A form of multivariate gamma distribution

Ann. Inst. Statist. Math. Vol. 44, No. 1, 97-106 (1992)

A FORM OF MULTIVARIATE GAMMA DISTRIBUTION A. M. MATHAI 1 AND P. G. MOSCHOPOULOS2 1 Department of M a t h e m a t i c s and Statistics, McOill University, Montreal, Canada H3A 2 K 6 2 D e p a r t m e n t of Mathematical Sciences, The University of Texas at El Paso, El Paso, T X 79968-0514, U.S.A.

(Received July 30, 1990; revised February 14, 1991)

A b s t r a c t . Let V,, i = 1 , . . . , k , be independent gamma random variables with shape ai, scale /3, and location parameter %, and consider the partial sums Z1 = V1, Z2 = 171 + V2, . • •, Z k = 171 + . • • + Vk. When the scale parameters are all equal, each partial sum is again distributed as gamma, and hence the joint distribution of the partial sums may be called a multivariate gamma. This distribution, whose marginals are positively correlated has several interesting properties and has potential applications in stochastic processes and reliability. In this paper we study this distribution as a multivariate extension of the three-parameter gamma and give several properties that relate to ratios and conditional distributions of partial sums. The general density, as well as special cases are considered.

K e y w o r d s a n d p h r a s e s : Multivariate gamma model, cumulative sums, moments, cumulants, multiple correlation, exact density, conditional density.

1.

Introduction

T h e t h r e e - p a r a m e t e r g a m m a with the density (x _ V)~_I exp ( x - 7 ) (1.1)

f ( x ; a, /3, 7) =

~

,

x>'7,

c~>O,

/3>0

stands central in the multivariate g a m m a distribution of this paper. Multivariate extensions of g a m m a distributions such t h a t all the marginals are again g a m m a are the most c o m m o n in the literature. Such extensions involve the s t a n d a r d g a m m a (/3 = 1, "y = 0), or the exponential (a = 1), see Johnson and Kotz (1972). O t h e r extensions include the multivariate chi-square (Miller e t al. (1958), Krishnaiah and Rao (1961) and Krishnaiah e t al. (1963)), while the particular case of a bivariate g a m m a received special consideration, see Kibble (1941), Eagleson (1964), Ghirtis (1967), M o r a n (1967, 1969) and Sarmanov (1970). T h e bivariate g a m m a is particularly useful in modeling the lifetimes of two parallel systems, 97

98

A. M. MATHAI AND P. G. MOSCHOPOULOS

see Freund (1961), Becker and Roux (1981), Lingappaiah (1984) and Steel and le Roux (1987). Other applications include uses of the distribution in rainmaking experiments (Moran (1970)). In this paper we consider a new form of a multivariate gamma that has potential applications in situations in which partial sums of independent, positive random variables are of interest• Such situations appear in the area of reliability and stochastic processes. Let V~, i = 1 , . . . , k be the times between successive occurrences of a phenomenon, and let Zi = Zi-1 + Vi, i = 1 , . . . , k, with Z0 = 0. Then, Zi is the total time required until the i-th occurrence. In stochastic processes, it is usually assumed that the times Vi are also identically distributed. In this case, the occurrence time Zi, i C N can be viewed as a renewal process, and the times Vi can be called renewal times, see for example ~inlar (1975). Note that Z~ is the i-th partial sum of the Vi's. In actual practice the V~'s may be times between arrivals, or, for example, time delays of an airplane at several airports. Then, Zi is the total waiting time for the i-th occurrence, or the total delay at the i-th airport. As another example, consider the following application from reliability. An item is installed at time Z0 = 0 and when it fails, it is replaced by an identical (or different) item• Then, when the new item fails it is replaced again by another item and the process continues. In this case Zi = Zi-1 + V/ where V/is the time of operation of the i-th item, and Z~ is the time at which the i-th replacement is needed. Zk denotes the time interval in which a total of k items need replacement. The applications mentioned above, motivate the new form of a multivariate gamma that we consider in this paper. This is given in the form of a theorem. THEOREM 1.1. Suppose V1,...,Vk are mutually independent where G(ai,/3, 7~), i = 1 , . . . , k (same/5). Let Z1 = g l , g 2

V~

~-- V1 nt- V 2 , . . . , Z k ~-- g l nt - ' ' " -I- gk.

Then, the joint distribution of Z = ( Z 1 , . . . , Zk)' is a multivariate gamma with density function

(1.2)

f(q,...,z~)

=

(2:1 -- ~ l ) a 1 - 1 /3ot~ Hik=l F(o~i) • (z2

-

zl

-

~/2) ~ 2 - 1 ' ' '

(zk

-

zk-1

-

~/k) ~k-1

• e-(Zk-(~l+--+~k))/~ for c~i > O, /3 > O, ~/i real, zi-1 +'~i < z~, i = 2 , . . . , k , ak = ~1 + "'" + ak, and zero elsewhere.

zk < oc, 71 < zl,

Many applications exist in which the Vi's represent independent and identically distributed times. Here, they are assumed to be distributed as in (1.1), thus allowing maximum flexibility in shape, scale and location. The requirement of equal scale parameter 3 is to ensure that the marginals are of the same form. Basic properties of the distribution of Z, including the moment

A F O R M OF M U L T I V A R I A T E G A M M A D I S T R I B U T I O N

99

generating function, means, variances, properties of the covariance matrix and the reproductive property are given in Section 2. In Section 3 we give the moments and cumulants, and in Section 4 we discuss conditional distributions and special cases. Before discussing the properties of the model in (1.2) a brief description of the various methods of construction of multivariate gamma distributions will be given here. After giving a brief sketch of the historical development Dussauchoy and Berland (1974) define a multivariate gamma random variable Z = (Z1,..., Zn), in terms of the characteristic function defined by

(1.3)

Cz(Ul,.-.,Un)

l~i ¢ ~ (uj + E ; : j + l / 3 j k u k )

= II

n

where Czj(Uj) = ( 1 - i u j / a j ) -~j, aj k /3jkak > O,

j = l,...,n,

i = x/~-f, /3jk >_ O,

j < k = 1 , . . . , n, 0 < et 0, i -----1 , . . . , k - 1 and ~-~-i=] Yi _< 1. Clearly, each Y/ is a B e t a type-1 (see (b)).

A FORM OF MULTIVARIATE GAMMA DISTRIBUTION

105

Finally we note the following results that concern the special case in which each of the V~'s is exponential (c~i = 1,/3 = 1, 7i = 0). From the density above we have: (e)

h(yl,Y2,...,Yk_l)

=(~

-- 1)!

and hence Y1,... ,Yk-1 a r e distributed like the order statistic from the uniform U(0, 1)-distribution. In this case, the joint density of Z 1 , . . . , Zk reduces to f(zl,...

,zk) = e -zk,

0 • z1 'Q z2 "~ "'" "~ Z k - 1 ,

0 < Zk "Q 00.

The transformation wl =

Z1

,w2 =

Z2

..,wk-1

-

Z k - 1 Wk = Zk zk

is one-to-one with Jacobian J(Z ---* W) = W£k-1 . Thus, the joint density of W I , . . . , W k _ I is g(Wl,...,Wk_l)

=--

wk-le-Wkdwk=

~0(X3

(k-

1)!.

Hence, W 1 , . . . , Wk-1 are also distributed like the order statistic from the uniform U(0, 1)-distribution.

Acknowledgements The authors would like to thank the referees for the constructive suggestions which enabled them to improve the manuscript. The first author would like to thank the Natural Sciences and Engineering Research Council of Canada for financial support. REFERENCES Becker, P. J. and Roux, J. J. J. (1981). A bivariate extension of the gamma distribution, South African Statist. J., 15, 1-12. ~inlar, E. (1975). introduction to Stochastic Processes, Prentice Hall, New Jersey. Dussauchoy, A. and Berland, R. (1974). A multivariate gamma type distribution whose marginal laws are gamma, and which has a property similar to a characteristic property of the normal case, Statistical Distributions in Scientific Works, Vol. 1 (eds. G. P. Patil, S. Kotz and J. K. Ord), 319-328, Reidel, Dordrecht. Eagleson, G. K. (1964). Polynomial expansions of bivariate distributions, Ann. Math. Statist., 35, 1208-1215. Freund, J. E. (1961). A bivariate extension of the exponential distribution, J. Amer. Statist. Assoc., 56, 971-977. Gaver, D. P., Jr. (1970). Multivariate gamma distributions generated by mixture, Sankhyd Ser. A, 32, 123 126. Ghirtis, G. C. (1967). Some problems of statistical inference relating to double gamma distribution, Trabajos de Estadistica, 18, 67-87. Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics, Vol. 4, Houghton Mifflin, Boston. Kibble, W. F. (1941). A two-variate gamma distribution, Sankhy& 5, 137-150.

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Kowalczyk, T. and Tyrcha, J. (1989). Multivariate gamma distributions-properties and shape estimation~ Statistics~ 20~ 465-474. Krishnaiah, P. R. and Rao~ M. M. (1961). Remarks on a multivariate gamma distribution, Amer. Math. Monthly, 68, 342-346. Krishnaiah, P. R., Hagis, P. and Steinberg, L. (1963). A note on the bivariate chi-distribution, SIAM Rev., 5~ 140~--144. Lingappaiah, G. S. (1984). Bivariate gamma distribution as a life test model, Applikace Matematiky, 29, 182-188. Mathai, A. M. and Moschopoulos, P. G. (1991). On a multivariate gamma~ J. Multivariate Anal., 39, 135-153. Miller, K. S., Bernstein, R. I. and Blumenson, L. E. (1958). Generalized Rayleigh processes, Quart. Appl Math., 16, 137 145 (Correction: ibid. (1963). 20, p. 395). Moran, P. A. P. (1967). Testing for correlation between non-negative variates~ Biometrika~ 54~ 385-394. Moran, P. A. P. (1969). Statistical inference with bivariate gamma distributions~ Biometrika, 56, 627-634. Moran~ P. A. P. (1970). The methodology of rain making experiments~ Review of the International Statistical Institute~ 38, 105-115. Sarmanov, I. O. (1970). Gamma correlation process and its properties~ Dokl. Akad. Nauk SSSR, 191, 30-32 (in Russian). Steel, S. J. and le Roux, N. J. (1987). A reparameterisation of a bivariate gamma extension, Comm. Statist. Theory Methods, 16~ 293-305. Wilks, S. S. (1962). Mathematical Statistics, Wiley, New York.