Download On a symptotic distributions in random sampling from finite

0
08
8
On a symptotic
o distribut
otic
distributions in
random sampling
mpling from finite
mpling
ni
populations
Paul Knottnerus
Discussion paper (09002)
Statistics Netherlands
The Hague/Heerlen, 2009
Explanation of symbols
.
*
x
–
–
0 (0,0)
blank
2005-2006
2005/2006
2005/’06
2003/’04–2005/’06
= data not available
= provisional figure
= publication prohibited (confidential figure)
= nil or less than half of unit concerned
= (between two figures) inclusive
= less than half of unit concerned
= not applicable
= 2005 to 2006 inclusive
= average of 2005 up to and including 2006
= crop year, financial year, school year etc. beginning in 2005 and ending in 2006
= crop year, financial year, etc. 2003/’04 to 2005/’06 inclusive
Due to rounding, some totals may not correspond with the sum of the separate figures.
Publisher
Statistics Netherlands
Henri Faasdreef 312
2492 JP The Hague
Prepress
Statistics Netherlands - Facility Services
Cover
TelDesign, Rotterdam
Information
Telephone .. +31 88 570 70 70
Telefax .. +31 70 337 59 94
Via contact form: www.cbs.nl/information
Where to order
E-mail: [email protected]
Telefax .. +31 45 570 62 68
Internet
www.cbs.nl
ISSN: 1572-0314
© Statistics Netherlands, The Haque/Heerlen, 2009.
Reproduction is permitted. ‘Statistics Netherlands’ must be quoted as source.
6008309001 X-10
On asymptotic distributions in random sampling
from nite populations
Paul Knottnerus
Summary
Existing proofs of central limit theorems in random sampling from nite
populations are quite lengthy. The present paper shows how the proof
for this kind of theorem in random sampling can be simpli ed by using
the central limit theorem for independent, random vectors. Furthermore,
use is made of the relationship between sampling without replacement and
Poisson sampling given the condition that the sample size is n. The theorems deal with both equal and unequal probability sampling. In the latter
case an approximation formula for the variance emerges as by-product
from the central limit theorem without using second order inclusion probabilities.
Keywords: Central limit theorem; Error normal approximation; Simple
random sampling; Unequal probability sampling; Variance estimation.
1. Introduction
Proofs of the central limit theorem for equal probability sampling from a nite
population can be found in Madow (1948), Erdös and Renyi (1959), and Hájek
(1960). For rejective Poisson sampling with varying probabilities a proof can
be found in Hájek (1964). All proofs are technically di-cult to demonstrate
and omitted in most texts. Often simulations are used for demonstration; see
Bellhouse (2001).
The main aim of this paper is to provide less intricate proofs for the central
limit theorems in probability sampling from nite populations. Proofs in this
paper are based on a generalization of the central limit theorem for a sequence
of mutually independent two-dimensional random variables.
The outline of the paper is as follows. Section 2 examines the asymptotic behaviour of estimators in simple random sampling. Section 3 examines the asymptotic behaviour of estimators in unequal probability sampling. Unlike Hájek
(1964) the author doesn’t use (approximate) second order inclusion probabilities.
3
2. Asymptotic distributions in simple random sampling
Consider a population U of N numbers U = {y1 , . . . , yN }. Suppose that as
2
Ny
N
the population mean Y N and the population variance
Y and
2,
y
respectively. Let y s denote the sample mean from a simple random
sample of
xed size n (= fN N) without replacement (SRS). The sampling
fraction fN is such that |fN
converge to
f | < 1/N where f is a xed number (0 < f < 1).
Next, de ne for the Poisson sampling design with equal probabilities the following unbiased estimators for the population mean Y N and n
1
N
Y PO =
N
ak
k=1
yk
fN
N
ns =
ak,
k=1
where the ak are independent and identically distributed random variables.
That is,
ak =
1
if the sample includes element k
0
otherwise
with P (ak = 1) = fN (k = 1, . . . , N).
In addition, de ne the random two-dimensional vector xk and the matrix
k
by
x1k
xk =
k
respectively.
k
x2k
= fN (1
yk
fN
= ak
fN )
1
yk2
2
fN
yk
fN
yk
fN
,
1
is the covariance matrix of xk . Other useful de nitions and
formulas in this context are
xN
1
=
N
ns /N
k=1
lim
N
N
= lim
N
(
= (1
Denote the typical element ij of
Y PO
xk =
YN
fN
E(xN ) =
=
N
f)
by
ij
1
N
N
k
k=1
2 +Y 2 )
y
f
Y
Y
f
and that of
1
by
ij
(1
i, j
2).
Applying the central limit theorem for vectors to the mutually independent xk
yields the following theorem.
4
Theorem 1. As N
the distribution of
N{xN
E(xN )} tends to the
bivariate normal distribution with zero expectation and covariance matrix
,
provided that the data satisfy the Lindeberg condition.
For the Lindeberg condition, see Hájek (1964, page 1500) and Feller (1971,
pages 262-3). Also note that the Lindeberg conditions for one dimension carry
over to two-dimensional vectors due to the Cramér-Wold device; see Basu (2004,
page 149).
In order to apply Theorem 1 to the problem of the limiting distribution of y s ,
de ne for an arbitrary constant u0
P0 = P (y s
Y N + u0 h1 )
h21 =
(=
1
= (1
N 11
f(1 f)
22
(=
)
N
N
1
1
exp( u2 )
2
2
N
h22 =
(u) =
f)
22
2
y
n
)
u
(u) =
(x; ) =
(x)dx
1 x/2)
exp( x
2 | |1/2
where in the last line x = (x1 , x2 ) and
that 1/
by
N
11
=cov(x). Recall from statistics theory
is the conditional variance of x1 given x2 . Furthermore, de ne
= µ3N /
3
N
and
by
=
where, given N,
2
N
N
and µ3N stand
for the variance and the third central moment of x2k , respectively. Note that
N
=
N (fN ).
The following theorem speci es the error of the approximation of the lattice
distribution of x2N by the normal distribution.
Theorem 2. For a midpoint x of the lattice distribution F2N of
E(x2N )}/
N
N{x2N
it holds that as N
F2N (x;
N)
(x) =
2N (x;
) + o(1/ N)
(1)
where
(2)
(1 x2 ) (x)
6 N
Proof. For the lattice distribution F2N it holds that for a midpoint x of the
2N (x;
)=
N)
(x) =
lattice
F2N (x;
2N (x;
N)
+ o(1/ N)
(3)
For a proof, see Feller (1971, page 540). Note that Feller assumes that
However, since
by
N.
N
=
{1 + o(1)} the proof keeps valid when
Moreover, since the derivatives of
N
and
2N
|
N
5
= .
is replaced
as functions of fN are
f| < 1/N, it holds that
continuous at the point f and |fN
N
| = O(1/N)
|
2N (x;
N)
2N (x;
)| = O(1/N)
from which (1) follows. This concludes the proof.
Expression (2) for the error
2N (x)
is based on a more re ned Taylor approx-
imation of the corresponding characteristic function
N
2rf (t/ N
N)
(1 +
3
N (it)
6 N
2 (t/ N
N) resulting in
) exp( t2 /2)
instead of exp( t2 /2). In addition, if F2N is not a lattice distribution (3) is
true for all x. Similar results can be derived for two-dimensional random vectors
or when the x2k have diBerent variances for a given N; see Feller (1971, pages
521-548). Now the following theorem on the asymptotic behaviour of y s can be
proved. The notation "A
N
B" is used to indicate that A/B tends to unity as
.
Theorem 3. As N
it holds under the Lindeberg condition that P0
(u0 ).
Proof. Denote Y N +u0 h1 by m0 . It follows from the equivalence of SRS sampling
and Poisson sampling conditional on a given n or fN that
P0 = P (x1N m0 |x2N = fN )
P (x1N m0 x2N = fN )
=
P {x2N = fN }
(4)
First we look at the relatively simple denominator of (4). Applying Theorem 1
yields
P {x2N = fN } = P {x2N
1/2
)
Nh2
(n + 1/2)/N}
(
2N (
1/2
)
Nh2
(n
1/2)/N}
1/2
1
) + o(
)
(5)
Nh2
N
where the error 2N (x) of the standard normal approximation at a midpoint x
= (
1/2
)+
Nh2
P {x2N
2N (
is given by (2). Since
(1
as x
x2 ){ (x)
( x)} = o(1)
0, the total error in (5) is o(1/ N) and hence, using the rst-order
Taylor expansion
(x) = (0) + x (0) + O(x2 ),
P {x2N = fN } =
1
1
(0) + o(
)
Nh2
N
(6)
Likewise, using Theorem 1 we get for the numerator of (4)
P (x1N
m0
x2N = fN ) =
u0 h1
1/2N
1/2N
Next, transform x according to z = T x with
T =
1
&
0
1
6
,
(x; /N)dx + o(
1
)
N
(7)
where & =
= Y /f. Since |T | = 1 and T T /N=diag(h21 , h22 ), (7) can
12 / 22
be rewritten as
u0 h1
1/2N
1/2N
z2
u0 h1
1/2N
1/2N
=
=
(z; T T /N)dz + o(
( hz11 ) ( hz22 )
z2
h1 h2
(u0 ) as N
Dividing (8) by (6) yields P0
dz2 + o(
1
)
N
(0)
1
+ o(
)
h2
N
u0 h1 0
h1
1
N
dz1
1
)
N
(8)
. This concludes the proof.
The following corollary is a slightly diBerent version of Theorem 3.
Corollary 1.
Let u1 and u2 be two arbitrary constants (u1 < u2 ). De ne
P12 = P (Y N + u1 h1
ys
Y N + u2 h1 ). Then under the Lindeberg condition
it holds that as N
P12
(u2 )
(u1 )
Comment. Although this corollary follows from Theorem 3, it should be noted
that a direct proof is as follows. By (6) and (7),
P12 =
u2 h1
u1 h1
1/2N
1/2N
(x; /N)dx + o(1/ N)
(0)/Nh2 + o(1/ N)
Because in the domain of integration x1 = O(1/ N) and x2 = O(1/N), x2 can
be set equal to zero without aBecting the order of the error in the numerator.
In fact, the change of the integrand thus introduced is of order
/N
that
1/2
= O(N). Hence, the corresponding change of the numerator
is of order 1/N. Setting x2 = 0 in the integrand and using N
1/2
/N
N; note
11
= 1/h21 and
= h1 h2 , we obtain
u2 h1
u1 h1
1
N
P12
=
exp( x21 /2h21 )dx1
2 h1 h2 (0)/Nh2
(u2 )
(u1 )
This concludes the proof.
3. Asymptotic distributions in rejective Poisson sampling
Consider now Hájek’s rejective Poisson sampling design. His estimator for the
population mean, say Y Haj , is de ned as Y P O given the condition that ns = n
where
Y PO
N
1
=
N
ak
k=1
N
ns =
ak .
k=1
7
yk
k
The
k
stand for the rst order inclusion probabilities and satisfy
N
k=1
k
= n.
For further details, see Hájek (1964).
In analogy with the previous section xk and
become
N
yk
xk = ak
N
1
N
=
Suppose that the matrix
N
k
1
yk2
N
k (1
2
k
yk
k
k)
k=1
yk
k
.
1
converges to the invertible matrix
as N
.
Furthermore, de ne PHaj by
PHaj = P (Y Haj
m0 )
m0 = Y N + u0 h1
h21 =
(=
N
22
1
11
N
),
The following theorem generalizes Theorem 3 for unequal inclusion probabilities.
Theorem 4. As N
it holds under the Lindeberg condition that PHaj
(u0 ). When N is su-ciently large, h21 can be approximated by
h21N
µy
=
=
1
N2
N
k (1
k )(
k=1
N
k )yk
k=1 (1
.
N
k)
k=1 k (1
yk
k
µy )2
Proof. The normality part of the proof runs along the same lines as that of
Theorem 3
PHaj = P (x1N m0 |x2N = fN )
P {x1N m0 x2N = fN }
=
P (Nx2N = n)
m0 Y N
= (u0 )
h1
In order to show that h21 can be approximated by h21N given in the theorem,
note that
h21 =
N
N
22
N
(9)
N,22
De ne
N
' =
k (1
k)
k=1
(k =
k (1
8
k )/'.
Furthermore, we have
yk2
'2
{
(
k
2
N2
k
=
N
'2
N2
=
N
N,22
( (k
yk
k
N
yk
(k
(k
k
k=1
)2 }
yk
2
(10)
k
= ',
(11)
Now it follows from (9)-(11) that
N
h21
N
=
N,22
1
N2
N
k (1
k)
2
yk
µy
k
k=1
= h21N .
This concludes the proof.
It is noteworthy that approximation h21N in Theorem 3 is equivalent to variance
approximation (8.10) of Hájek (1964). His derivation is based on an approximation of the second order inclusion probabilities
kl
leading to some tedious
algebra. Also note that the actual inclusion probabilities, say
ditional Poisson sampling need not be equal to the
k
k,
for con-
corresponding to the
unconditional Poisson sampling design. As pointed out by Hájek (1964), only
asymptotically it holds that
k/ k
1 as '
. This is a maybe somewhat
counterintuitive diBerence with equal probability sampling described in Section
2 where
k
=
k
= fN (k=1,...,N).
References
Basu, A.K. (2004). Measure Theory and Probability. Prentice Hall of India,
Delhi.
Bellhouse, D.R. (2001). The central limit theorem under simple random sampling.
The American Statistician, 55, 352 357.
Erdös, P. and Rényi, A. (1959). On the central limit theorem for samples
from a nite population. Magyar Tudoanyos Akademia Budapest Matematikai Kutato Intezet Koezlemenyei, Trudy Publications, 4, 49 57.
Feller, W. (1971). An Introduction to Probability Theory and its Applications,
Vol. II. Wiley and Sons, New York.
Hájek, J. (1960). Limiting distributions in simple random sampling from a nite population. Magyar Tudoanyos Akademia Budapest Matematikai
Kutato Intezet Koezlemenyei, Trudy Publications, 5, 361 374.
Hájek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a nite population. Annals of Mathematical Statistics, 35,
1491 1523.
Madow, W.G. (1948). On the limiting distributions of estimates based on
samples from nite universes. Annals of Mathematical Statistics, 19, 535 545.
9