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Doubly Singular Matrix Variate Beta Type I and II and Singular Inverted Matricvariate $ T $ Distributions

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Doubly Singular Matrix Variate Beta Type I and II and Singular Inverted Matricvariate $ T $ Distributions Doubly singular matrix variate beta type I and II and singular inverted matricvariate t distributions Jos´eA. D´ıaz-Garc´ıa ∗ Department of Statistics and Computation 25350 Buenavista, Saltillo, Coahuila, Mexico E-mail: [email protected] Ram´on Guti´errez J´aimez Department of Statistics and O.R. University of Granada Granada 18071, Spain E-mail: [email protected] Abstract In this paper, the densities of the doubly singular beta type I and II distributions are found, and the joint densities of their corresponding nonzero eigenvalues are provided. As a consequence, the density function of a singular inverted matricvariate t distribution is obtained. 1 Introduction Let A and B be independent Wishart matrices or, alternatively, let B = YY′ where Y has a matrix variate normal distribution. Then the matrix variate beta type I distribution is arXiv:0904.2147v1 [math.ST] 14 Apr 2009 defined in the literature as (A + B)−1/2B(A + B)−1/2, U = B1/2(A + B)−1B1/2, (1) ′ −1 Y (A + B) Y. Analogously, the matrix variate beta type II distribution is defined as A−1/2BA−1/2, F = B1/2A−1B1/2, (2) ′ −1 Y A Y. These can be classified as central, noncentral or doubly noncentral, depending on whether A and B are central, B is noncentral or A and B are noncentral, respectively; see D´ıaz-Garc´ıaand Guti´errez-J´aimez ∗Corresponding author Key words. Singular distribution, matricvariate t distribution, matrix variate beta type I distribution, matrix variate beta type II distribution. 2000 Mathematical Subject Classification. 62E15, 15A52 1 (2007, 2008b). In addition, such a distribution can be classified as nonsingular, singular or doubly singular, when A and B are nonsingular, B is singular or A and B are singular, respectively, see D´ıaz-Garc´ıaand Guti´errez-J´aimez (2008a). Under definitions (1) and (2) and their classifications, mentioned above, the matrix variate beta type I and II distribu- tions have been studied by different authors; see Olkin and Rubin (1964), Dickey (1967), Mitra (1970), Khatri (1970), Srivastava and Khatri (1979), Muirhead (1982), Uhlig (1994), Cadet (1996), D´ıaz-Garc´ıaand Guti´errez-J´aimez (1997, 2006, 2007, 2008a,b), among many others. However, the density functions of doubly singular beta type I and II distributions have not been studied. In such cases the inverses that appear in definitions (2) and (1) must be replaced by the Moore-Penrose inverse. The doubly singular beta type I distribution was briefly considered by Mitra (1970), but this distribution has been the object of considerable recent interest. Observe that this distribution appears in a natural way when the number of observations is smaller than the dimension in multivariate analysis of variance; see Srivastava (2007). Furthermore, observe that if Z is any random matrix m × r, it is generically called as matrix variate if the kernel of its density function is in terms of the trace operator. But if the kernel of the density function is in terms of the determinant alone or in terms of both, then the determinant and the trace operator, Z, are generically called matricvariate, see Dickey (1967) and D´ıaz-Garc´ıaand Guti´errez-J´aimez (2008c). Alternatively, when the random matrix A : m × m is symmetric by definition (e.g. Wishart and beta matrices, etc.), they are generically referred to as matrix variate. In Section 2 of the present study, we review singular matricvariate t distributions and determine the distribution of a linear transformation and the joint density function of its nonzero singular values. In Section 3, an expression is provided for the density function of the doubly singular matrix variate beta type II distribution, and the joint density function of its nonzero eigenvalues is determined. Similar results are provided for the doubly singular matrix variate beta type I distribu- tion; see Section 4. Finally, in Section 5, we study the inverted matricvariate t distribution, also termed the matricvariate Pearson type II distribution. 2 Preliminary results Let Lr,m(q) be the linear space of all m × r real matrices of rank q ≤ min(m, r) and let + Lr,m(q) be the linear space of all m × r real matrices of rank q ≤ min(m, r), with q distinct ′ singular values. The set of matrices H1 ∈ Lr,m such that H1H1 = Ir is a manifold denoted as Vr,m, termed the Stiefel manifold. In particular, Vm,m is the group of orthogonal matrices O(m). The invariant measure on a Stiefel manifold is given by the differential form r m ′ ′ (H1dH1) ≡ hj dhi i=1 j=i+1 ^ ^ written in terms of the exterior product ( ), where we choose an m × (m − r) matrix H2 . such that H is an m × m orthogonal matrix,V with H = (H1.H2) and where dh is an m × 1 vector of differentials; see Muirhead (1982, Section 2.1.4). Moreover 2 r mr/2 m m /2 ′ 2 π ′ 2 π (H1dH1)= and (H dH)= (3) H Γ [m/2] H Γ [m/2] Z 1∈Vr,m r Z ∈O(m) m Denote by Sm, the homogeneous space of m × m positive definite symmetric matrices; and by Sm(q), the (mq − q(q − 1)/2)-dimensional manifold of rank q positive semidefinite 2 + m × m symmetric matrices; and by Sm(q), the (mq − q(q − 1)/2)-dimensional manifold of rank q positive semidefinite m × m symmetric matrices with q distinct positive eigenvalues. Assume A ∈ Sm; then chi(A) denotes the i-th eigenvalue of the matrix A. Moreover, let A+ and A− be the Moore-Penrose inverse and any symmetric generalized inverse of A, respectively; see Rao (1973). Finally, A1/2 is termed a non-negative definite square root of A if A1/2 is such that A = A1/2A1/2. Using the notation of D´ıaz-Garc´ıaet al. (1997) for the singular matrix variate nor- mal and Wishart and pseudo-Wishart distributions, from D´ıaz-Garc´ıaand Guti´errez-J´aimez (2008c) we have: + Lemma 2.1. Let T ∈ Lr,m(rΞ ) be the random matrix + T = A1/2 Y + µ 1/2 1/2 + q where A A = A ∈ Sm(q) ∼Wm (n, Θ), Θ ∈ Sm(rΘ ) with rΘ ≤ m and q = min(m,n), + m,rΞ independent of Y ∈ Lr,m(rΞ ) ∼ Nm×r (0, Im ⊗ Ξ), Ξ ∈ Sr(rΞ ) with rΞ ≤ r and m ≥ q ≥ rΞ > 0. Then the singular matricvariate T has the density rα c(m,n,q,q1, rΘ , rΞ , rα ) − − ′ −(n+rΞ )/2 ch Θ + (T − µ)Ξ (T − µ) (dT), (4) rΞ rΘ l m/2 n/2 l=1 chi(Ξ) chj (Θ) Y i=1 j=1 Y Y with n(q−rΘ )/2−(n+rΞ )(q1−rα )/2−mrΞ /2 π Γq1 [(n + rΞ )/2] c(m,n,q,q1, rΘ , rΞ , rα )= (mr +nr )/2−(n+r )r /2 , (5) 2 Ξ Θ Ξ α Γq[n/2] − − ′ where q1 = min(m,n + rΞ ), rα = rank [Θ + (T − µ)Ξ (T − µ) ] ≤ m, and (dT) denotes the Hausdorff measure. In particular observe that if Ξ = Ir, i.e. rΞ = r, rΘ = rα = m, Θ = Im, q = n and q1 = n + r, we have −r(r+2n)/2 Γn+r[(n + r)/2] ′ −(n+r)/2 dFT(T)= π |Im + (T − µ)(T − µ) | (dT) (6) Γn[n/2] + where T ∈ Lr,m(r) and now (dT) denotes the Lebesgue measure. ′ + + Theorem 2.1. Under the condition of Lemma 2.1, let µ = 0 and X = TC ∈ L (rΞ ), rΞ ,m ′ + where Ξ = CC such that C ∈ L (rΞ ). Then the density function of X is rΞ ,r rα − c(m,n,q,q1, rΘ , rΞ , rα ) − ′ (n+rΞ )/2 chl Θ + XX (dX), (7) rΘ n/2 l=1 chj (Θ) Y j=1 Y − ′ where q1 = min(m,n+rΞ ), rα = rank (Θ + XX ) ≤ m, and now (dX) denotes the Lebesgue measure. ′ Proof. Make the change of variables T = XC as described by D´ıaz-Garc´ıa(2007) rΞ rΞ ′ m/2 m/2 (dT)= chi(CC ) (dX)= chi(Ξ) (dX). (8) i=1 i=1 Y Y 3 Also, observe that ′ ′ + + + ′ + + + ′ + C ΞC = C (CC )C = C C(C C) = C C = IrΞ , (9) + + since C and C C have the same rank rΞ and C C is of the order rΞ × rΞ . Finally, note that by (9) ′ ′ TΞ−T′ = XC′Ξ−CX′ = X(C−ΞC −)−X′ = X(C+ΞC+ )−X′ = XX′. (10) By substituting (8) and (10) in (4) we obtain the desired result. Now let κi, i =1,...,r be the nonzero singular value of T. Then taking µ = 0 in (6) we have Theorem 2.2. The joint density function of κ1,...,κr, the singular values of T is r r 2r Γ [(n + r)/2] κm−r π−r(2n−m)/2 n+r i κ2 − κ2 dκ (11) Γ [n/2]Γ [r/2]Γ [m/2] (1 + κ2)(n+r)/2 i j i n r r i=1 i i<j i=1 ! Y Y ^ where κ1 > ··· >κr > 0. ′ Proof. Let T = Q1DTP be the nonsingular part of the singular value decomposition (SVD), where Q1 ∈ Vr,m, P ∈ O(r) and DT = diag(κ1,...,κr), κ1 > ··· >κr > 0. Then, from D´ıaz-Garc´ıaet al. (1997) −r m−r 2 2 ′ ′ (dT)=2 |DT| κi − κj (dDT)(Q1dQ1)(P dP). i<j Y The result is then obtained immediately from (6), noting that ′ ′ 2 |Im + Q1DTP PDTQ1| = |Ir + DT| and integrating over Q1 ∈Vr,m and P ∈ O(r) using (3). 3 Doubly singular beta type II distribution 1/2 1/2 + qA Theorem 3.1.
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