# Matrix Variate Pareto Distribution of the Second Kind

Authors: Daya K. Nagar; Lata Joshi; Arjun K. Gupta
Journal: ISRN Probability and Statistics
 (2012)

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## Abstract

We generalize the univariate Pareto distribution of the second kind to the matrix case and give its derivation using matrix variate gamma distribution. We study
several properties such as cumulative distribution function, marginal distribution of submatrix, triangular factorization, moment generating function, and expected values of the Pareto matrix. Some of these results are expressed in terms of special functions of matrix arguments, zonal, and invariant polynomials.


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## Body

## 1. Introduction

The Lomax distribution, also called the Pareto distribution of the second kind is given by the p.d.f.(1.1)βλ(1+vλ)-(β+1),v>0,
where shape parameter β>0 and location parameter λ>0. The Lomax distribution, named after Lomax, is a heavy-tail probability distribution often used in business, economics, and actuarial modeling. The standard Pareto Distribution of the second kind has λ=1 with the p.d.f.(1.2)β(1+v)-(β+1),v>0,β>0.
Although a wealth of results on Pareto distribution is available in the literature (see Johnson et al. [1]) nothing appears to have been done to define and study matrix variate Pareto distribution.Therefore, in this paper, we define matrix variate Pareto distribution and study several of its properties.We will use the following standard notations (cf. Gupta and Nagar [2]). Let A=(aij) be an m×m matrix. Then, AT denotes the transpose of A; tr(A)=a11+⋯+amm;etr(A)=exp(tr(A));det(A)=determinantofA;∥A∥=normofA;A>0 means that A is symmetric positive definite and A1/2 denotes the unique symmetric positive definite square root of A>0. The submatrices A(α) and A(α),1≤α≤m, of the matrix A are defined as A(α)=(aij),1≤i,j≤α, and A(α)=(aij),α≤i,j≤m, respectively.The multivariate gamma function which is frequently used in multivariate statistical analysis is defined by(1.3)Γm(a)=∫X>0‍etr(-X)det(X)a-(m+1)/2dX=πm(m-1)/4∏i=1m‍Γ(a-i-12),Re(a)>m-12.
The multivariate generalization of the beta function is given by
(1.4)Bm(a,b)=∫0Im‍det(X)a-(m+1)/2det(Im-X)b-(m+1)/2dX=Γm(a)Γm(b)Γm(a+b)=Bm(b,a),
where Re(a)>(m-1)/2 and Re(b)>(m-1)/2. Further, by using the matrix transformation X=(Im+Y)-1Y in (1.4) with the Jacobian J(X→Y)=det(Im+Y)-(m+1) one can easily establish the identity
(1.5)Bm(a,b)=∫Y>0‍det(Y)a-(m+1)/2det(Im+Y)-(a+b)dY.The beta type 1 and beta type 2 families of distributions are defined by the density functions (Johnson et al. [1]):
(1.6){B(α,β)}-1uα-1(1-u)β-1,0<u<1,(1.7){B(α,β)}-1vα-1(1+v)-(α+β),v>0,
respectively, where α>0,β>0, and
(1.8)B(α,β)=Γ(α)Γ(β)Γ(α+β).
Recently, Cardeño et al. [3] have defined and studied the family of beta type 3 distributions. A random variable w is said to follow a beta type 3 distribution if its density function is given by
(1.9)2α{B(α,β)}-1wα-1(1-w)β-1(1+w)-(α+β),0<w<1.If a random variableu has the p.d.f. (1.6), then we will write u~B1(α,β), and if the p.d.f. of the random variable v is given by (1.7), then v~B2(α,β). The distribution given by the density (1.9) will be designated by w~B3(α,β). The matrix variate generalizations of (1.6), (1.7), and (1.9) are defined as follows (Gupta and Nagar [2, 4, 5]).Definition 1.1.
Anm×m random symmetric positive definite matrix U is said to have a matrix variate beta type 1 distribution with parameters (α,β), denoted as U~B1(m,α,β), if its p.d.f. is given by
(1.10)det(U)α-(m+1)/2det(Im-U)β-(m+1)/2Bm(α,β),0<U<Im,
where α>(m-1)/2 and β>(m-1)/2.Definition 1.2.
Anm×m random symmetric positive definite matrix V is said to have a matrix variate beta type 2 distribution with parameters (α,β), denoted as V~B2(m,α,β), if its p.d.f. is given by
(1.11)det(V)α-(m+1)/2det(Im+V)-(α+β)Bm(α,β),V>0,
where α>(m-1)/2 and β>(m-1)/2.Definition 1.3.
Anm×m random symmetric positive definite matrix W is said to have a matrix variate beta type 3 distribution with parameters (α,β), denoted as W~B3(m,α,β), if its p.d.f. is given by
(1.12)2mαdet(W)α-(m+1)/2det(Im-W)β-(m+1)/2Bm(α,β)det(Im+W)α+β,0<W<Im,
where α>(m-1)/2 and β>(m-1)/2.


## 2.  The Density Function

First we define the matrix variate Pareto distribution of the second kind.Definition 2.1.
Anm×m random symmetric positive definite matrix V is said to have a matrix variate Pareto distribution of the second kind, denoted as V~Pm(β),β>(m-1)/2, if its p.d.f. is given by
(2.1)Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(Im+V)-β-(m+1)/2,V>0.Definition 2.2.
Anm×m random symmetric positive definite matrix U is said to have a matrix variate Lomax distribution with parameters Λ and β, denoted as U~Lm(Λ,β), if its p.d.f. is given by
(2.2)Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(Λ)-(m+1)/2det(Im+Λ-1U)-β-(m+1)/2,U>0,
where Λ is an m×m symmetric positive definite matrix and β>(m-1)/2.From Definitions2.1 and 2.2 it is clear that if V~Pm(β), then for an m×m symmetric positive definite constant matrix Λ,Λ1/2VΛ1/2~Lm(Λ,β) and if U~Lm(Λ,β), then Λ-1/2UΛ-1/2~Pm(β).Form=1, the matrix variate Pareto distribution and matrix variate Lomax distribution reduce to their respective univariate forms.The matrix variate Pareto distribution can be derived by using independent gamma matrices. A random matrixY is said to have a matrix variate gamma distribution with parameters Ψ(>0) and κ(>(m-1)/2), denoted by Y~Ga(m,κ,Ψ), if its p.d.f. is given by
(2.3)etr(-Ψ-1Y)det(Y)κ-(m+1)/2Γm(κ)det(Ψ)κ,Y>0.Theorem 2.3.
LetY1 and Y2 be independent, Y1~Ga(m,(m+1)/2,Im) and Y2~Ga(m,β,Im). Then, Y2-1/2Y1Y2-1/2~Pm(β).Proof.
The joint density function ofY1 and Y2 is given by
(2.4)etr[-(Y1+Y2)]det(Y2)β-(m+1)/2Γm[(m+1)/2]Γm(β),Y1>0,Y2>0.
Transforming W=Y2-1/2Y1Y2-1/2 with the Jacobian J(Y1→W)=det(Y2)(m+1)/2 in the joint density of Y1 and Y2, we obtain the joint density of W and Y2 as
(2.5)etr[-(Im+W)Y2]det(Y2)β+(m+1)/2-(m+1)/2Γm[(m+1)/2]Γm(β),0<W<Im,Y2>0.
Now, the desired result is obtained by integrating Y2 using (1.3).The cumulative distribution function ofV is obtained as
(2.6)P(V<Ω)=Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]∫0Ω‍det(Im+V)-β-(m+1)/2dV=Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(Ω)(m+1)/2∫0Im‍det(Im+ΩW)-β-(m+1)/2dW,
where the last line has been obtained by substituting W=Ω-1/2VΩ-1/2 with the Jacobian J(V→W)=det(Ω)(m+1)/2. Now, writing
(2.7)det(Im+ΩW)=det(Im+Ω)det(Im-(Im+Ω)-1Ω(Im-W)),
the above expression is rewritten as
(2.8)P(V<Ω)=Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(Ω)(m+1)/2det(Im+Ω)β+(m+1)/2×∫0Im‍det(Im-(Im+Ω)-1Ω(Im-W))-β-(m+1)/2dW.
Finally, using the integral representation of the Gauss hypergeometric function (Herz [6], Constantine [7], James [8], and Gupta and Nagar [2]), namely,
(2.9)F21(a,b;c;X)=Γm(c)Γm(a)Γm(c-a)∫0Imdet(R)a-(m+1)/2×det(Im-R)c-a-(m+1)/2det(Im-XR)-bdR,
where Re(a)>(m-1)/2,Re(c-a)>(m-1)/2, and X<Im, we obtain
(2.10)P(V<Ω)=Γm[β+(m+1)/2]Γm[(m+1)/2]Γm(β)Γm(m+1)det(Ω)(m+1)/2det(Im+Ω)β+(m+1)/2×F12(m+12,β+m+12;m+1;(Im+Ω)-1Ω).The moment generating function ofV is derived as
(2.11)MX(Z)=Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]∫V>0‍etr(ZV)det(Im+V)-β-(m+1)/2dV,
where Z(m×m)=((1+δij)zij/2). Now, evaluating the above integral, we obtain
(2.12)MX(Z)=Γm[β+(m+1)/2]Γm(β)Ψ(m+12,-β+m+12;-Z),
where the confluent hypergeometric function Ψ, with m×m symmetric matrix X as argument, is defined by the integral
(2.13)Ψ(a,c;X)=1Γm(a)∫R>0‍etr(-RX)det(R)a-(m+1)/2×det(Im+R)c-a-(m+1)/2dR
valid for Re(X)>0 and Re(a)>(m-1)/2.


## 3. Properties

In this section, we give several properties of the matrix variate Pareto distribution of the second kind defined in the previous section.Theorem 3.1.
LetV~Pm(β) and let A be an m×m constant nonsingular matrix. Then, the density of X=AVAT is
(3.1)Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(AAT)βdet(AAT+X)-β-(m+1)/2,X>0.Theorem 3.2.
LetV~Pm(β) and let H be an m×m orthogonal matrix, whose elements are either constants or random variables distributed independent of V. Then, the distribution of V is invariant under the transformation V→HVHT. Further, if H is a random matrix, then V and H are distributed independently.Theorem 3.3.
IfV~Pm(β), then, V-1~B2(m,β,(m+1)/2),(2Im+V)-1V~B3(m,(m+1)/2,β) and (Im+2V)-1~B3(m,β,(m+1)/2). Further, if W~B3(m,(m+1)/2,β), then 2(Im-W)-1W~Pm(β).Theorem 3.4.
LetV=(V11V12V21V22), where V11 is a q×q matrix. Define V11·2=V11-V12V22-1V21 and V22·1=V22-V21V11-1V12. If V~Pm(β), then (i) V11 and V22·1 are independent, V11~B2(q,(m+1)/2,β-(m-q)/2) and V22·1~Pm-q(β); (ii) V22 and V11·2 are independent, V22~B2(m-q,(m+1)/2,β-q/2) and V11·2~Pq(β).Proof.
From the partition ofV, we have
(3.2)det(Im+V)=det(Iq+V11)det(Im-q+V22·1+V21V11-1(Iq+V11)-1V12).
Now, making the transformation V11=V11,X=V21V11-1/2 and V22·1=V22-V21V11-1V12=V22-XXT with the Jacobian J(V11,V22,V21→V11,V22·1,X)=det(V11)(m-q)/2 in the density of V, we get the joint density of V11,V22·1, and X as
(3.3)Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(V11)(m-q)/2det(Im+V11)-β-(m+1)/2×det(Im-q+V22·1+X(Im+V11)-1XT)-β-(m+1)/2.
Further, transforming Y=(Im-q+V22·1)-1/2X(Iq+V11)-1/2 with the Jacobian J(X→Y)=det(Im-q+V22·1)q/2det(Iq+V11)(m-q)/2, the joint density of V11,V22·1, and Y is derived as
(3.4)Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(V11)(m-q)/2det(Im+V11)-β-(q+1)/2×det(Im-q+V22·1)-β-(m-q+1)/2det(Im-q+YYT)-β-(m+1)/2,
where V11>0, V22·1>0 and Y∈ℝ(m-q)×q. From the above factorization, it is clear that V11 and V22·1 are independent, V11~B2(q,(m+1)/2,β-(m-q)/2) and V22·1~Pm-q(β). The second part is similar.Theorem 3.5.
LetA be a q×m constant matrix of rank q(≤m). If V~Pm(β), then
(3.5)[(AAT)-1/2AV-1AT(AAT)-1/2]-1~Pq(β),(AAT)-1/2AVAT(AAT)-1/2~B2(q,m+12,β-m-q2).Proof.
WriteA=M(Iq,0)G, where M is a q×q nonsingular matrix and G is an m×m orthogonal matrix. Now,
(3.6)(AV-1AT)-1=(M(Iq0)GV-1GT(Iq0)TMT)-1=(MT)-1[(Iq0)Y-1(Iq0)]-1M-1=(MT)-1(Y11)-1M-1,
where Y=(Y11Y12Y21Y22)=GVGT~Pm(β), Y11 is a q×q matrix, and Y11=(Y11-Y12Y22-1Y21)-1=Y11·2-1. From Theorem 3.4, Y11·2~Pq(β), and Theorem 3.1, Z=(MT)-1Y11·2M-1 has the p.d.f. proportional to
(3.7)Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(MMT)-βdet((MMT)-1+Z)-β-(m+1)/2,Z>0.
Now, noting that MMT=AAT and making the transformation S=(AAT)1/2Z(AAT)1/2 with the Jacobian J(Z→S)=det(AAT)-(m+1)/2 in the above density, we get the desired result. The proof of the second part is similar.From Theorem3.5, it is clear that if V~Pm(β) and a∈ℝm,a≠0, then aTa(aTV-1a)-1~P(β). Further, if y(m×1) is a random vector, independent of W, and P(y≠0)=1, then it follows that yTy(yTV-1y)-1~P(β) and (yTy)-1(yTVy)~B2((m+1)/2,β-(m-1)/2).From the above results, it is straightforward to show that ifc(m×1) is a nonzero constant vector or a random vector independent of V with P(c≠0)=1, then
(3.8)cTV-1ccT(V-1+Im)c~B1(β,1),cTccT(V-1+Im)c~B1(1,β),cTV-1ccTc~B2(β,1),cTVccTc~B2(m+12,β-m-12).The expectation ofV, E(V), can easily be obtained from the above result. For any fixed c∈ℝm,c≠0,
(3.9)E[cTVccTc]=E(v),
where v~B2((m+1)/2,β-(m-1)/2). Hence, for all c∈ℝm,
(3.10)cTE(V)c=cTcE(v)=m+12β-m-1cTc,β>m+12
which implies that
(3.11)E(V)=m+12β-m-1Im,β>m+12.Theorem 3.6.
IfV~Pm(β) and V=WWT, where W=(wij) is a lower triangular matrix with positive diagonal elements, then w11,…,wmm are all independent, wii2~B2((m+2-i)/2,β-(m-i)/2),i=1,…,m.Proof.
Making the transformationV=WWT with the Jacobian J(V→W)=2m∏i=1m‍wiim+1-i in (2.1), the density of W is derived as
(3.12)2mΓm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(Im+WWT)-β-(m+1)/2∏i=1m‍wiim+1-i,
where wii>0, i=1,…,m and -∞<wij<∞ for 1≤j<i≤m. Now, partition W as
(3.13)W=(w110wW22),
where w is an (m-1)×1 vector and W22 is an (m-1)×(m-1) lower triangular matrix. Then
(3.14)det(Im+WWT)=det(1+w112w11wTw11wIm-1+wwT+W22W22T)=(1+w112)det(Im-1+W22W22T)×[1+11+w112wT(Im-1+W22W22T)-1w].
Now, make the transformation
(3.15)y=1(1+w112)1/2(Im-1+W22W22T)-1/2w
with the Jacobian J(w→y)=(1+w112)(m-1)/2det(Im-1+W22W22T)1/2 in (3.12) to get the joint density of w11,W22, and y as
(3.16)2mΓm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]w11m(1+w112)-β-1×det(Im-1+W22W22T)-β-m/2∏i=2m‍wiim+1-i(1+yTy)-β-(m+1)/2.
From the above factorization, it is clear that w11,W22, and y are all independent, w112~B2((m+1)/2,β-(m-1)/2) and the density of W22 is proportional to
(3.17)det(Im-1+W22W22T)-β-m/2∏i=2m‍wiim+1-i
which has the same form as the density (3.12) with m replaced by m-1. Repeating the argument given above on the density function of W22, we observe that w222~B2(m/2,β-(m-2)/2) and is independent of w33,…,wmm. Continuing further with the same argument, we get the desired result.Corollary 3.7.
IfV~Pm(β), then the distribution of det(V) is the same as the distribution of the product of m independent beta type 2 variables, that is, det(V)~∏i=1m‍vi where vi~B2((m+2-i)/2,β-(m-i)/2),i=1,…,m.Corollary 3.8.
IfV~Pm(β), then
(3.18)det(V(1))det(V(0)),det(V(2))det(V(1)),…,det(V(m))det(V(m-1))(det(V(0))≡1)
are independently distributed. Further, for i=1,…,m, det(V(i))/det(V(i-1))~B2((m+2-i)/2,β-(m-i)/2).Theorem 3.9.
IfV~Pm(β) and V=WWT, where W=(wij) is an upper triangular matrix with positive diagonal elements, then w11,…,wmm are all independent, wii2~B2((i+1)/2,β-(i-1)/2),i=1,…,m.Proof.
Making the transformationV=WWT with the Jacobian J(V→W)=2m∏i=1m‍wiii in (2.1), the density of W is derived as
(3.19)2mΓm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]det(Im+WWT)-β-(m+1)/2∏i=1m‍wiii,
where wii>0, i=1,…,m and -∞<wij<∞ for 1≤i<j≤m. Now, partition W as
(3.20)W=(W11w0wmm),
where w is an (m-1)×1 vector and W11 is an (m-1)×(m-1) upper triangular matrix. Then
(3.21)det(Im+WWT)=det(Im-1+wwT+W11W11TwmmwwmmwT1+wmm2)=(1+wmm2)det(Im-1+W11W11T)×[1+11+wmm2wT(Im-1+W11W11T)-1w].
Now, make the transformation
(3.22)y=1(1+wmm2)1/2(Im-1+W11W11T)-1/2w
with the Jacobian J(w→y)=(1+wmm2)(m-1)/2det(Im-1+W11W11T)1/2 in (3.19) to get the joint density of W11,y, and wmm as
(3.23)2mΓm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]wmmm(1+wmm2)-β-1×det(Im-1+W11W11T)-β-m/2∏i=1m-1‍wiii(1+yTy)-β-(m+1)/2.
From the above factorization, it is clear that W11, wmm, and y are all independent, wmm2~B2((m+1)/2,β-(m-1)/2) and the density of W11 is proportional to
(3.24)det(Im-1+W11W11T)-β-m/2∏i=1m-1‍wiii
which has the same form as the density (3.19) with m replaced by m-1. Repeating the argument given above on the density function of W11, we observe that wm-1,m-12~B2(m/2,β-(m-2)/2) and is independent of wm-2,m-2,…,w11. Continuing further with the same argument, we get the desired result.Corollary 3.10.
IfV~Pm(β), then the distribution of det(V) is the same as the distribution of the product of m independent beta type 2 variables, that is, det(V)~∏i=1m‍vi where vi~B2((i+1)/2,β-(i-1)/2), i=1,…,m.Corollary 3.11.
IfV~Pm(β), then
(3.25)det(V(1))det(V(2)),det(V(2))det(V(3)),…,det(V(m))det(V(m+1))(det(V(m+1))≡1)
are independently distributed. Further, for i=1,…,m,det(V(i))/det(V(i+1))~B2((i+1)/2,β-(i-1)/2).We conclude this section by deriving moments ofdet(V) and det(Im+V)-1.Theorem 3.12.
LetV~Pm(β), then
(3.26)E[det(V)rdet(Im+V)s]=Γm[β+(m+1)/2]Γm[(m+1)/2+r]Γm(β+s-r)Γm[β+(m+1)/2+s]Γm[(m+1)/2]Γm(β),
where Re(r)>-1 and Re(β+s)>(m-1)/2.Proof.
By definition(3.27)E[det(V)rdet(Im+V)s]=Γm[β+(m+1)/2]Γm(β)Γm[(m+1)/2]∫V>0‍det(V)(m+1)/2+r-(m+1)/2dVdet(Im+V)β+(m+1)/2+s.
Now, evaluating the above integral using (1.5), we get the result.Corollary 3.13 .1.
IfV~Pm(β), then
(3.28)E[det(V)r]=Γm[(m+1)/2+r]Γm(β-r)Γm[(m+1)/2]Γm(β),Re(β-r)>m-12,E[det(Im+V)-s]=Γm[β+(m+1)/2]Γm(β+s)Γm[β+(m+1)/2+s]Γm(β),Re(β+s)>m-12.By writing multivariate gamma functions in terms of ordinary gamma functions, expressionsE[det(V)r] and E[det(Im+V)-s] can be simplified as
(3.29)E[det(V)r]=∏j=1m‍Γ[(m+1)/2+r-(j-1)/2]Γ(β-r-(j-1)/2)Γ[(m+1)/2-(j-1)/2]Γ(β-(j-1)/2),(3.30)E[det(Im+V)-s]=∏j=1m‍Γ[β+(m+1)/2-(j-1)/2]Γ[β+s-(j-1)/2]Γ[β+(m+1)/2+s-(j-1)/2]Γ[β-(j-1)/2].
Substituting r,s=1,2, the first and second order moments of det(V) and det(Im+V)-1 are calculated as
(3.31)E[det(V)]=(2)m(2β-m-1)m,β>m+12,E[det(V)2]=(2)m(4)m(2β-m-1)m(2β-m-3)m,β>m+32,E[det(Im+V)-1]=(2β-m+1)m(2β+2)m,β>m-12,E[det(Im+V)-2]=(2β-m+1)m(2β-m+3)m(2β+2)m(2β+4)m,β>m-12,
where the Pochhammer notation (a)k is defined by (a)k=a(a+1)⋯(a+k-1),k=1,2,…with (a)0=1.


## 4. Results Involving Zonal and Invariant Polynomials

LetCκ(X) be the zonal polynomial of an m×m symmetric matrix X corresponding to the partition κ=(k1,…,km),k1+⋯+km=k,k1≥⋯≥km≥0. Then, for small values of k, explicit formulas for Cκ(X) are available as (James [8])
(4.1)C(1)(X)=tr(X),C(2)(X)=13[(trX)2+2tr(X2)],C(12)(X)=23[(trX)2-tr(X2)],C(3)(X)=115[(trX)3+6(trX)(trX2)+8tr(X3)],C(2,1)(X)=35[(trX)3+(trX)(trX2)-2tr(X3)],C(13)(X)=13[(trX)3-3(trX)(trX2)+2tr(X3)].
From the above results, it is straightforward to show that
(4.2)tr(X2)=C(2)(X)-12C(12)(X),tr(X3)=C(3)(X)-14C(2,1)(X)+14C(13)(X),tr(X)tr(X2)=C(3)(X)+16C(2,1)(X)-12C(13)(X).For an ordered partitionρ of r, ρ=(r1,…,rm),r1≥⋯≥rm≥0,r1+⋯+rm=r, Γm(a,ρ) and Γm(a,-ρ) are defined by
(4.3)Γm(a,ρ)=(a)ρΓm(a),Γm(a,0)=Γm(a),Γm(a,-ρ)=(-1)rΓm(a)(-a+m+1/2)ρ,Re(a)>r1+m-12,
where the generalized hypergeometric coefficient(a)ρ is defined by
(4.4)(a)ρ=∏i=1m‍(a-i-12)ri.
Further, det(Im-X)-a, in terms of zonal polynomials, can be expanded as
(4.5)det(Im-X)-a=∑k=0∞‍∑κ‍(a)κCκ(X)k!,∥X∥<1,
where ∑κ‍ denotes summation over all ordered partitions κ of k.For properties and further results, the reader is referred to Constantine [7] and Gupta and Nagar [2].Lemma 4.1.
LetT be an m×m arbitrary complex symmetric matrix. Then
(4.6)∫0Im‍det(R)a-(m+1)/2det(Im+R)-(a+b)Cκ(RT)dR=Γm(a,κ)Γm(b,-κ)Γm(a+b)Cκ(T)=(-1)kΓm(a,κ)Γm(b)(-b+(m+1)/2)κΓm(a+b)Cκ(T),Re(b)>k1+m-12.Davis [9, 10] introduced a class of polynomials Cϕκ,λ(X,Y) of m×m symmetric matrix arguments X and Y, which are invariants under the transformation (X,Y)→(HXHT,HYHT), H∈O(m). For properties and applications of invariant polynomials, we refer to Davis [9, 10], Chikuse [11], and Nagar and Gupta [12]. Let κ, λ, ϕ, and ρ be ordered partitions of the nonnegative integers k, ℓ, f=k+ℓ, and r, respectively. Then
(4.7)Cϕκ,λ(X,X)=θϕκ,λCϕ(X),θϕκ,λ=Cϕκ,λ(Im,Im)Cϕ(Im),Cϕκ,λ(X,Im)=θϕκ,λCϕ(Im)Cκ(X)Cκ(Im),(4.8)Cκκ,0(X,Y)≡Cκ(X),Cλ0,λ(X,Y)≡Cλ(Y),Cκ(X)Cλ(Y)=∑ϕ∈κ·λ‍θϕκ,λCϕκ,λ(X,Y),
where ϕ∈κ·λ denotes that irreducible representation of Gl(m,R), the group of m×m real invertible matrices, indexed by 2ϕ, appears in the decomposition of the tensor product 2κ⊗2λ of the irreducible representation indexed by 2κ and 2λ. Further,
(4.9)∫0Im‍det(R)t-(m+1)/2det(Im-R)u-(m+1)/2Cϕκ,λ(AR,A(Im-R))dR=Γm(t,κ)Γm(u,λ)Γm(t+u,ϕ)θϕκ,λCϕ(A),(4.10)∫0Im‍det(R)t-(m+1)/2det(Im-R)u-(m+1)/2Cϕκ,λ(AR,BR)dR=Γm(t,ϕ)Γm(u)Γm(t+u,ϕ)Cϕκ,λ(A,B).From the density ofV, we have
(4.11)E[Cκ(BV)]=Γm[β+(m+1)/2]Γm(β)Γm[m+1/2]∫V>0‍Cκ(VB)det(Im+V)-β-(m+1)/2dV=(-1)k((m+1)/2)κ(-β+(m+1)/2)κCκ(B),Re(β)>k1+m-12,
where the last line has been obtained by using (4.6).Using results– on zonal polynomials, it is easy to see that
(4.12)E[C(1)(BV)]=m+12β-m-1C(1)(B),E[tr(BV)]=m+12β-m-1tr(B),β>m+12,E[C(2)(BV)]=(m+1)(m+3)(2β-m-1)(2β-m-3)C(2)(B)=(m+1)(m+3)3(2β-m-1)(2β-m-3)[(trB)2+2tr(B2)],β>m+32,E[C(12)(BV)]=m(m+1)(2β-m-1)(2β-m)C(12)(B)=2m(m+1)3(2β-m-1)(2β-m)[(trB)2-tr(B2)],β>m+12,E[C(3)(BV)]=(m+1)(m+3)(m+5)(2β-m-1)(2β-m-3)(2β-m-5)C(3)(B)=(m+1)(m+3)(m+5)15(2β-m-1)(2β-m-3)(2β-m-5)×[(trB)3+6(trB)(trB2)+8tr(B3)],β>m+52,E[C(2,1)(BV)]=m(m+1)(m+3)(2β-m)(2β-m-1)(2β-m-3)C(2,1)(B)=3m(m+1)(m+3)15(2β-m)(2β-m-1)(2β-m-3)×[(trB)3+(trB)(trB2)-2tr(B3)],β>m+32,E[C(13)(BV)]=(m-1)m(m+1)(2β-m)(2β-m+1)(2β-m-1)C(13)(B)=(m-1)m(m+1)3(2β-m+1)(2β-m)(2β-m-1)×[(trB)3-3(trB)(trB2)+2tr(B3)],β>m+12,E[tr(BV)2]=E[C(2)(BV)]-12E[C(12)(BV)]=2β(m+1)(2β-m)(2β-m-1)(2β-m-3)(trB)2+(m+1)[(m+2)(2β-m-1)+2](2β-m)(2β-m-1)(2β-m-3)(trB2),β>m+32,(4.13)E[tr(BV)3]=E[C(3)(BV)]-14E[C(2,1)(BV)]+14E[C(13)(BV)]=1+m(2β-m+1)(2β-m)(2β-m-1)(2β-m-3)(2β-m-5)×[110{(m-1)m(3+m)(5+m)-4(m3+5m2+m-5)β+4(m2+3m+10)β2}(trB)3+110{(m2+33m+60)β2(m-1)m(3+m)(5+m)-4(m(3+m)(17+m)-30)β+4(m2+33m+60)β2}(trB)(trB2)+45{(m-1)m(3+m)(5+m)+20β-m(4m2+25m+29)β+2(2m2+11m+20)β2}tr(B3)110],β>m+52,E[tr(BV)tr(BV)2]=E[C(3)(BV)]+16E[C(2,1)(BV)]-12E[C(13)(BV)]=1+m(2β-m+1)(2β-m)(2β-m-1)(2β-m-3)(2β-m-5)×[-115{(m-1)m(3+m)(5+m)-2(2m3-5m2-48m+15)β+4(m2-12m-15)β2}(trB)3+215{(7m2+21m+45)β27(m-1)m(3+m)(5+m)+90β-2m(14m2+70m+39)β+4(7m2+21m+45)β2}×(trB)(trB2)+215{(m2+33m+60)β2(m-1)m(3+m)(5+m)-4(m(3+m)(17+m)-30)β+4(m2+33m+60)β2}tr(B3)115],β>m+52.
Further, using the invariance of the distribution of V and the above results, one obtains
(4.14)E(V)=m+12β-m-1Im,β>m+12,E(V2)=(m+1)[4(m+1)β-m(m+3)](2β-m)(2β-m-1)(2β-m-3)Im,β>m+32,E(V3)=1+m10(2β-m+1)(2β-m)(2β-m-1)(2β-m-3)(2β-m-5)×[(m-1)m(3+m)(5+m)(m2+m+8)-4(m5+6m4+29m3+96m2+28m-40)β+4(m4+4m3+51m2+104m+80)β2]Im,β>m+52,E[(trV2)V]=-1+m15(2β-m+1)(2β-m)(2β-m-1)(2β-m-3)(2β-m-5)×[(m-1)m(3+m)(5+m)(m2-14m-2)+2(-2m5+33m4+192m3+143m2+114m-120)β+4(m4-26m3-59m2-156m-120)β2]Im,β>m+52.Theorem 4.2.
LetV1 and V2 be independent, Vi~Pm(βi),i=1,2. Define S=V1+V2 and R=(V1+V2)-1/2V2(V1+V2)-1/2. Then, the density of S is given by
(4.15)Γm[β1+(m+1)/2]Γm[β2+(m+1)/2]Γm(β1)Γm2[(m+1)/2]Γm(β2)det(S)(m+1)/2det(Im+S)-(β1+β2+m+1)×∑k=0∞‍∑l=0∞‍∑κ‍∑λ‍(β1+(m+1)/2)κ(β2+(m+1)/2)λk!l!×∑ϕ∈κ·λ‍(θϕκ,λ)2Γ((m+1)/2,κ)Γ((m+1)/2,λ)Γ(m+1,ϕ)Cϕ((Im+S)-1S),S>0.
Further, the density of R is derived as
(4.16)Γm[β1+(m+1)/2]Γm[β2+(m+1)/2]Γm(β1)Γm(β2)Γm2[(m+1)/2]×∑k=0∞‍∑l=0∞‍∑κ‍∑λ‍(β1+(m+1)/2)κ(β2+(m+1)/2)λk!l!×∑ϕ∈κ·λ‍Γ(m+1,ϕ)Γm(β1+β2)Γm(β1+β2+m+1,ϕ)θϕκ,λCϕκ,λ(R,Im-R),0<R<Im.Proof.
The joint density ofV1 and V2 is given by
(4.17)Γm[β1+(m+1)/2]Γm[β2+(m+1)/2]Γm(β1)Γm(β2)Γm2[(m+1)/2]×det(Im+V1)-β1-(m+1)/2det(Im+V2)-β2-(m+1)/2,V1>0,V2>0.
Making the transformation V2=S1/2RS1/2 and V1=S1/2(Im-R)S1/2 with the Jacobian J(V1,V2→S,R)=det(S)(m+1)/2 in (4.17), the joint density of R and S is derived as
(4.18)Γm[β1+(m+1)/2]Γm[β2+(m+1)/2]Γm(β1)Γm(β2)Γm2[(m+1)/2]det(S)(m+1)/2det(Im+S)-(β1+β2+m+1)×det(Im-S1R)-β1-(m+1)/2det(Im-S1(Im-R))-β2-(m+1)/2,
where S1=(Im+S)-1S, 0<R<Im, and S>0. Since, ∥S1R∥<1 and ∥S1(Im-R)∥<1, using (4.5), we can write
(4.19)det(Im-S1R)-β1-(m+1)/2=∑k=0∞‍∑κ‍(β1+(m+1)/2)κk!Cκ(S1R),det(Im-S1(Im-R))-β2-(m+1)/2=∑l=0∞‍∑λ‍(β2+(m+1)/2)λl!Cλ(S1(Im-R)),
where κ and λ are the ordered partitions of k and l, respectively. Now, the application of  yields
(4.20)det(Im-S1R)-β1-(m+1)/2det(Im-S1(Im-R))-β2-(m+1)/2=∑k=0∞‍∑l=0∞‍∑κ‍∑λ‍(β1+(m+1)/2)κ(β2+(m+1)/2)λk!l!×∑ϕ∈κ·λ‍θϕκ,λCϕκ,λ(S1R,S1(Im-R)).
Finally, substituting (4.20) in (4.18), the joint density of R and S is obtained as
(4.21)Γm[β1+(m+1)/2]Γm[β2+(m+1)/2]Γm(β1)Γm(β2)Γm2[(m+1)/2]det(S)(m+1)/2det(Im+S)-(β1+β2+m+1)×∑k=0∞‍∑l=0∞‍∑κ‍∑λ‍(β1+(m+1)/2)κ(β2+(m+1)/2)λk!l!×∑ϕ∈κ·λ‍θϕκ,λCϕκ,λ(S1R,S1(Im-R)),0<R<Im,S>0.
Now, the integration of R in (4.21) using (4.9) yields the density of S. The density of R is obtained by substituting S1=(Im+S)-1S with the Jacobian J(S→S1)=det(I-S1)-(m+1) in (4.21) and integrating S1 by using (4.10).



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*Source: 789273-2012-10-17.xml*