A Two-Variate Gamma Type Distribution

A Two-Variate Gamma Type Distribution Author(s): W. F. Kibble Source: Sankhyā: The Indian Journal of Statistics (1933-1960), Vol. 5, No. 2, Proceedings of the Indian Statistical Conference 1940 (1941), pp. 137-150 Published by: Indian Statistical Institute Stable URL: http://www.jstor.org/stable/25047664 Accessed: 04-08-2017 15:31 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms

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THEORETICAL STATISTICS Chairman: Professor HAROLD HOTELLING

A TWO-VARIATE GAMMA TYPE DISTRIBUTION By Dr. W. F. KIBBLE Madras Christian College

? 1. Introduction That the two-variate distribution function for correlated vari?tes which have each

a Gaussian distribution can be expressed as a series bilinear in Hermite polynomials was shown by Mehler1, who showed that

^r/(l_p2)exp\-i(x2-2pxy + y2)/(\-p')} = -?- exp |-i(x2 + r)} [1 + pH.W H1(y) + -|^rH2(x)H2(y) + ...] More recently analagous series bilinear in the appropriate orthogonal polynomials have been found for certain discontinuous distributions.10* n

In this paper a distribution function bilinear in Laguerre polynomials is found for two vari?tes each of which has a 'Type IIP, or Gamma Type, distribution In the special case in which the two vari?tes have similar distributions (i. e., have the same coefficient of variation), the distribution function can also be expressed in terms of a modified Bessel function.

The approach to the problem is as follows : the variance, in a sample of given size from a normally distributed population, has a distribution of type III : we inves tigate the distribution of the variances of two vari?tes in samples from a normal two variate distribution. I have to thank Dr. A. C. Aitken, F.R.S., for suggesting this problem, and methods of approaching it*. A subsequent paper will give an extension of the main result of this paper to any number of vari?tes ; and also of Mehler's theorem, and those for the binomial, hypergeometric and Poisson distribution. In this paper an extension is made to a two-variate distribution of Normal type in one variate and Gamma type in the other. * The bulk of the paper was included in my thesis for the degree of Ph.D. accepted by the University of Edinburgh in 1938.

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V?. 5] SANKHYA : THE INDIAN JOURNAL OF STATISTICS [Part 2 ?2. Distribution of the Variances in a Normal Correlated Two-Var?ate Distribution It is well known that if a var?ate X has a normal distribution, so that, with a suitable choice of the unit, its frequency is given by

then the distribution of square of X can be given by the equation,

d& = T7JT x~~* e"xdx and that of the mean variance in a sample of n is

1 hi - 1 - x j

dp = -x ? e dx9

r(i)

with suitable choice of the unit.

For, putting x = JX2, dX= ?prX 5 dx and equation (1) gives

dp= 2 / x~^ e~x ?x but two values of X, one positive and one negative, correspond to one value of x, so that the distribution for x is

(2) #--4"v77 *-* e~x dx= T-7TT *~? e~* dx r(i) and the moment generating function is

1 [xi ? J ?* -x,#,,=. ax , ?-* r(i)-m\o Jo (i-?)i (3) x > e e dx

If *1} x2, x3, .. xn each have a distribution of th

function of x1 + x2 + x3 + ... +x?, is (I?a) 2 and the resultant distribution function is

(4) dp = ?- a***-1 e~x dx n

r(-r)

Except for the fact that the index of x is half an integer, this is a form into which the most general Type III equation

/c\ i. km+1 kc , xWt ?kx ,

W) dp l(m+l) = ?-?- e (x-c) e dx

(where m is any real number > - 1), can be put, by a suitable choice of origin and


TWO VARIATE GAMMA TYPE DISTRIBUTION ?3. We therefore expect that a similar discussion of the distribution of the variances in a two-variate normal correlated distribution will lead to a two-variate di? tribution which is the natural extension of the Type III distribution. Consider, then, two vari?tes X and Y, with frequency function

(6) dp = 2v/{i_pt) exp { - 2(1!_p2) (X?-2pX Y + Y'j} dXdY and put x = |X2, y = ?Y2, so that (6) becomes

(7) dp = ? ?^ exp {--____.(%_2p^(x3,) + 3,) j dxdy but there are four pairs of values of X, Y corresponding to one pair of values of x, y; two with positive and two with negative XY, so that the true expression for dp in terms of x and y is not (7) but

(8) dp = 1>(x, y)dx jy = - {**y) 'exp \-(x-29*J(xy) + y)\ dx dy 2ir y ( 1 - p )

+ 2ljX(\- 2)eXP J-^ + W^ + yH dxdy in which the positive sign must be taken with each root. We shall now find the moment generating function (m. g. f.) for this distribution.

I: is defined as

G(?,j8) = J \ + r--2)

r!ft(ft-M)(ft + 2)...(? + r-l) " W*, ft) Lr-xCv, ft) "^

as r tends to infinity, and thus the ratio of any term in (11) to the preceding term tends to p2. The series is therefore absolutely convergent, if \ p \ < 1, for all values

of x and p9 and is divergent if | p | >1. For the purpose of statistical theory we are interested only in the case in which | p \ < 1. If | p | ' = 1 the series will prob ably be found to converge4 within some region including all real positive values of x

and y.


TWO var?ate gamma type distribution ?51. Expression of the Series of Laguerre Polynomials as a Bessel Function The series in Laguerre polynomials can be reduced to a simple form involving a Besssl function, by two different processes ; first, directly, using a result of Hardy's

and secondly, by a process of integration from probability distributions discussed by Wishart and Bartlett and others. It is of interest to give both, and connect the theory of the distribution function (?3) with various analytical results.

Hardy has shown that6, 7

1 ,_?. X,.W x,(,? _!__ exp I -Ux + y) î }j?{*?Li } where i_x e*K / d \?

xAx)=[n\r(n+\+2a)Y*^-(^) (e~* x*+*?); in our

notation this is Xn(x)= [n ! P(w + ft)] "* e~*x x'i(p~l) Ln(x9 p)9

r i,

?v ' n\Y(n + p) 1 + i I 1 + M

t \2*/{xy

Jpl V l +

Putting t=-p29 this shows at once that the

_ ~m~ "R^ i a i-p2 ? v i_p2 )

l\1\ P"""" fa: vW~'> exo \- X + y II /*W(*jO

forIp.1(3) = e-^-1)iJp.1(^' ). ?5*2. The second method of deducing this result starts from results of Wishart and Bartlett8 who have discussed the distribution of the variances and bivariances in

a sample of n sets from a population with p vari?tes ; that is, the distribution function of the |ft(ft+l) variables:

û = (*2n + *2i2+ ... +*2in)/w, v22 = (x22l + x222 + ... + x22n)/n9 1*12=Vîl X2y + X\2 X22+ ... + Xytx x2n)?n where x119 x129 ... xin9 x2l9 x229 ... x2n9 ... xnl, xn29 ... xnn, are sample sets from a normally

correlated population. They first find the moment generating function of the joint distribution of x19 x29 x39 ... xp and xxx29 xxxl% etc. It is

(13) [|R|/|R-A|]*exp[B(?,?)] 141 This content downloaded from 193.194.76.5 on Fri, 04 Aug 2017 15:31:04 UTC All use subject to http://about.jstor.org/terms

Vol. 5] SANKHY? : THE INDIAN JOURNAL OF STATISTICS [Part 2 where

A =

R =

the frequency distribution with which the discussion starts being

(14) f(xlfx2.xp) = tt-& IRI* exp[-R(%, x)]

where R(x9 x) denotes the quadratic form Xr^x^Xy ; and B = (R-A)_1 ;* B(a, a) be

a quadratic form in a19 a2, a3 .ap which correspond to the first order vari? X\> x2, x39 .Jtp. It then follows that the moment generating function for variances, bivariances, and means is

(15)

[ | R | / | R-A | ]?n exp [nB(a, a)]

The distribution function for these p + \p(p?l) variables is then to be found by an integral inversion, and the integration with respect to the p first-order variables can be carried out and leaves iCC iOO +?CX)

(16) /(.?-) = (_?)-? *??-'> I R | ?("-? J j"... j | R-A \-">-l> exp [-Sîy] -i-X) -?OO -*3C /?' V

x dalx dal2 ... daw as the distribution function for vix, vl2, v13,...v22, v23,...vPV. This differs only in having n ? 1 for n from the integral which would be obtained by putting ax = a2 = a3= ... ?p in (15) so that the factor exp[n B(a, ?)] disappears, (thereby giving the m. g. f. for the distribution of the variances and bivariances, and finding the distribu tion function for that m. g. f.); thus showing that the distribution of the variances and bivariances from the means, in a sample of n is the same as the distribution of the true variances and bivariances in a sample of n? 1. The integral (16) has been evaluated by Ingham3 (1933) and by Ledermann. The result is

II-\ p(w> | r | .(-i) | v | _C"-p-?>expH,{ v=l 2 R?v v,v

(17)

f ( Vuv. ) =

n irti(n-v)]l

where Rpv = | r?v | , v = iV /(