Multivariate Liouville distribution of second kind. Acknowledgment. References. Summary. Riassunto. Key words. 1. Introduction. Let W( , λ) denote the Wishart ...
A.K. GUPTA – D.G. KABE (*)
On multivariate Liouville distribution Contents: 1. Introduction. — 2. Multivariate Liouville distribution of first kind. — 3. Multivariate Liouville distribution of second kind. Acknowledgment. References. Summary. Riassunto. Key words.
1. Introduction Let W (, λ) denote the Wishart density of a p × p symmetric positive definite (SPD) matrix X , i.e., if X ∈ W (, λ), then its density function is |2|−λ 1 −1 (1) exp − tr X |X |λ−q p (λ) 2 where 2q = p + 1, and p (·) is the multivariate gamma function (e.g. see Gupta and Varga, 1993). The joint density of (X 1 , . . . , X n ), with independent marginal random matrices X i ( p × p) ∈ W (, λi ), i = 1, . . . , n is n 1 |2|−(λ1 +...+λn ) −1 |X i |λi −q . (2) exp − tr (X 1 + . . . + X n ) p (λ1 ) . . . p (λn ) 2 i=1
Now (2) can be used to derive multivariate Dirichlet distribution and certain multivariate beta distributions of the first and second kind as stated by properties A1 and A2 below. The property A2 is a new application of a certain Jacobian of matrix transformation to the statistical distribution theory.
(*) Bowling Green State University and St. Mary’s University.
174 PROPERTY A1 . Define the sequence of ( p × p) SPD random matrices Y1 , . . . , Yn , by the relations −1
−1
Yi = Yn 2 X i Yn 2 , i = 1, . . . , n − 1, Yn = (X 1 + . . . + X n ) .
(3)
Then using (2), the joint density of Y1 , . . . , Yn−1 is the first kind of Dirichlet density, D p (λ1 , . . . , λn−1 ; λn ) given by f (Y1 , . . . , Yn−1 ) =
p (λ1 + . . . + λn ) n−1 × p (λ1 ) . . . p (λn ) i=1
(4)
× |Yi |λi−q |I − Y1 − . . . − Yn−1 |λn−q and Yn is independently distributed as
|2|−(λ1 +...+λn ) 1 exp − tr −1 Yn |Yn |λ1 +...+λn−q = (5) p (λ1 + . . . + λn ) 2 = W (, λ1 + . . . + λn ) .
f (Yn ) =
Proof. We first transform from X n to Yn , and then keeping Yn fixed we transform from X i to Yi , i = 1, . . . , n − 1. Thus the required Jacobian is J (X 1 , . . . , X n ; Y1 , . . . Yn ) = =
n−1 i=1 n−1
J (X i : Yi ) [J (X n : Yn )] =
|Yn |
q
(6) (n−1)q
[1] = |Yn |
.
i=1
It follows that (2) changes to the density
1 f (Y1 , . . . , Yn ) = K exp − tr −1 Yn |Yn |(n−1)q × 2 ×
n−1
|Yn1/2 Yi Yn1/2 |λi−q ×
i=1
× |Yn1/2 (I − Y1 − . . . − Yn−1 )Yn1/2 |λn−q = 1 −1 = K exp − tr Yn |Yn |λ1 +...+λn−q × 2 × |I − Y1 − . . . − Yn−1 |λn−q
n−1 i=1
|Yi |λi−q .
(7)
175 where K as a generic letter denotes the normalizing constants of density functions in this paper. Now the results (4) and (5) follow. In case n = 2, (4) reduces to the multivariate beta density of the first kind, B p(1) (λ1 , λ2 ), given by f (Y1 ) =
1 |Y1 |λ1 −q |I − Y1 |λ2 −q B p (λ1 , λ2 )
(8)
where B p (λ1 , λ2 ) = p (λ1 ) p (λ2 )/ p (λ1 + λ2 ). PROPERTY A2 . By using (2), and setting X 1 = P1 P1 ,
X 2 = P1 Y1 P1 ,
X 1 + X 2 = P1 (I + Y1 )P1 = G 2 = P2 P2 ,
(9) X 3 = P2 Y2 P2 ,
X 1 + X 2 + X 3 = P2 (I + Y2 )P2 = G 3 = P3 P3 ,
X 4 = P3 Y3 P3 , (11)
= G n−1 = Pn−1 Pn−1 , X 1 + . . . + X n−1 = Pn−2 (I + Yn−2 )Pn−2 , X n = Pn−1 Yn−1 Pn−1
(10)
(12)
X 1 +. . .+ X n = Yn = Pn−1 (I +Yn−1 )Pn−1 , (13)
where P matrices are ( p × p) nonsingular, and Y matrices are ( p × p) SPD, we find that Y1 , Y2 , . . . Yn are independent and that f (Yi ) =
1 |Yi |λi+1 −q = B p (λ1 + . . . + λi , λi+1 ) |I + Yi |λ1 +...+λi+1
(14)
= B p(2) (λ1 + . . . + λi , λi+1 ) , i = 1, . . . , n − 1, and f (Yn ) is given by (5). Proof. First observe that , X i = Pi Pi − Pi−1 Pi−1
i = 1, . . . , n;
P0 = 0 ,
I + Yi = Pi−1 Pi+1 Pi+1 Pi −1 , i = 1, . . . , n − 1 ,
(15) (16)
and then note the following Jacobians. For a p × p nonsingular P and a p × p SPD A, we have that J (P −1 : P) = |P|−2 p ; J (P A P : P) = 2 p |A|q |P|; J (P
−1
AP
−1
: P) = 2 |A| |P| p
q
−2 p−1
(17) .
176 Now using (17) we first transform the X variates density to the P variates density,
1 f (P1 , . . . , Pn ) = K exp − tr Pn Pn −1 × 2 ×
n i=1
|Pi Pi − Pi−1 Pi−1 |λi −q |Pi | =
(18)
1 = K exp − tr −1 Pn Pn |P1 |2λ1 −2q |Pn |× 2 ×
n−1
|Pi−1 Pi+1 Pi+1 Pi−1 − I |λi+1 −q |Pi |2λi+1 − p .
i=1
Next by using (16) and (18) we transform the P variates density to the Y variates density. This is easily done by noting that J (Yi : Pi ) = 2 p |Pi+1 |2q |Pi |−2 p−1 , i = 1, . . . , n − 1; J (Yn : Pn ) = 2 p |Pn | ,
(19)
and reducing (18) to the density n−1 1 f (Y1 , . . . , Yn ) = K exp − tr −1 Yn |Yi |λi+1 −q × 2 i=1
× |P1 |
2λ1
n−1
|Pi |
2λi+1
|Pn |
(20)
,
i=1
where the curly brackets {. . . } must be expressed in terms of Y variates by the relations 1
|Pi | = |Pi+1 ||I + Yi |− 2 , i = 1, . . . , n − 1;
1
|Pn | = |Yn | 2 ,
(21)
and hence we find that {. . . } = |Yn |λ1 +...+λn −q
n−1
|I + Yi |−λ1 −...−λi+1 ,
(22)
i=1
which is the property A2 . Obviously it is possible to pass from (4) to (14) by using certain Jacobians of matrix transformations.
177 2. Multivariate Liouville distribution of first kind Starting with (2), the joint density of X 1 , . . . , X n is assumed to be Kg
n
Xi
i=1
n
|X i |αi −q .
(23)
i=1
Following Sivazlian (1981), we denote (23) by (X 1 , . . . , X n ) ∈ L (1) pn [g(U ); α1 , . . . , αn ] ,
(24)
and say that X 1 , . . . X n have a joint Liouville distribution of the first kind, where U = X 1 + . . . + X n . Now the Properties A1 and A2 are generalized as follows. PROPERTY B1 . Let (X 1 , . . . , X n ) ∈ L (1) pn [g(U ); λ1 , . . . , λn ], and Y1 , . . . , Yn be defined by (3). Then (Y1 , . . . , Yn−1 ) ∈ D p (λ1 , . . . , λn−1 ; λn ); Yn is independent of (Y1 , . . . , Yn−1 ) and Yn ∈ L (1) p [g(U ); λ1 + . . . + λn ], i.e., (25) f (Yn ) = K g (U )|U |λ1 +...+λn −q . PROPERTY B2 . Let (X 1 , . . . , X n ) ∈ L (1) pn [g(U ); λ1 , . . . , λn ], and Y1 , . . . , Yn be defined by (9), (10), (11), (12), and (13). Then Yi ∈ B p(2) (λ1 +. . .+λ1 , λi+1 ), i = 1, . . . , n −1, Yn ∈ L (1) p [g(U ); λ1 +. . .+λn ], and Y1 , . . . , Yn−1 , Yn are independent. 3. Multivariate Liouville distribution of second kind
n The joint density (23), with the additional restriction (I − i=1 Xi ) (2) = (I − U ) > 0, is termed L pn [g(U ); λ1 , . . . λn ], and in this case the density of U is
f (U ) = K g (U )|U |λ1 +...+λn −q , I − U > 0 . If, however, U is defined as U = X 1 + . . . + X j , j ≤ n, then the density of U is f (U ) = K |U |
λ1 +...+λ j −q
where (I − U − T ) > 0.
g(U + T )|T |λ j+1 +...+λn −q dT ,
178 In conclusion we remark that there is no unique definition of the multivariate beta distribution in multivariate normal statistical analysis. Thus e.g., the Property A2 can be generalized in several different ways. For example, when n = 2, we can set X 1 = L L ,
X 2 = L QY1 Q L ,
Y2 = X 1 + X 2 = L Q(I + Y1 )Q L ,
where Q is any p × p orthogonal matrix, and L is a p × p nonsingular, or lower (or upper) triangular matrix. Then also Y1 and Y2 are independent, Y1 ∈ B p(2) (λ1 , λ2 ), and
|2|−(λ1 +λ2 ) 1 f (Y2 ) = exp − tr −1 Y2 |Y2 |λ1 +λ2 −q . p (λ1 + λ2 ) 2
Acknowledgment We thank one of the referees for the short proof of the Property A2 presented in this paper.
REFERENCES Gupta, A.K. and Varga, T. (1993) Elliptically Contoured Models in Statistics, Kluwer Academic Publisher, Dordrecht. Sivazlian, B.D. (1981) On a multivariate extension of the gamma and beta distribution, SIAM J. Appl. Math., 41, 205–209.
On multivariate Liouville distribution Summary The concept of Liouville distributions of the first and the first and the second kind provides certain generalization of the multivariate Dirichlet and beta distributions, as noted by Sivazlian (1981). The present paper studies some multivariate generalizations of univariate Liouville distribution results given by Sivazlian (1981). Starting with n independent Wishart densities certain multivariate beta densities associated with them are derived.
179 Sulla distribuzione multivariata di Liouville Riassunto La distribuzione di Liouville del primo e secondo tipo suggerisce alcune generalizzazioni delle distribuzioni multivariate di Dirichlet e Beta, come osservato da Sivazlian (1981). Il presente articolo studia alcuni risultati su estensioni multivariate della distribuzione univariata di Liouville presentate da Sivazlian (1981). Partendo da n densit`a di Wishart indipendenti vengono derivate alcune densit`a Beta multivariate ad esse associate.
Key words Wishart distributions, Multivariate Liouville distributions, Multivariate beta distribution.
[Manuscript received June 1994; final version received November 1998.]
