subroutines:
PACKR
packs a rectangular matrix as a column vector
HULT
multiplies two rectangular mio"'.trices
PACKLS: packs a lower symmetric matrix as a column vector
ATRBA : performs the matrix multiplication AlBA
were written by Dr. T. G. Donnely for the Hatrix Subroutines Pac.kage
of the
Dep~rtment
of Biostatistics, University of North Carolina at
Chapel Hill, North Carolina.
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C. NillmRlCAL RESllLTS OF THE H01'1'fE CARLO STUDY
TABLE
RELATIVE MEAN SQUARE ERROR OF THE FIVE ESTIMl\.TORS
-
_..
n
m
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I
Estimator
Corre~ation
-0 :1.L-=-Q.!..~
Parameter Value
5
0
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.1
I
C.1
1
3
-0.25
.-:.:~.=:::::==::::~:.::.~
Coefficient
;-,
._--
0.0
0.25
O'.~__ ~_.'
1. 5625
1.25
1.0625
1.000
1.0625
1.25
1.5625
1
4.225
2.655
2.100
1.476
1. 795
2.200
3.146
2
7.061
3.390
2.435
1.394
2.085
2.918
4.944
2.655
2.100
1.476
1. 795
2.200
3.1l16
3
.4.225
4
2.743
1.769
1.381
0.998
1.188
1./173
2.101
5
11.820
5~776
4.439
2.802
3.813
5.052
8.572
1
5.925
3.527
2.083
1.656
2.352
4.442
3.492
2
8.635
4.111
2.111
1.517
2.125
4.743
5.160
3
5.879
3.222
1.994
1.606
2.077
4.132
3.571
4
3.779
2.107
1.295
1.058
1.371
2.687
2.346
5
12.951
7.396
4.588
3.222
4.768
6.566 10.652
1
6.063
4.200 . 5.237
2.449
3.274
5.183
6.802
2
4.594
3.169
2.736
1. 780
2.041
4.010
5.650
3
3.634
2.721
2.468
1.833
1. 929
3.627
4.685
4
2.390
1. 782
1.611
1.183
1.268
2.374
3.031
5
11.285
6.859
6.741
3.673
4.791
7.058 13.142
8°u
TABLE C.I--Continued
.
..
=-----==--=--.:::.;.::;.=.;.:::.::.~"":::::;..:.~.:;:::_:::~,.;;:R_:::.:::.-=;=:::;=_...=_:= ..:::.::_:::;.;=_~_=:::::.~:·=:'::::=:=';':'::~;..~_ .:";=:-:.:,::;-:::-:-;=-~~:-=-,,=,._
n
m
-:::..-=:.---==
Estimator _________ -..£9rr_~.latic.?lL(,~()_(;d)2-,-c:;i~1l.L_.
:_____._____
" ..:-0 .1.5__:.9_~5__ -=.Q: 25___~ ._Q___ ._._.iJ~?-2. __.9~5______9_~? 5 __
5
10
15
10
0
1
6.580
4.013
4.369
1.527
2.617
3.524
7.711 6
2
3.353
2.666
3.834
1. 62)
2.188
2.507
'•. 745
3
3.3.53
2.666
3.83/1
1.621
2.188
2.507
4.745
-4
2.:115
1.748
2.497
1.085
1.438
1.652
3.098
5
11.539
7.879
4.508
2.834
5.221
5.682 10.853
1
7.260
4.586
2.308
1.392
2.139
2.809
9.668
2
4.11/.
2.910
1.818
1.373
1.641
1.710
6.095
3
4.114
2.910
1.818
1.373
i.641
1.710
6.095
4
2.709
1.919
1.231
0.934
1.121
1.200
4.017
5
8.367
6.312
3.465
2.517
3.438
4.248 10.931
1
15.252
5.525
2.242
1.655
2.244
I~. 770
9.756
2
5.771
3.136
1. 776
1.537
1. 679
2.612
3.573
3
5.771
3.136
1.776
1.537
1.679
2.612
3.573
4
3.7"61
2.098
1.191
1.051
1.154
1.753
2.350
5
15.560
6.890
2.972
2.387
3.802
5.318 10.238
1
2.459
1.200
0.992
1.067
1.021
1.266
1. 726
2
3.811
1. 784
1.188
1.03l
1.147
1.896
2.989
3
2.459
1.200
0.992
1.067
1.021
1.266
1. 726
4
2.006.
0.'997
0.813
0.863
0.826
1.043
1.431
.5
2.666
1.839
1.302
0.974
1.047
1. 991.
2.696
-------_.-
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89
TABLE C.1--Col'. t inucd
n
m Estitlator
_
____-~o::..;._I2-_-=.9_~5
1
3
5
10
-0. 25,,--_0_._0
0.:J.5_ _.Q.~__Q.. 75 __
1
2.137
1.~82
1.030
0.782
1.039
1.203
1.693
2
3.353
2.152
1.173
0.742
1.142
1.652
2.688
3
2.088
1. 498
1.014
0.772
1.027
1.191
1.667
4
1. 707
1.224
0.826
0.638
0.846
0.986
1.362
5
2.988
2.156
1.354
0.938
1.267
1.797
2.389
1
1. 709
1.495
1. 064
0.901
1.034
1.327
2.180
2
2.354
1.933
1.106
0.885
1.105
1.617
2.879
3
1.665
1.464
0.999·
0.905
0.978
1.241
2.067
4
1.373
1.186
0.815
0.737
0.795
1. 024
1. 686
5
2.397
1.931
1.193
0.926
1.260
1.873
2.586
1
2.397
2.120
1.167
0.868
1.291
2.186
2.582
2
2.872
2.197
1.153
0.786
1.150
2.453
2.821
3
2.180
1.907
1.078
0.797
1.110
2.127
2.276
4
1.768
1.547
0.881
0.653
0.904
1. 742
1.860
5
3.118
1.780
1.274
1.025
1.052
1. 852
2.602
1
3.824
2.404
1. 022
0.966
1.330
1. 632
3.691
2
2.280
1.501
0.851
0.956
1.119
1.050
1.889
3
2.280
1.501
0.851
0.956
1.119
1.050
1. 889
4
1.864
1.228
0.710
.0.782
0.923
0.881
1.552
5
2.566
2.179
1.220
0.903
1.170
1. 669
3.335
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I
90
TABLE C .l··-Collt:imwd
=----==~--=-~_.:.:.::=
n
m
~~=:::=.:_~=':;:..=--~--.:.:::-~.~-:::==.;..-:':.:::::':':'-:~
Estintator
_ .__
~ _ _.___Qp.xL~L~tion C(2.S'J.tt..~i5'nt_·
-0.75
15
15
0
1
3
.=..:~:~"::7_'7;."'.::::':-':'=:-=::-;'-":'=-....;:;;;:.;.:,,:::::.~.~-==-=~:'.':==:=:--=:'";';:::
-0.5
.-O_,.? ~_..Q,_O ____~~..L__
O"-~ ..___-.Q:.} 5 "~"
1
4.265
3.045
1.405
0.865
1.154
2.636
4.451
2
2.282
1. 946
1.031
0.792-
0.906
1 ,1133
2.237
3
2.282
1. 946
1.031
0.792
0.906
l. l133
2.237
q
1.880
1.590
0.864
0.663
0.762
1.178
1.832
5i
3.290
2.182
1.303
0.948
1.175
2.71+8
3.557
1
1.343
0.957
0.600
0.537
0.671
0.790
1.075
2
2.328
1.396
0.716
0.528
0.780
1.169
2. all.
3
1.343
0.957
0.600
0.537
0.671
0.790
1.075
4
1.178
0.839
0.527
O.ll72
0.592
0.702
0.944
5
1.457
0.968
0.620
0.569
0.655
0.980
1.438
1.
1.484
0.907
0.691
0.463
0.776
1.043
1.432
2
2.478
1.312
0.795
0.466
0.843
1.502
2.292
3
1.487
0.903
0.697
0.461
0.775
1. 0117
1.423
4
1.2"94
0.791
0.612
0.412
0.676
0.915
1.250
S
1.549
0.977
0.561
0.504
0.568
1.135
1.4'17
1.
1.655
0.967
0.676
0.453
0.633
1.099
1.465
2
2.412
1.274
0.760
0.445
0.689
1.493
2.3 l13
3
1.625
0.939
0.681
0.449
0.625
1. 09 /+
1.491
4
1.417
0.817
0.597
0.398
0.548
0.956
1.304
5
1.450
0.9l.2
0.573
0.381
0.590
1.157
1.545
-.
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LIST OF REFERENCES
1. AHchinson, J. and J. Bro\VIl.
l'he_L2.,g1~o..rmaJ-l?JsJ:_I}bution,
CmLbriclgo University Press, 1957, pp. 8-9.
2.
AJ~oian,
L. A.
liThe Probability Function of the Product of T\vo
:Normally Distributed Variables." An.!:l.al!i..-of Mat:..hcma_tiq!.:J:.
St_ati§tic.s, Vol. XVIII, 1947, pp. 265·-277.
3. Baney, N. T. J. Th£ Elem_eE.ts of St:.QEJ..1_"!.~.!ic Proc_~.§.:~8s..\Vi_~.h.
APplications to the Natu,ra1 Sciences, John lViley & Sons, Inc.,
New York, 1964, pp. 110-115.
4. Bartlett, 'H. s.
Stocllastic Processell, Notes of a course given
at the University of North Carolina in the Fall quarter, 1946,
pp. 28-42.
5. Bell, J. R. "Algorithm 334, Normal Random Deviates," Communications
of the IICM, July 1968, p. 498.
6. Central Statistical Committee.
Population Trends in the U. A. R.,
Government Printing Office, Cairo, 1965.
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