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MATHEMATICS OF COMPUTATION
VOLUME 42. NUMBER 166
APRIL 19K4.PAGES 601-616
Calculation of the Moments
and the Moment Generating Function
for the Reciprocal Gamma Distribution
By Arne Fransen and Staffan Wrigge
Abstract. In this paper we consider the distribution G(x) = F~ lfo(T(t)) ' dt. The aim of the
investigation is twofold-, first, to find numerical values of characteristics such as moments,
variance, skewness, kurtosis,etc; second, to study analytically and numerically the moment
generating function <p(i) = /" e~'x/T(x) dx. Furthermore, we also make a generalization of
the reciprocal gamma distribution, and study some of its properties.
Introduction. In [4] we considered, among many other things, the distribution
G(x) = F~x^(Y(t))~xdt,
which was suggested as a distribution of possible use in
reliability theory by Dr. Gustaf Borenius, the former head of our section. He
suggested this distribution because its shape looked like that of the gamma distribution, only with a thinner tail. Moreover, the problem as such was a mathematical
challenge. The normalization constant F was calculated and presented to 60D by
Fransen and Wrigge[4],to 80D by Fransen [3],and to 300D by W. A. Johnson [10].
We were asked by Dr. Samuel Kotz if we could calculate analytically and
numerically the moments of the reciprocal gamma distribution. To our surprise, we
managed to find not only a useful recurrence relation for the moments a„ = E(Xn)
but also several analytical expressions.
The aim of this investigation is twofold: first, to find numerical values of
characteristics such as moments, variance, skewness, kurtosis, etc.; second, to study
analytically and numerically the moment generating function
<p(t) =
Jo
(e-'VT(x))dx.
Furthermore, we also make an obvious generalization of the reciprocal gamma
distribution and study some of its properties.
When starting this study we did not know much about the work of our predecessors on similar problems. We learned that Ramanujan [8] as well as Wyman and
Wong [17] and Doming et al. [1] had studied related problems from a purely
analytical standpoint. In Erdélyi et al., Vol. 3 [2, pp. 217-224] several functions close
to the one studied by us are mentioned. Paley and Wiener used these functions,
when studying inversion formulae for the Laplace transform; see [13, p. 39]. We also
learned a lot when reading a doctoral thesis from 1887 by A. Lindhagen [11] on the
gamma function.
Received September 21, 1982; revised June 20, 1983.
1980MathematicsSubjectClassification.Primary33A15,65D20,65U05; Secondary44A10,60E15.
Key words and phrases. Reciprocal gamma distribution, population characteristics, generating function.
©1984 American Mathematical Society
0025-5718/84 $1.00 + $.25 per page
601
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602
ARNE FRANSEN AND STAFFAN WRIGGE
1. A General Formula for the Mean Value E(xp( X)), Where X Has a Generalized
Reciprocal Gamma Distribution. We begin with a
Definition. By a generalized reciprocal gamma distribution we mean a distribution
with a density function /(x) = \/g(x), where g(x) satisfies g(x + 1) = xg(x) and
x e [0, oo).
We may then formulate
Theorem 1. Let \/g(x) be a generalized reciprocal gamma probability density
function, and let ^(x) be a continuous function such that
Hx)
E(HX))=f^dx
Jo g(x)
exists. Then
-i uF(u)
Jo g(u + 1)
where
F(u) = t(u)+
£
\\
k-i nk4(u+j)
'
=M+i4^(îHu+\:j+k)
£oß(u+J)
k\
A=0
and the series for uF(u) is assumed to converge uniformly in [0,1].
G
Proof.
rk + \f(x)
£
k-oJk
S(x)
J0 g(u+
1)
fi^(u
+ k)
du
k=0Jo g{u + k)
The "partial fraction" is deduced in the ordinary way using the identity
_I_-£("-!)
(-1)"
u(u
+ 1) ■■■(u
n- 1)
í(«+l)--(u + n-l)
JrA
n^0\
m l(n-l)
/(«-!)!«
m
1
„=
+m
Of special importance is the case when 4>(x) is periodic with a period equal to one.
We may, e.g., determine the constant c2n such that c2„sin2"(77\x)/r(jc) becomes a
probability density. (See Section 3b.)
It should be noted that one might as well define the generalized reciprocal gamma
distribution demanding that the density function f(x) satisfies f(x + 1) = f(x)/x.
We have also studied the more general case where the probability density function
f(x) satisfies the functional relation
(1.1)
f(x + 1) = r(x)f(x),
where r(x) is a rational, or even more general, function. It is then easy to see how to
generalize Theorem 1.
An example of such a function/(x), which we have studied in some detail, is
(1.2)
f(x) = cYlr(m,x)/Ur(nJx),
,=1
y=l
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RECIPROCAL GAMMA DISTRIBUTION
603
where m, and «y are positive integers, chosen such that the integral /0°°f(x) dx exists,
and c is a normalization constant. A simple example is
(1.3)
f(x) = cT(x)/T(2x),
giving r(x) = j(2x + I)1.
We plan to present our studies of the more general density functions in (1.1) and
(1.2), and corresponding moments and moment generating functions, in a future
paper.
2. Some Formulae for the Moments and the Factorial Moments.
a. The General Case. Let X be a random variable with the same distribution as in
Theorem 1. We want to obtain a useful formula for the moments a„ = E(X").
Let us define the numbers Xn as Xn = ¡Q(x"/g(x)) dx. We may then formulate
Theorem 2. Let an = j^(x"/g(x))dx
and Xn = fo(x"/g(x))
dx. Then the moments an satisfy the recurrence relation an = 1 + X„ - X0 + 'L"k=l("k)ak_1, with starting value a0 = 1. D
Proof.
n
x"
J
a„ = / —r^dx
Jo g(x)
fx
+ /
J\
x"
J
.
f*>(x + 1)" ,
-7—rdx = Xn + /
g(x)
J0
-——dx.
xg(x)
A simple use of the binomial theorem and some manipulations easily yield Theorem
2. D
We may describe the solution of the equation in Theorem 2 in
Theorem 3. The solution of the recurrence relation in Theorem 2 may be written in
the following way
n
«n = *„+i+ Ldk(n)Xk;
d0(0) = 0, d„(n) = 1, bx = 1,
k=0
where bn is the Bell number of order n and the integers dk(n) satisfy
"
dm(» + 1) = L
I
4- 1 \
1+ 1 )dJj)-som,
j = m \-'
dm(j) = 0 forj<m.
O
'
Proof. The numbers bn satisfy the equation
(2.1)
bn+l=Í(l)bk,
b0= l
(see, e.g., Lunnon, Pleasants and Stephens [12, p. 2]), which is also obtained from the
recurrence relation in Theorem 2. Now, consider
(2.2)
a„+x = i + xn+l-x0+
E";i
j=0\J
n + r/
¿,+1+ Ldk(mf
I \
k= 0
Identifying coefficients of Xmon both sides of (2.2), we get
(2.3)
<*„(»
+1)- j i= m Í"tl)dm(j)-S0m.
\J
'
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604
ARNE FRANSEN AND STAFFAN WRIGGE
The obvious advantage of knowing Theorem 3 is that in order to calculate E( X")
we only have to know the numbers Xn. If \/g(x) may be expanded as \/g(x) =
Lf=lckxk,then
£
1
A„ = ' =1 kn + k + 1 '
(2-4)
Putting \¡/(x) = (x - l)(x - 2) ■■■(x - n) in Theorem 1, we may state the following theorem for the factorial moments, viz.
Theorem 4. Put xP(x) = U"=l(x - i). Then
rlL"kZlon?Zk(x-i)
Jo
Jo
+ l J
^
g(x)
gyx)
[I
XP(X)
J0 g(x
g(x ++ l)
->o
i)
-dx,
where
p(x)=
t ,(,f,;
„=o"!(* + ")
eP(x) = - + —.-—
+ —-—,-—
+ ■• • .
'
x
x(x + 1)
x(x + l)(x + 2)
(Concerning the function P(x) see, e.g., Lindhagen [11, pp. 21-22].)
We note the special case « = 1, which yields
(2.5)
a, = 1 + A, + e C f^.dx.
Jo g{x + 1)
From Theorem 2 we obtain a1 = 2 + A1-A0. We conclude that
b. The Special Case. In this case we put
1 fx x"
H;/"Jn T(x)
and define y„ as y„ = FXn, i.e
'1
(2.7)
x'
W0r7^
-)
The recurrence relation in Theorem 2 then becomes
(2-8)
an = l+^j^+
L (nk)«k-i,
«o = l-
k= l
When considering the moment generating function E(e~'x) the numbers ßn = an/n\
are more interesting. Thus we get, from a numerical point of view, the more useful
recurrence relation
(2.9)
A_¿(1 + ^)+£.R-i_A_1.
*_,.
To obtain the moments an we first calculate ßn using (2.9) and after that we use the
identity a„ = n\ßn.
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RECIPROCAL GAMMA DISTRIBUTION
605
When calculating the numbers y„ we use the expansion
1
°°
rW=£oW+1,
The coefficients ak+l converge very fast towards zero and are tabulated by Fransen
and Wrigge [4] with addendum by Fransen [3], in both cases to 80D. From (2.4) we
get the formula
00
(2.10)
a
Yb E_^L_,
„ = 0,1,2,....
K=0
The values of an, ßn, and yn are presented in Table I for « = 0,1,2,...,
15 to 30D.
We also deduced a formula for the coefficients an using the Euler-Maclaurin
summation formula and applied it for checking purposes.
Therefore consider the function
/•OO p
<p(t) = FE(e-'x)=(
tx
-^dx.
J0
i (x)
The Euler-Maclaurin summation formula with step-length h = 1 yields (B2k are
the ordinary Bernoulli numbers)
(2.11)*(r)= F L ^-a}= e-'+'-'+ E L<L
j=o
->'
y-o
■'•
£
/c-[(y+i)/2]+i
ff«2,_w.
(Cf. Section 3a.)
A remaining problem is how to calculate the coefficients 8j in the expansion
(2.12)
e"*'-
= Q(t) = t £«,•
7 = 0 ■'*
Differentiating (2.12) with respect to t we get Q(t) = _(1 + e~')Q(t), which after
identification of coefficients yields the relation
(2.13)
s,+1 = - Í«, + (-\y
t(-\)k[j^s\,
S0
e.
From (2.11) we get the relation
(2.14)
Faj = (-ï)Jôj+
B
E
-#"2*-!-,,
; = 0,1,2,....
fc=[(7+ l)/2] + l
Formulae (2.13) and (2.14) were used to check the numerical values of a, calculated
by (2.9) for small values of y (j = 0,1,2,..., 6).
From Hardy's lectures on Ramanujan [8, p. 196] or from his collected works,
Volume IV [9, p. 544],we learn that
\
o T(l + x)
dx = e*-
\
\
J0 <n2+ ln2x
dx,
s>0.
Differentiating (2.15) n + 1 times with respect to s and putting s = 1, we get a
formula for the factorial moments
(2.16) r®$£4*-.
Jo
1 (x)
+i-irr-r5r*J0
7TZ+ hr *
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606
ARNE FRANSEN AND STAFFAN WRIGGE
Putting s = e'1 in (2.15), we get, after differentiation with respect to t, the formula
JfOO e~'x
o
_,
^—sdx = e-'+'- +(
1 (x)
/.°c p~'~xe
e
Jo
77z + hr x
dx.
(See also Section 3.)
To obtain an analytical expression for the moments a„ in this case, we must first
consider a certain generating function, viz.
(2.18)
e»"-
£ck{x)(-l)k£.
A= 0
For the functions Ck(x) we easily obtain the relations
(2.19)
Ck(x) = t^x»,
n —0
(2.20)
Ck(x) = xCk_x(x),
(2.21)
Ck(x) = exPk(x),
k>\,
where Pk(x) is a polynomial of order k with positive integer coefficients,
(2.22)
Pk(x) = Z S(k, j)XJ;
7= 0
S(k, j) denotes Stirling's number of the second kind (Riordan [14, p. 192]),
(2.23)
C„(u + v)= A=0V/C/
Í(nk)ck(u)C„_k(v).
Differentiating (2.17) n times with respect to t and putting t = 0, we obtain
(2.24)
Fa„= i(l)ck(l)+f
^zZU0(nk)Ck(-x)
■n2+ ln2x
dx.
The value of 0.(1) is ebk, where bk is the Bell number of order k (see Riordan [14, p.
193]). We note also that the numbers 8j defined by (2.12) are related to Q(l) in the
following way
(2.25)
Sn = (-!)"
i
(n)ck(l)
= (-l)"ebn+1.
Finally, comparing (2.14) and (2.24), we obtain
ata
(2-26)
y
2,
k = [(n + l)/2] + l
B2kn
-2k<i2k-i-n=J
r^o("k)ck(-x)
,
it1 + ln2x
Gautschi [5] considered polynomials orthogonal with respect to the reciprocal
gamma distribution. Let the polynomials be {irk(x)), normalized so that the
coefficient of xk equals 1. Gautschi tabulated the coefficients á^ and ßk (k = 0(1)39,
18D) of the three-term recursion formula
(2.27)
irk+ l(x)
= (x - âk)irk(x)
- ßkirk_x(x);
ir0(x) = 1,
■n_l(x) = 0.
These polynomials could be used either to calculate or to check the moments of the
reciprocal gamma distribution. No discrepancies were found.
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RECIPROCAL GAMMA DISTRIBUTION
607
c. Some Population Characteristics in the Special Case. For the sake of completeness only we give the numerical values of the population characteristics mentioned
above to 30D. The moments around zero, an, are given in Table I, and from these we
get
mean value = ax =
1.934567042147884721183714704369,
variance = a2 = p2 =
(2
.
1.093936334068611315479743413155,
skewness coefficient = p1 = ju3/a3 =
0.815914578635844856917857543121,
coefficient of kurtosis = v2 = p.4/a4 - 3 =
0.802301598314383564696933612202
and the moment-ratios
(2.29)
g3 = f~i=
3 V'
g4 = v2 + 3 =
0.903279900493664730430818109121,
3.80230 15983 1438356469 69336 12202.
3. Further Analysis of the Function y(t).
a. Some Analytical Expressions. Consider the function <p(t) = FE(e~'x) =
f0x(e~'x/T(x)) dx. Ramanujan, as mentioned by Hardy in [8], proved the formula
W'J-tT(l+x)
fwT^^+r^Wcosi^-^ln,}
Jo
{
m
I
-rfh—e',
7T2+ In2 x
when y > 0 and I > 0, using an ingenious method. Putting | = 1, we get
/OO
yx
-iT(l
+ x)**
/-OO
~¡o
p ~yx
-dx
TT2+ ln2x'
which may be written
-yx
(3.2)
/
L—dx-e'+j
Jo
1 (x )
-
IT2 + In2 x
J0
dx.
We proved (3.2) from scratch showing that the Laplace transform of the L.H.S.
equals the Laplace transform of the R.H.S. Thereby we used probability, as well as
residue, calculus.
Setting y = e~' in (3.2) yields
(3.3)
„(/)-/
-oo
Jo
p~'x
l—dx = e-'^,
1 (x)
_,
/-OO p-t-xe
+f
e
Jo 7T + hr x
dx.
We will return to some different forms of Eq. (3.3) in Section 3c.
Several interesting analytical expressions of <p(i) may be obtained using the
Euler-Maclaurin summation formula. Thus we calculate
(3'4)
*'(,)-j?,rW-<""'"'
and
(3-5)
cc
-/(A + l/2)
-1/2
<p2(0= E 77-—T = V
k=0r(k
+ 2-)
+ (2N(2l/2e~'/2)- I)«""'
H
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.
ARNE FRANSEN AND STAFFAN WRIGGE
608
where N( ■) denotes the standardized normal distribution function, i.e.
JV(/)= -Lf
e-'^dx.
V277 ■'-00
A formal use of the summation formula yields the equations
00
(3.6)
.j
00
<p(0= e-'+e"'+L(-l)y7,
y=0
D
E
jt'u-i-j
J' * = [(y+l)/2] + l
and
i
°°
(-,\J
°°
(3.7) <p(/)= i(m1(o + <p2(0)+Ei-rL
y-o •'■
R
/ ^ \2k
E
If i
*=[(y+i)/2]+i z/c vz;
a»-i-y
(Cf. Section 2b.) The more complicated case with the step-length h = \ is treated in
a short note (Wrigge [16]).
We may generalize these results in
Theorem 5. Let f(x) = \/g(x) be a generalized reciprocal gamma probability
density function such that /(0) = 0, /(0) is finite and nonzero and suppose that
!/#(*) = E*-iC*x*. Define k(í) = E(e',x).
forn = 0,1,2,...,
Then formally we have, iff(n)(oo) = 0
00 t-,\J
°°
«(')-«i(0+IVy= 0
R
E
■/•
«(f)=1(^(0
+«,(/))+7=0
E ^#■'•
z
t-[(y+l)/2]
-#c"-i-y
+l
E
/t=[(y+i)/2]+i
^(èr«a*-i-y.
z" vz,;
vv/iere
K^ = e-J^r
S(l)
and «2(0 = £7TÇ+ ^7(2M21/V'/2)-l)e-'«(i)
D
Sil)
Using Theorem 1, we may establish the following result.
Theorem 6. Let f(x)=
\/g(x)
be a generalized reciprocal gamma probability
density function. Define n(t) = E(e~'x). Then
p.-m;.;.-)
Jn
eiu)
where ^(a;
y; x) denotes the confluent hypergeometric function (see Sneddon [15, p.
35]). An alternative expression is
k(0 = illT^du
Jo g(u)
+ e-'+e" f1 /
'\.du
J0 g(u + l)
+ e-'+e" (l^rG(u)
where
52 <--\\kp-'k
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Jo g(u)
du,
RECIPROCAL GAMMA DISTRIBUTION
609
The function y(t) is related to the beta function in a natural way. Consider the
well-known identity
(3.8)
rwa-o'-1*-^^.
J0
T(x + y)
We divide both sides by T(x)T(y), multiply by e~s(x+y) and integrate with respect
to x and y between 0 and oo. Formally we get
(3.9)/■
»--♦(,)■
•'o '<- ■■<'»*<''(1 - t) '""- "' * - ff^U
Jo Jo T(x + y)
The last part of (3.9) will be proved using Eq. (3.12).
We will prove that
roo roo
(3.10)
<p(0=/
p~t(x+y)
I ^¡——-Trdxdy.
Jo Jo T(x + y + 1)
Therefore consider the auxiliary function
(3.11)
(?,(,)-
//
f0^-dxdy.
x+y<is;x,y>0
^^^
i)
Using the mean value theorem of integral calculus, we get
G,(s + As) - G,(s) = T(s + eAs + l)
//
dxdy;
N[0,1],
This easily yields
d_r(
v
ds^5'
se~"
e~'s
T(s + \)
T(s)-
We finally get
/•OO p~'x
3.12)
G,(oo) = «p /)=/
^f^dx=
^o i(x)
/-OO /-OO
/
o_'(-*+.)')
_—-_—<&-#.
^o •'o L(x + j + 1)
The last part of (3.9) may now be proved using similar methods or simply by
differentiating (3.10) with respect to t.
b. Some Interesting Inequalities. We start with Weierstrass' formula for 1/T(x),
i.e.
(3.13)
_2_,^n(1+i)e-/..
However, since (1 + x/n)e'x/"
(3-14)
^ 1, we get
1
r < e XVkY, Cj kxj+1,
T(x))
7=0 '
where vk = E n 1 ~ Y>
«=1
and the numbers eyt are related to the Stirling numbers of the first kind, viz.,
cjk = (s(k + 1, ; +\)/k\)(-iy+k.
(See e.g. Riordan [14, p. 90].) This yields the
estimate
(3-15)
<P(0< E" cM/U
+ 1}l
\J+2
+ vky
j-=0~J'K it(t _L
In Table II we give the values of the upper limit for k = 24 and t = 0.0(0.1)5.0.
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610
ARNE FRANSEN AND STAFFAN WRIGGE
Another inequality for <p(t) may be obtained using the multiplication theorem for
the gamma function. We see that
,00
(3.16)
<p(/) = n/
p-n'x
——-dx,
« = 1,2,3.
•>o i(nx)
But it is known that T(nx) - (2tT)(X-")/2n''x-l/2T(x)Y\nkz\ T(x + k/n). This yields
.00
-x(n»
<P(0 = «3/2i
+ nlogn)
-n—T7^-dx.
Jo (2'nf")/1T(x)nrJlT(x
+ k/n)
Putting T(x0) = minj:>0 T(x) and c = T(x0)/ {2ti , we may write
(3.17)
<p(«/ +«log«)
>—r—(p(/),
« = 1,2,3,....
n '
Similar inequalities may be deduced for the more general transformation
/
^
r^e-axf(tx).
Before continuing the analysis of <p(t) we note the obvious inequality (from (3.3)),
(3.18)
<p(0 > e~t+e".
We will now deduce a different lower limit for <p(i) using a variant of Jensen's
inequality, viz.,exp(/„"+1 ln(/(x)) dx) < /„"+1 f(x) dx, where f(x) is supposed to be
continuous and positive.
We put
{t<u)=Clnfu¡jdx
r(x)
and get
(3.19)
G(t, u) = -t(u
+ \) -ulnu
+ u-
lnv^r".
But
•w-<rfe*f(>
ru + l e
Summing over u = 0,1,2,...,
we obtain
,t/2
(3-20)
«
v(/)>f3_|i+
£
27T \
t=i
eA(l-0
A;
The simpler inequality (3.18) is slightly better in the interval (0,0.6), see Table III.
We may easily generalize the result in (3.20), viz.,
Theorem 7. Let f(x) = \/g(x)
be a generalized reciprocal gamma probability
density function. Define k(í) = E(e~'x). Then
I
K(t)>Ke-'2
\
where K=
e-)°Xn(gM)dx.
oo
<i(l-r)
1+ E^TTk= l
K
D
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RECIPROCAL GAMMA DISTRIBUTION
611
An interesting example, which includes (3.20) as a special case, is given by the
density function
Jln(X)
C2n
sm2"(iTx)
T{x)
(c2„ is a normalization constant and c0 = 1/F),
E(e~'x) in this case. Then
p-'/T-
(3-21)
I
n = 0,1,2.
Define K2n(t) =
°° pk(l-t)
K2n(t)^c2„——
1+ E
22nv/2^r\
¿fi
kk
Proof. Simply note that f¿ ln(sin(7rx)) dx = -In2.
D
c. Numerical Calculation of(p(t). We set ourselves the task to compute <p(t) to 6D
for t = 0.0(0.1)5.0. In all we considered more than 10 different methods to compute
tp(t) to the required accuracy. The methods used could roughly be divided into
Gaussian quadrature rules and series expansions.
For small values of t there are several useful expansions. We may, e.g., consider
<P(t)=
ri
e
,x
r00
e
,x
rFT^dx + e~' \
dx,
Jo í(x)
J0 xT(x)
from which we get the expansion (cf. Section 2b)
oo
/
-i\",n
oo
(_-,\"tn
(3.22)„(0 = E H^"T" + e~'<F- *>+ Fe" E Mr"4"1'
n=0
'
n= l
More rewarding from a computational point of view was to use the Euler-Maclaurin
expansions with step-lengths h = 1 resp. h = \ (formulae (3.6) and (3.7)). These
expansions may in this case be regarded as the Taylor series expansions in disguise.
Thus, (3.6) gave <p(t) correct to 6D in the interval [0,2.0]. Note that the somewhat
complicated coefficients which occur in (3.6) may be calculated from or checked
against (2.14). Equation (3.7) gave <¡d(í)correct to 6D in the entire interval [0,5.0].
The coefficients in (3.7) were truncated with an upper limit equal to 30 and
calculated using high-precision techniques. (Note that we only know the numerical
values of ax, a2,..., a6l; see [3] and [4].) The achieved numerical results of <p(r)
using the Euler-Maclaurin methods are presented in Table IV.
For larger values of t it is useful to use Watson's Lemma. Applied to q>(t),the
lemma yields
(3.23)
00 a
<p(0 - E
(n -
"-,„
n=2
1)'
h , where -^
'
X
1
== £ an^x-\
V\X)
n=2
Values of y(t) correct to 6D were obtained from t = 3.8.
Finally, we turn our attention to the Gaussian quadrature rules. To be able to use
a Gauss-Laguerre quadrature rule we rewrite (3.3) in the following two ways, viz.
(3.24)
<p(0= e-'+*-' + H
/""
Jo 7T2+ (r + ln(w))2
du
and
(3.25) <p(t)-e-+'-+f
/•OO
p
'
' «
-f---2dz+f
J0> IT2 + (t-
z)
,00
J0
p
-z.Zz'/a
c
-L-Î--dz.
IT2 + (t + z)
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612
ARNE FRANSEN AND STAFFAN WRIGGE
We evaluated numerically the integrals occurring in (3.24) and (3.25) using GaussLaguerre quadrature rules. However, none of the formulae were sufficient to yield
6D even when using a 15-point formula. The best result was achieved using (3.25)
and a 15-point formula. The maximum absolute error was then about 4 x 10"4 in
the required interval.
We considered the Gauss-Christoffel quadrature rule with respect to the weight
distribution w(x) = 1/T(x) on [ 0, oo), which is also mentioned by Gautschi [5]. In
Table V we give weights p¡(n) and abscissae x¡(n) together with the remainder term
coefficients c„ for n = 14, 15. Thus we get
(3.26) fPr\dx
= E P,(n)f(x,(n))+ cJ™{S),
0 < | < *o.
When calculating the weights and abscissae we used the well-known methods
presented by Golub and Welsch [6] and Gustafson [7].
Using (3.26) with f(x) = e~'x, n = 14 and 15, and comparing the results with
each other (and with other results), we could determine q>(t) to 6D in the required
interval. The numerical result for n = 15 appears in Table VI.
Note that the data in Table VI are subject to a progressively increasing error
(beginning with <p(2.8)) so that the final entry is correct only to 6D. On the other
hand, comparing Table IV with Table I in [16], we see that Table IV is correct to
10D.
The discrepancies between Table IV and Table VI may be interpreted in the
following way. The function <p(?) does not behave like an exponential polynomial
for large values of the argument t. When using (3.26) with f(x) = e~'x, we in fact
put
(3.27)
<p(0= ipMe-'^,
i=i
and it is obvious from a comparison of Table IV and Table VI that this approximation is good only for "small" values of t. Therefore we think it is valuable to give
Table VI with 10 decimals even if only 6 are correct throughout.
4. Tables. In this section we present the tables previously mentioned, i.e.,
I. The values of
1r
T(x) -dx,
' ß„
n = %,
n! ' yn-Cff-,dx,
'- J0 r(jc)
for« = 0,1,2,..., 15 to30D.
II. Tabulation of the upper limit
ULt(,)-£c,
' j'k t. <¿±^
i
W+2
o
(t + vky
of <p(t)for k = 24 and t = 0.0(0.1)5.0((3.15)).
III. Tabulation of the lower limits
LL1(0 = e-'+'
1/2 /
and LL2(/) = =-=
oo ek(i-
1+ E
v2tt \
of <p(/)for t = 0.0(0.1)5.0((3.18)and (3.20)).
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t=1
RECIPROCAL GAMMA DISTRIBUTION
Table I
Values ofan,ß„
o)
0)
0)
andyn to 30D for nupto\5
1.00000 00000 00000 00000 00000 00000
1.00000 00000 00000 00000 00000 00000
0.54123 573^3 28670 53011 95373 28880
1)
1)
1.93156 70121 47884 72118 37147 04369
1.93456 70421 47884 72118 37147 04369
0.35751 50221 23591 90711 31309 21362
2)
2)
2)
1.83618 59716 33126 89173 63606 92321
2.11821 29873 16713 11736 81803 16161
0.26581 66316 55185 73050 08966 15819
3)
3)
3)
11.52263 23631 36761 25202 21329 96310
2.12013 87271 89160 70867 03551 99390
0.21116 90393 30388 01296 03091 51101
4)
4)
4)
50.31552 58868 21618 33058 26933 58920
2.09773 02152 81360 31710 76122 23288
O.17I95 I7281 85158 64133 35562 67979
5)
5)
5)
196.5296O 95489 16802 68500 46071 32870
1.63771 67162 11223 35570 83717 26107
0.11924 76440 33000 57064 19815 32525
6) =
6) =
6) =
849.04403 89161 01902 55653 61461 70944
1.17922 78318 27919 30910 63002 03015
0.13007 56150 34949 37665 14166 11576
7) =
7) =
7) =
4008.05067 83135 15751 45650 77329 55599
0.79524 81504 59030 90306 67674 07332
0.11523 86555 19321 63140 50448 25187
8) r
8) =
8) =
20473.02093 44326 72186 66432 03391 11034
0.50776 31160 32556 35383 59206 43331
0.10342 17446 83890 73215 80751 09877
9) =
9) =
9) =
1 12278.68745 79147 99966 36906 13390 83314
0.30940 99632 32789 90290 55584 80322
0.09379 13992 27884 14684 90322 19711
[10) =
:10) =
MO) r
6 56942.77694 75960 16930 75176 45786 40082
O.18IO3 58181 62366 62724 06172 19075
O.O8579 41856 48139 30346 42449 43023
:11) =
;11) =
;i1) =
40 79302.42315 67110 12970 46671 96004 30938
0.10219 51264 41916 96260 64831 52948
0.07904 84426 07206 82303 34826 57878
:12) =
;12) =
:i2) =
267 63361.94933 30895 88698 48695 73219 27461
0.05587 32203 59458 27652 49604 32193
O.07328 25617 57236 85865 02003 52079
;13) =
:13) =
;13) =
1848 15036.57385 64721 49672 31971 02966 00716
0.02967 95277 40433 51070 06021 75556
0.06829 80580 98503 72710 45091 76512
;i4) ;14) =
;14) =
13388 81303.23715 11770 64912 67976 41416 82370
0.01535 79668 14990 18848 70268 33118
O.O6394 65552 27707 72516 40691 75135
;15) =
:i5) =
:15) =
1 01461 22183.57558 18409 17108 15898 20647 91856
0.00775 89057 58085 16714 83684 75292
0.06011 49366 30457 01942 63331 98573
1)
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613
614
ARNE FRANSEN AND STAFFAN WRIGGE
Table II
Upper limit of<p(t) to 6D
0.0
0.1
0.2
0
1
1.2
1.3
1.4
1.5
1.6
UL
2D
t
UL24
3.104091
2.545533
2.111334
1.769449
I.497054
1.7
1.8
O.307953
0.281396
0.257935
O.237128
0.218605
1.277650
1.099143
0.952553
831128
729740
644448
572196
510592
0.457746
0.412154
0.372611
0.338142
1.9
2.0
2.1
2.2
UL24
3.4
3.5
3.6
3.7
3.8
3.9
0.202057
0.187224
2.3
4.0
0.173885
0.161855
0.150973
2.4
2.5
2.6
2.7
2.8
2.9
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
0.141103
0.132128
0.123946
0.116470
0.109623
0.103338
0.097559
3.0
3.1
3.2
3.3
O.O92232
O.O87315
0.082767
0.078553
0.074642
0.071006
0.067621
0.064465
O.O61519
0.058764
0.056184
0.053766
0.051497
0.049364
0.047358
0.045468
0.043687
Table III
Lower limits of<p(t) to 6D
LL
ll2
o.o 2.718282
0.1
0.2
2.236333
1.856570
1.553978
1.310384
1.112412
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0.950100
0.815940
0.704215
0.610528
0.531464
0.464344
0.407055
0.357912
0.315561
0.278909
0.247065
LL
LL2
LL1
1.7 O.21930O 0.266516
1.8 0.195011 O.243837
2.625200
2.163212
1.800680
1.513054
1..282521
1. 095987
0. 943715
0. 818377
0.,714406
0.627526
0.554424
O.492513
2.3
2.4
2.5
2.6
2.7
2.8
0.439753
O.394525
O.35553I
0.321739
2.9
3.0
3.1
3.2
O.292296
3.3
1.9
2.0
2.1
2.2
0.223793
3.4
3.5
0.173699
0.154948
0.138409
O.123787
0.110832
0.099333
O.O89107
O.O80OOO
O.I63I89
0.151685
0.141273
O.131817
4.3
0.071877
O.123202
4.4
0.064623
0.058136
0.052329
0.047125
0.042458
0.038269
O.II5328
0.108112
0.101479
O.O95367
0.089721
0.084494
4.5
4.6
4.7
1.8
4.9
0.206001
0.190140
0.175944
3.6
3-7
3.8
3.9
4.0
4.1
4.2
5.0
034506
031123
028081
025342
022877
.020656
,018654
.016850
0.079644
O.O75135
0.070935
0.067016
0.063353
0.059926
0.056713
0.053698
,015222 0.050865
,013754 0.048200
,012429
0.045692
,011233 0.043328
,010153 0.011098
OO9178 0.038993
,008298 0.037005
007502 O.O35127
006784 0.033350
Table IV
Values of(p(t)to
<P(0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.1
1.5
1.6
2.8077702120
2.3262370471
1.9167718218
1.6113584989
1.1008236962
1.2027934333
1.0103059617
0.9058566158
O.7937313323
0.6995357300
O.619858I1II
O.5520275I72
O.I9393298II
0.II38950131
0.1005665617
0.3628597079
O.3298899227
10D by the Euler-Maclaurin formula
«P(t)
1.7
1.8
1.9
2.0
2.3
2.1
2.5
2.6
.3009335120
.2753918016
.2527806955
.2326807977
.2117517803
.1987050126
.18I2967120
.1713200562
.1595988329
.1189822986
.1393109958
.1305633342
.1225527820
.1152255491
.1085086671
.1023383935
.0966588753
<P(t)
3.1
3.5
3.6
3.7
3.8
3.9
1.0
1.1
1.2
1.3
1.1
1.5
1.6
4.7
4.8
1.9
5.0
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LL2
,0911210330
,0865816181
,0821021199
,0779195893
.0710930688
,0705061033
,0671618226
0610478828
0611361584
0584121762
0558613851
0534689567
0512226118
0491109691
OI71237134
0152514796
0134857514
RECIPROCAL GAMMA DISTRIBUTION
615
Table V
Abscissae x¡(n) and weights p,(n)
to 25S for n = \Aand\5
14
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1.622858744052096651180154E-01
5.229208541663484816942969E-01
1.051449279617592254069644E+00
1.726854794707082013337275E+00
2.537139970455747796982883E+00
3.477139770547848558819249E+00
4.547183105381307655844180E+00
5.752809218181122223580066E+00
7.105517354977O53201482566E+00
8.62I916OI1709722666870543E+00
1.031350430938327785568881E+01
1.231779194476O2826OI12930E+01
1.465920463327447787732363E+01
1.765918673901601163881919E+01
4.666936308723003950950625E-02
2.646109639584808299341717E-01
6.215442051821438630110274E-01
8.142350693171694265817131E-01
6.414191022297807547724504E-01
3.099598643455755089011735E-01
9.148075324998564823913511E-02
1.612928241047227855479475E-O2
1.630797462009361385772756E-03
8.849849557915662269500680E-05
2.317750836324959203635878E-06
2.446556157452933003045058E-08
7.366805939154725353566559E-11
2.653882898025302661903184E-14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1.490070588640254308122436E-01
4.819351400658091188356546E-01
9.723272335870747628473832E-01
1.601001667819464486376536E+00
2.356286743334194675877424E+00
3.2323507821652031 HI36O36E+OO
4.227983287203939526165386E+00
5.346137300427159830530296E+00
6.591175177757317293887338E+00
7.981927181287131330907394E+00
9.539032836171919476943656E+00
1.128985951391608900789341E+01
1.329475808401580688647499E+01
1.566620236805261061598493E+01
1.869771150641537821275855E+01
3.924675165725691451042447E-02
2.260891662923499176678823E-01
5.527923916882637016023382E-01
7.759201771782941429763990E-01
6.757927480583987655263235E-01
3.73674657IOOO929295131507E-01
1.312282470710664383753163E-01
2.885258016254480346810935E-02
3.860321080491039705432555E-03
3.002822090665240753851297E-04
1.265843407897613382797982E-05
2.589555120399712100012903E-07
2.136113605313739301854631E-09
4.988209679385246841957279E-12
1.356702577187496841722868E-15
•14
•15
3.8031842E-16
1.3723708E-17
Table VI
Values of<p(t) to 10D by Gaussian quadrature rule
9(0
0.0
0.1
0.2
2.8077702420
2.3262370474
1.9467718218
1.7
1.8
0.3
1.6443584989
1.4008236962
1.2027934333
1.0403059617
0.9058566158
O.7937313323
0.6995357300
0.6198584141
2.0
2.1
2.2
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.5520275472
0.4939329844
O.443895OI31
0.4005665647
0.3628597079
O.3298899227
«PÍO
9(t)
1.9
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
0.3009335120
0.2753948016
0.2527806955
O.23268O7977
O.2147517803
0.1987050126
0.1842967120
O.17132OO562
0.1595988329
0.1489822986
0.1393409958
0.1305633341
0.1225527819
0.1152255490
0.1085086671
O.IO23383931
0.0966588746
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
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0.0914210319
0.0865816168
0.0821024175
O.O779495857
0.0740930635
0.0705060958
0.0671648119
0.0640478678
0.0611361378
0.0584124481
O.O558613473
0.0534689064
0.0512225456
O.04911O883O
0.0471236023
0.0452513376
0.0434855716
616
ARNE FRANSEN AND STAFFAN WRIGGE
IV. <p(0 = ff(e-'x/T(x))dx
tabulated to lOD for t = 0.0(0.1)5.0 by means of
the Euler-Maclaurin expansion in (3.7).
V. The values of the numbers p,(n) and x¡(n) corresponding to the weight
function l/T(x) (cf. (3.26)), n = 14 and 15.
VI. q>(t) tabulated to 10D for / = 0,0(0.1)5.0 by means of Gaussian quadrature
(cf. (3.26)).
Comparing the values of Tables IV and VI,we thus may rely on all of them up to
6D.
Acknowledgment. We want to thank Dr. Jacques Dutka, who gave us several very
valuable references and comments.
National Defence Research Institute
Division I, Section 123
S-102 54 Stockholm,Sweden
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Clarendon Press, 1969.
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H.A.
LlNDHAGEN,Studier bfver Gamma-Funktionen och Nàgra Beslàgtade Transcendenter (Studies of
the gamma function and of some related transcendentals), Doctoral Thesis, B. Almqvist & J. Wiksell's
boktryckeri, Upsala, 1887.
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New York, 1934.
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15. I. N. Sneddon,
Special Functions of Mathematical Physics and Chemistry, Oliver and Boyd,
Edinburgh and London, 1961.
16. S. Wrigge, "A note on the moment generating function for the reciprocal gamma distribution,"
Math. Comp.,v. 42, 1984,pp. 617-621.
17. M. Wyman & R. Wong, "The asymptoticbehaviour of ¡i(z, ß, o)," Cañad. J. Math., v. 21, 1969,
pp. 1013-1023.
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