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A Note on the Existence of the Multivariate Gamma Distribution
Thomas Royen Fachhochschule Bingen, University of Applied Sciences e-mail: [email protected]
Abstract. The p - variate gamma distribution in the sense of Krishnamoorthy and Parthasarathy exists for all positive integer degrees of freedom and at least for all real values pp 2, 2. For special structures of the “associated“ covariance matrix it also exists for all positive . In this paper a relation between central and non-central multivariate gamma distributions is shown, which implies the existence of the p - variate gamma distribution at least for all non-integer greater than the integer part of (p 1) / 2 without any additional assumptions for the associated covariance matrix.
1. Introduction
The p-variate chi-square distribution (or more precisely: “Wishart-chi-square distribution“) with degrees of 2 freedom and the “associated“ covariance matrix (the p (,) - distribution) is defined as the joint distribu- tion of the diagonal elements of a Wp (,) - Wishart matrix. Its probability density (pdf) has the Laplace trans- form (Lt)
/2 |ITp 2 | , (1.1) with the ()pp - identity matrix I p , , T diag( t1 ,..., tpj ), t 0, and the associated covariance matrix , which is assumed to be non-singular throughout this paper.
The p - variate gamma distribution in the sense of Krishnamoorthy and Parthasarathy [4] with the associated covariance matrix and the “degree of freedom” 2 (the p (,) - distribution) can be defined by the Lt
||ITp (1.2) of its pdf
g ( x1 ,..., xp ; , ). (1.3)
2 For values 2 this distribution differs from the p (,) - distribution only by a scale factor 2, but in this paper we are only interested in positive non-integer values 2 for which ||ITp is the Lt of a pdf and not of a function also assuming any negative values. These values are called here “admissible values”. The admissibility of all values 21 p follows from the existence of the Wp (2 , ) - distribution, and the ad- missibility of 2 (pp 2, 1) follows from formula (1.4) below. Smaller values of are admissible at least with some additional assumptions for . Sufficient and necessary conditions for entailing infinite divisibility Key words and phrases: multivariate gamma distribution, non-central multivariate gamma distribution, Wishart distribution 2010 Mathematics Subject Classifications: 60E05, 62E10 - 2 -
1 1 of the Lt ||ITp (i.e. all 0 are admissible) are found in [1] and [3]. According to [1], ||ITp is 1 infinitely divisible if and only if there exists any signature matrix S diag( s1 ,..., spj ), s 1, for which SS is an M-matrix. For the infinite divisibility of a more general class of multivariate gamma distributions see [2].
At least all values 2 m 1, m , m p , are admissible for “ m- factorial” covariance matrices
2 T T j W AA ( W W Ip BB, B WA with rows b ) with a suitable matrix W diag( w1 ,..., wp ),
wj 0, and a real ()pm - matrix A of the lowest possible rank m. This follows from the representation
p g(,...,;,) x x E w22 g ( w x , 1 bjj Sb T (1.4) 1 p j1 j j j 2 of the p (,) - pdf (see [7] and [8]), where
yn g(,)() x y e y g x n0 n n! is the non-central gamma density with the non-centrality parameter y and the central gamma densities g n , 2 and the expectation refers to the WImm(2 , ) - Wishart matrix S. With WI p , where is the lowest eigenvalue of , it follows mp1. A special case – entailing infinite divisibility – is a one-factorial W2 aaT with a real column a.
In the following section it will be shown that all values 2 [(p 1) / 2] (the integer part of (p 1) / 2 ) are admissible without any further assumptions for . This result is obtained as a corollary of a relation between central and non-central multivariate gamma densities given in theorem 1. This relation was already derived in a similar form for integer values 2 in [6], but the non-integer values require a different proof by means of the Lt.
2. A Sufficient Condition for the Existence of the p (,) - Distribution
Let be given any partition
11 12 (2.1) 21 22 of the non-singular ()pp - covariance matrix with ()ppii - matrices ii and set
1 0 11 12 22 21, (2.2) which is a non-singular covariance matrix too.
For 2 max(pp 1, 1) the Wishart W (2 , ) - pdf and the non-central W (2 , ,) - pdf exist 12 p2 22 p1 0 where the symmetrical positive semi-definite ()pp11 - matrix is any “non-centrality matrix” of rank kp 1. The diagonal of a random matrix Z has a non-central (,,) - distribution if 2Z has a W (2 , , ) - p1 0 p1 0 distribution, which exists (apart from integer values 2 with rank( ) min(p1 ,2 )) for all (p1 1) / 2 with rank( ) p1 (see [5]). - 3 -
The corresponding (,,) - pdf p1 0
g( x ,..., x ; , , ) (2.3) 1 p1 0 has the Lt
1 |ITTIT1 0 1 | etr( 1 ( 1 0 1 ) ) (2.4) with T diag( t ,..., t ). (In the literature the “non-centrality matrix” is frequently defined in a different non- 1 1 p1 symmetrical way.) Let denote the set of all ()pp - correlation matrices C and let p2 22
22Y X1/2 CX 1/2 (2.5) be a W (2 , ) - matrix with a random C and X diag( X ,..., X ), where the elements X p2 22 p2 p11 p j 1 1 1 1 have the gamma-densities jjg ( jj x j ) jj x j exp( jj x j )/ ( ) with the jj from the diagonal of
22. The density of Y is given by
11 ( p 1)/2 ( ( )) | | |YY |2 etr( ) (2.6) p2 22 22
p2 j1 with the multivariate gamma function (). pp22( 1)/4 p2 j1 2
Now, with the notations from (2,1), (2.2), (2.3), (2.5), (2.6), we show for the p (,) - pdf from (1.3):
Theorem 1. If p p12 p and 2 max(pp12 1, 1), then
g( x ,..., x ; , ) ( ( ))1 | | 1p p2 22
gxx( ,...,; , , 11/2 XCX 1/21 ) | XC | 1 | | (p2 1)/2 etr( 11/2 XCXdC 1/2 ) . 1 p1 0 12 22 22 21 22 p2
Remark. The simple special case with pC2 1, 1 (and 22 1) was already given by theorem 2 in [11].
As a corollary we obtain
Theorem 2. The function g ( x1 ,..., xp ; , ) with the Lt ||ITp from (1.2) is a p (,) - pdf at least for 2 ([( p 1) / 2], ).
Proof of theorem 2. Choose for p1 or p2 in theorem 1 the value [(p 1) / 2].
Proof of theorem 1. The equation in theorem 1 will be verified by the Lt of both sides. The left side has the Lt
||ITp and we get with the Schur complement for determinants
1 ITTT1 0 1 12 22 21 1 12 2 ||ITp 21TIT 1 2 22 2 - 4 -
11 |ITITTTITT2 222 | | 1 01 1222211 122 ( 2 222 ) 211 |
11 |ITITITITT2 222 | | 1 01 12222() 2222 ( 222 ) 211 |
11 |ITITITT2 222 | | 1 01 12222 ( 222 ) 211 |
1 1 1 |ITITIITTIT1 01 || 2 222 || 1 12222 ( 222 )( 2111 01 )|. (2.7)
With (2.4), (2.5) and (2.6) we find for the right side the Lt
|IT | ( ( )) 1 | | 1 0 1p2 22
1 1 1( p 1)/2 1 etr(T ( I T ) Y ) | Y |2 etr( Y T Y ) dY Y 0 1 1 0 1 12 22 22 21 22 2
|IT | ( ( )) 1 | | 1 0 1p2 22
( p 1)/2 1 1 1 1 |Y |2 etr() T ( I T ) T Y dY Y 0 22 21 1 1 0 1 12 22 22 2
11 |ITTITIT1 01 | | 2111 ( 01 ) 1222 2 222 | ,
(see e.g. formula (2.2.6) in [12]), and
1 1 1 |ITITITITIT1 01 | | 2 222 | | 2 2111 ( 01 ) 12222 ( 222 ) |
1 1 1 |ITITIITTIT1 01 || 2 222 || 1 12222 ( 222 )( 2111 01 )|, which coincides with formula (2.7), where the last identity follows from the general equation
IA1 12 |IBAIAB2 21 12 | | 1 12 21 |. BI21 2
Some further remarks. If W2 AAT is m- factorial, where A – and consequently BWA – may con- tain any mixture of real or pure imaginary columns, then the function with the representation from (1.4) has again the Lt ||ITp and it is a p (,) - pdf at least for all values 2 mp 1 [( 1) / 2]. For special structures of smaller values of 2 are possible, e.g. with an m- factorial W2 AAT with a
2 T real matrix A and mp1 [( 1) / 2]. Furthermore, let be, e.g., p2 p 1, 0 W 0 A 0 A 0 with a real
()pm10 - matrix A0 of rank m0 and mp12rank( 12 ) 2 . Then, at least 2 max(m0 m 12 1, p 2 1) is admissible and max(m0 m 12 1, p 2 1) [( p 1)/ 2] is possible for low values of m0 and m12.
On the other hand it is at present an open question if there exist some ()pp - covariance matrices for which 2 is inadmissible for some values 2 ( [(pp 3) / 2],[( 1)/ 2] ),p 5.
A consequence of theorem 2 is the extension of the inequality - 5 -
G( x ,..., x ; , ) G ( x ,..., x ; , ) G ( x ,..., x ; , ), p1 p p1 1 p 111 p p 1 p1 1 p 22
xx1,...,p 0, rank( 12 ) 0, (2.8)
for the p - variate cumulative p (,) - distribution function Gp . This inequality was proved for 2 and for all values 22 p in [9] (see also [10]), and it implies the famous Gaussian correlation inequality for 2 1. Now, this inequality can be extended to all non-integer values 2 [(p 1) / 2] without any further assumptions for .
References
[1] Bapat, R. B., Infinite divisibility of multivariate gamma distributions and M-matrices, Sankhyā Series A 51 (1989), 73-78. [2] Bernardoff, P., Which multivariate gamma distributions are infinitely divisible?, Bernoulli 12 (2006),169-189. [3] Griffiths, R. C., Characterization of infinitely divisible multivariate gamma distributions, J. Multivarate Anal. 15 (1984), 13-20. [4] Krishnamoorthy, A. S. and Parthasarathy, M., A multivariate gamma type distribution, Ann. Math. Stat. 22 (1951), 549-557. [5] Letac, G. and Massam, H., Existence and non-existence of the non-central Wishart distribution, (2011), arXiv:1108.2849. [6] Miller, K. S. and Sackrowitz, H., Relationships between biased and unbiased Rayleigh distributions, SIAM J. Appl. Math. 15 (1967), 1490-1495. [7] Royen, T., On some central and non-central multivariate chi-square distributions, Statistica Sinica 5 (1995), 373-397. [8] Royen, T., Integral representations and approximations for multivariate gamma distributions, Ann. Inst. Statist. Math. 59 (2007), 499-513. [9] Royen, T., A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions, Far East J. Theor. Stat. 48 (2014), 139-145. (preprint: arXiv:1408.1028) [10] Royen, T., Some probability inequalities for multivariate gamma and normal distributions, Far East J. Theor. Stat. 51 (2015), 17-36. [11] Royen, T., Non-central multivariate chi-square and gamma distributions, (2016), arXiv:1604.06906. [12] Siotani, M., Hayakawa, T. and Fujikoshi, Y., Modern Multivariate Statistical Analysis: A Graduate Course and Handbook, American Science Press, Inc. (1985), Columbus, Ohio, U.S.A.