function, the distribution of the sample mean of a simple random sample from a statistical population designated by this general probability function, distributions ...
On a Generalized Hypergeometric Distribution By A. M. MATHAI and R. K. SAXENA, Jodhpur 1)
1. Introduction and Summary In this article we introduce a general family of statistical probability distributions from which almost all the classical probability distributions are obtained as special cases. The distribution function, the characteristic function, the distribution of the sample mean of a simple random sample from a statistical population designated by this general probability function, distributions of some order statistics and the distribution of the ratio of two independent stochastic variables having the probability functions in this family of probability distributions, are investigated. Some special and interesting properties enjoyed by this general distribution are also pointed out. A number of statistical distributions is studied by many authors from time to time because of the practical applications of these special distributions. Some general classes of statistical distributions, such as the general exponential family, the general exponential type family, the series distribution type, the general gamma distribution, are studied by KHATRI (1959), PATIL (1961), STACY (1962), MATHAI (1966) and others. Consider the probability distribution with the probability density function, da°t ~ I'(~) F(fl) I'(r - - c/d)
p(x) = r(cld) I'(r)
c/d)
x c - 1 , F , (~, r ; r; - - a x ~)
--c/d)
for x > 0, c > 0, ,¢ - - c/d > O, fl - - c/d > 0 and p(x) = 0 elsewhere,
(1)
where ~FI(g, r ; r; - - a x a) is the hypergeometfic function (ERD~I.VI, 1953, p. 56). The parameters g, ~, y are restricted to take those values for which p(x) is positive, p(x) also has the very interesting property that, the non-constant part, x) Prof. Dr. R. K. Saxena, Dept. of Mathematics, University of Jodhpur, Jodhpur/India
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A.M. MATHAI and R. K. SAXENA
x * - I ,FI(~, 8; r; - - a x a) ---- x~- 1(1 + a x a). . . . ----= x c - - l ( 1
a 2Fl(r __ o:, r - - 8 ; r; - - a x a) =
+ a xd)-~zFl(o:, fl - - r ; r; a xa/(1 + a xa))
(ERI)~LYI, 1953, p. 105). So far it is n o t k n o w n t h a t this p r o p e r t y is enj o y e d b y a n y o t h e r function. T h u s this p r o p e r t y m a k e s p(x) v e r y i m p o r t a n t .
2. Special Cases T h e following i n t e r e s t i n g special cases are o b t a i n e d . Also n o t i c e t h a t a l m o s t all classical statistical d i s t r i b u t i o n s are t h e r e in these special cases. (a) The General Hypergeometric Distribution: I n (1) replace a b y a/~ a n d t a k e t h e limit w h e n ~ t e n d s to i n f i n i t y a n d using t h e formula, lim ,Fl(o:, 8; r; - - x/o:) = xFl(fl; r; - - x),
(2)
CX"--¢- ~ 0
we get,
da old 1"(8) P(r --c/d) xc- 1 ,FI(/~; r; - - a x ~) /(x) = r(cld) F(r) r ( ~ - - c / d ) for x > 0 a n d [(x) = 0 elsewhere. (b) The Generalized G a m m a Distribution: ]l(x) = dac/dx~-le-a'a/N(c/d) (c) The (d) T h e the normal (e) The
(3)
In (3) p u t fl = r, we get, for
x > 0.
G a m m a Distribution: I n (3) p u t fl ~ r a n d d = 1. N o r m a l Distribution: I n (3) p u t fl ~ r, c = 1, d ---- 2, we get d i s t r i b u t i o n w r i t t e n in a m o d i f i e d form. Generalized F-distribution: I n (1) p u t fl = r, we get, da cla F(o~) x ~- 1 fg.(x) - - F(c/d) F ( a - - c/d) (1 + a x a)- ~ for
x > 0.
(f) The F-distribution: I n (3) p u t fl ---- r, d = 1. (g) The Student-t distribution: I n (3) p u t fl = r, c---- 1, d = 2 we get t h e S t u d e n t - t d i s t r i b u t i o n w r i t t e n in a m o d i f i e d form. (h) The Beta Distribution: In (3) p u t ~ - r, a----d----- 1, we get the B e t a d i s t r i b u t i o n w r i t t e n in a m o d i f i e d form. (i) The E x p o n e n t i a l Distribution: I n (3) p u t fl = r, c----d = 1. (j) The Cauchy Distribution: I n (3) p u t fl = r, d = 2, c = 1 ~- a = we get t h e C a u c h y d i s t r i b u t i o n w r i t t e n in a m o d i f i e d form. (k) T h e Weibull Distribution: T h i s is w i d e l y used in life testing problems a n d we o b t a i n this b y p u t t i n g fl ---- r, c = ~, a ---- c, d = ~ in (3). (1) The Raleigh Distribution: T h i s is w i d e l y u s e d in t h e o r e t i c a l physics a n d we o b t a i n this b y p u t t i n g fl = r, c = 1/2, a = 1/2a*, d = 1 in (3). (m) The W a i t i n g T i m e Distribution w h i c h is u s e d in q u e u e i n g t h e o r y a n d in o t h e r s t o c h a s t i c models, is o b t a i n e d b y p u t t i n g d = 1, fl = r, c = 1 in (3).
On
a Generalized
Hypergeometric
Distribution
129
(n) The Logistic Distribution which is extensively used in life testing and ballistic problems, is obtained by putting d -----2, and x -----ey in (3).
3. The Characteristic Function The characteristic function ¢(t) of p(x) is given by, ao t!
¢(t) = E(e-X) = /e"" p(x) dx,
(4)
o
where E denotes 'mathematical expectation' and i ---- (-- If/z. By using the result, (GuPTA, 1965),
i
e
pt Xe - 1
2 E l ( C , d , v , - - z x a) d x =
o
P - " I~(v) . , , s [ . ~ . - a = F(c)/'(d) ,~s,2~- ~
(1 - - a , d ) , (1 - - c , 1), (1 - - d , 1}) (0, 1), (1 - - v , 1) _
where d > O , R(p) > 0 , R(c) > 0 , v ¢ O, - - 1 , - - 2 We get the characteristic function as,
(5)
.....
da */a F(r - - c/d) ( - - i t)-"
,~(t) = z'(cld) r(~-- c/d) 1~(~- c/d)"
(
• Hz:2 1 3 a ( - - ~ "t ) -
d
(1 c ; d ) , ( 1 - - ~ ; 1 ) , ( 1 (0; 1), (1 - - r ; 1)
)
fl,1)
(0)
.
For a definition of the H-function see at the end of this article. Since the derivatives of the H-function exist, we can easily evaluate the moments and related measures for the general class of distributions defined in (1).
4. The Distribution of the Sample Mean Since a set of independently and identically distributed stochastic variables is defined as a simple random sample, we can obtain the distribution of the sample mean by taking the Fourier transform of (¢(t/n)) ~ where n is the sample size. An interesting special case can be obtained here when fl = r and ,t tends to infinity then ¢(t) becomes, ~l(t) = (1 - - i t/a)-cla
and
($l(t/n))" = (1 - - i tin a)-"~la.
(7)
In this case the distribution of the sample mean has the property that the distribution of the sample mean belongs to the same family of distribution as the parent distribution with the parameters a and c scaled by the sample size n. 9
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A.M. MATHAI and R. K. SAXENA
5. T h e Distribution F u n c t i o n The distribution function or the cumulative probability function for the probability density function p(x) can be obtained by using the result,
x~-lzFl(a, fl;r; xz) d x = F(a) aFz(=,fl, a ; a + 1, r;z)lI'(a+ 1)
(8)
(see also ERD]~LYI, 1954, p. 399). In (8) put x----y/p we get, P
(l/p") I Y"-I~FI(°~' 8; r; y z/p) dy -~ F(a) aF~(o~,8, a; a + 1, r; z)lr(~ + 1). o
(9) P u t y = P and in the resulting relation replace p~/a by p, we get, P
.r(~/a) (a/~,/d po) I~,-1 ~r~(~, 8; r; e'z) -- l'(,r/d + 1) 8F,(o~, 8, ,r/d; cr/d + 1, r; z p ~) o
00) and therefore, X
(11)
F(x) = I p(t)at = o
r(a) p ( , - c/a) r(r) I'(o~ -- c/d) F([t - - c/d) I'(c/d + 1)
x ° ~F,(e, 8, c/d; c/d + 1, r; a x~).
6. T h e Distribution of t h e L a r g e s t Order Statistic In the general case it is difficult to obtain the distribution of the order statistics, in a simplified compact form. However, in a special case for a sample of size 2 we can obtain the distribution of the largest order statistic in a simplified form. Consider the special case, d = 1, fl = c + 1, r = ~ + c + 1/2, then the probability function p(x) of (1) reduces to, a c/'(~) .r'(c + 1) F(0: + 112) x'-X q(x) = _v(c) I'(~ + c + 1/2) r ( ~ - - c) sFa(~, c + 1, ~ + c + 1/2, - - a x) (1~) X
I q(t) d t = A(a x)c~Fl(c, ~; ~ + c + 1/2; - - a x) o
(13)
On a Generalized Hypergeometric Distribution
131
where P(2~) 21 - 2~1/2 A = F(~) F(~ + 1/2) / r ( ~ + ~ + 1/2) r(~ - - c) = ; ( ~ _ ~) F(~ + ~ + 1/2) The density function for the largest order statistic for a sample of size n from a population /(x) is given by,
g(x) = n
t) d
(14)
t(x).
W h e n [(x) is p1(x) and when n = 2, we get the density function as, x
0
= 2 A 2 a c ( a x)2~-laF~(2c + 1, 2e, e + c; 2 ~ + 2 c , ~
+ c-t- 1 / 2 ; - - a x )
b y virtue of (ERDI~LYI 1953, p. 186).
7. The Distribution of the Ratio We will consider the distribution of the ratio of two independent stochastic variables having the probability density functions belonging to the same family as p(x). This will enable us to deduce the distributions of the ratio of two stochastic variables having one of the classical distributions m e n t i o n e d in section 2 with different parameters, from this general ratio distribution. L e t X 1 a n d X~ be two independent stochastic variables with p r o b a b i l i t y functions,
d i a;# aj F(aj) F(flj) F(r i - - q/di)
pdx) = I'(cj/4) F(rj) r ( ~ j - c~/4) r ( ~ j - cd4)
x~j-12Fl(o~i, flj; rj; - - ai x ai)
for x > 0, ci > 0, :9 --ci/di > O, fij --ci/d s > 0, r i ~ 0, - - 1, - - 2 . . . . and ~"= 1, 2. Let W = X a / X 2 t h e n V = log W -----log X 1 - - log Xa. function for V is,
The characteristic
oO a0
6v(t) ---~E ei'g --~ f t #'(logxl -logxJ # l ( x l ) #z(x2) d x l dx2 = oo ~.
x t 1 Xl d x I " xe o o
2 x~
x 2 --~
=a(al, G ~ l , ~ l , c + iO "a(a~,G~,~..c, - i t )
G(a~,dj,~,N, cj) 9*
132
A . M . MATHAI and R. K. SAXENA
where
C(aj, d~, ~j, ~j, c~) = r(cddj) r ( ~ - - eddy) r(~j - - c#dj) a~/'J F(rj -- cjldj) The Fourier transform of ¢~(g) gives the density function of V and by making the transformation, W = ev the density function of W is obtained. The density function is given by, X--Ct -- 1
i,~(x)
=
a7
~
(°' + ~ ' ) t ~ '
x
I-I a(aj, dj, ~j, ~, ci) d~ d~
j=l
t
1
-;zd'?',;zJ '}
cl+c~ 1 /
t"
and p~(x)
=
~,+c,. 1\ l
z, ,z,l /
0 elsewhere, where the H-function is defined by, m , ~
H P,q
t (al; At), (a2; A~),..., (ap;Ap)1
x (b~; B~), (b2; Bz) . . . . , (bq; Bq) ] -~-
i =~i
1
f l F(bk + B~s) f i F ( a k - - A k s ) x -s k=l ds q P L Y I F ( b k - - B k s ) I-I F ( a k + A ~ s ) k= 1
h=m+l
k=n+l
where an e m p t y product is interpreted as unity; 0 =< m =< q, 0 ~ n ~ p; Ak's and Bk's are positive; all the poles of the integrand are simple; the contour L extends from ~ - i oo to ~ + i oo such that all the poles of F(bk - - Bk" s), k = 1, 2 , . . . , m lie on the right of the contour and those of F(ak + Ak" s), k --- 1 . . . . , n lie on the left of it. References
[1] ERD]~LYI, et al. : Higher Transcendental Functions. McGraw-Hill, N . Y . (1953). [2] ERD~LYI, et al. : Tables of Integral Transforms. McGraw-Hill, N . Y . (1954). [3] GUPTA, K. C. : On H-Functions. Annales de la Soci6t6 Scientifique de Bruxelles, T. 79, I I (1965), pp. 97--106. [4] MATHAI, A. M. : Some Characterization of one p a r a m e t e r family of distributions. Canadian Math. Bull. (to appear). [5] PATIL, G. P. : A characterization of the exponential t y p e distribution. Biometrika 50 (1961), p. 205--207. [6] STACY, E. W . : A generalization of the g a m m a distribution. Ann. Math. Statist. 33 (1962), p. 1187--1192.
