Unitary invariant and residual independent matrix distributions

Abstract

Define Z13 = A½Y(A½)H (A and Y are independent) and Z15 = B½Y(B½)H (B and Y are independent), where Y, A and B follow inverted complex Wishart, complex beta type I and complex beta type II distributions, respectively. In this article several properties including expected values of scalar and matrix valued functions of Z13 and Z15 are derived.

beta distribution; inverted complex Wishart; complex random matrix; Gauss hypergeometric function; residual independent; unitary invariant; zonal polynomial

Unitary invariant and residual independent matrix distributions

Arjun K. Gupta^I; Daya K. Nagar^{II, *}; Astrid M. Vélez-Carvajal^II

^IDepartment of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403-0221, USA, E-mail: [e-mail]

^IIDepartamento de Matemáticas, Universidad de Antioquia, Calle 67, No. 53-108, Medellín, Colombia E-mail: [e-mail]

ABSTRACT

Define Z₁₃ = A^1/2Y(A^1/2)^H (A and Y are independent) and Z₁₅ = B^1/2Y(B^1/2)^H (B and Y are independent), where Y, A and B follow inverted complex Wishart, complex beta type I and complex beta type II distributions, respectively. In this article several properties including expected values of scalar and matrix valued functions of Z₁₃ and Z₁₅ are derived.

Mathematical subject classification: Primary: 62E15; Secondary: 62H99.

Key words: beta distribution, inverted complex Wishart, complex random matrix, Gauss hypergeometric function, residual independent, unitary invariant, zonal polynomial.

1 Introduction

This paper deals with complex random quadratic forms involving complex Wishart, inverted complex Wishart, complex beta type I and complex beta type II matrices. To define these distributions we need some notations from complex matrix algebra. Let A = (a_ij) be an m × m matrix of complex numbers. Then, A^H denotes conjugate transpose of A;

tr(A) = a₁₁+···+ a_mm;

etr(A) = exp(tr(A));

det(A) = determinant of A;

A = A^H > 0 means that A is Hermitian positive definite; 0 < A = A^H < I_m means that A and I_m - A are Hermitian positive definite and A^1/2 denotes square root of A = A^H > 0.

Now, we define complex Wishart, inverted complex Wishart, complex matrix variate beta type I, complex matrix variate beta type II distributions.

Definition 1.1. The m × m random Hermitian positive definite matrix X is said to have a complex Wishart distribution with parameters m, ν( > m) and Σ = Σ^H > 0, written as X ~ W_m(ν, Σ), if its p.d.f. is given by

{

_m(ν) det (Σ)^ν}^-1 det (X)^ν-m etr (– Σ^-1X), X = X^H > 0,

where m (a) is the complex multivariate gamma function defined by

Definition1.2.The m × m random Hermitian positive definite matrix Y is said to be distributed as inverted complex Wishart with parameters m, µ(> m) and Ψ = Ψ^H > 0, denoted by Y ~ I

W_m(µ, Ψ), if its p.d.f. is given by

{

_m (µ) }^-1 det(Ψ)^µ det(Y) ^–(µ+m) etr (-Y^-1Ψ), Y = Y^H > 0.

Note that if Y ~ IW_m(µ,Ψ), then Y^-1 ~ W_m(µ,Ψ^-1). The inverted complex Wishart distribution was first derived by Tan [17] as posterior distribution of Σ in a complex multivariate regression model. Later Shaman [16] studied some of its properties and applied it to spectral estimation. For m = 1, the inverted complex Wishart density slides to an inverted gamma density given by

{

_m (µ) }^-1ψ^µ y^-(µ+1)exp (–ψy^-1), y > 0, µ > 0, ψ > 0.

This distribution is designated by y ~ IG (µ, ψ).

Definition 1.3. The m × m random Hermitian positive definite matrix X is said to have a complex matrix variate beta type I distribution, denoted as X ~ (a, b), if its p.d.f. is given by

{

_m(a, b)}^-1 det (X)^a-m det (I_m – X)^b-m, 0 < X = X^H < I_m,

where a > m - 1, b > m-1 and _m(a, b) is the complex multivariate beta function defined by

Definition 1.4. The m × m random Hermitian positive definite matrix Y is said to have a complex matrix variate beta type II distribution, denoted as Y ~ (a,b), if its p.d.f. is given by

{

_m(a, b)}^-1 det (Y)^a-m det (I_m – Y)^-(a+b), Y = Y^H < 0,

where a > m - 1, and b > m - 1.

The complex matrix variate beta distributions arise in various problems in multivariate statistical analysis. Several test statistics in complex multivariate analysis of variance and covariance are functions of complex beta matrices. These distributions can be derived using complex Wishart matrices (Tan [18]). The relationship between beta type I and type II matrices can be stated as follows. If X ~ (a, b), then (I_m-X)^-1/2X (I_m- X)^-1/2 ~ (a, b). Further, if Y ~ (a, b), then Y^-1 ~ (b, a) and (I_m+ Y)^-1/2Y (I_m+ Y)^-1/2 ~ (a, b).

It is well known in the statistical literature that the complex matrix variate beta type I and type II, complex Wishart (Σ = I_m), and complex inverted Wishart (ψ = I_m) distributions are unitary invariant and residually independent (Khatri [9], Tan [18], Nagar, Bedoya and Arias [14], Bedoya, Nagar and Gupta [1], Goodman [4], Shaman [16]) and therefore belong to the class of unitary invariant and residual independent matrix (UNIARIM) distributions, _m, defined as follows (Khatri, Khattree and Gupta [12]).

Definition 1.5. The m × m random Hermitian positive definite matrix X is said to have an UNIARIM distribution if

(i) for any m×m unitary matrix U, distributions of X and U XU^H are identical, and

(ii) for any lower triangular factorization X = TT^H, T = (T_ij), T_ii (m_i × m_i), i = 1,..., k are independent, for any partition {m₁, m₂, ..., m_k} of m.

When X has UNIARIM distribution we will write X ∈ _m. Next, we state several properties of UNIARIM distributions.

Theorem 1.6. Let X ∈ _m. Partition X as X = , X₁₁ (q × q). Then X₁₁ and X_22·1 = X₂₂ - X₂₁ X₁₂ are independent, X₁₁∈ _q, and X_22·1∈ _m-q.

Theorem 1.7. Let X ∈ _m and Y ∈ _m be independent. Then, for any square root T of Y, the distribution of Z = TXT^H belongs to _m. Further, if T₁ and T₂ are two different square roots of Y, then and have identical distributions.

From the above theorem it follows that if U = (U₁, U₂), U_i (m × m_i), i = 1,2, m₁ + m₂ = m is a random unitary matrix independent of Z ∈ _m, then ∈ _m1and ∈ _m2are independent. Further, for c ∈ ^m, c ≠ 0, (i) c^H Zc/ c^Hc has same distribution as z₁₁, where Z = (z_ij), and (ii) c^Hc/ c^H Z^-1c has same distribution as 1/z¹¹ where Z^-1 = (z^ij). Furthermore, if E(Z), E(Z^-1), and E(Z^α), α an integer, exist, then E(Z) = aI_m, E(Z^-1) = bI_m and E(Z^α) = ^α I_m, where a = E(x₁₁y₁₁), b = E(x¹¹) E (y¹¹), and the constant ^α depends on moments of order less than or equal to α of X and Y.

Let Z^[i] = (z_αβ), 1 < α, β < i, i < m be a submatrix of Z = (z_αβ), 1 < α, β < m and Y = TT^H, X = RR^H be lower triangular factorizations. Then

where det(Z^[0]) = 1, are independent and E[det(Z)^α] = provided the expectations involved exist.

In Section 2, we give several UNIARIM distributions by using Theorem 1.7. Section 3 gives a number of results on random quadratic forms A^1/2Y(A^1/2)^H (A and Y are independent) and B^1/2Y(B^1/2)^H (B and Y are independent), where Y ~ IW_m(µ, I_m), A ~ (a, b) and B ~ (c, d) by exploiting the fact that they too belong to the class of UNIARIM distributions and using properties that are available for UNIARIM distributions. Finally, in Appendix, we give certain known results on complex matrix variate beta type I and type II, complex Wishart and inverted complex Wishart distributions, confluent hypergeometric functions and zonal polynomials of Hermitian matrix argument.

2 Generating UNIARIM Distributions

In the previous section it is stated that if X ∈ _m and Y ∈ _m are independent, then for any square root T of Y, the distribution of Z = T X T^H belongs to _m. In this section we exploit this property to generate a number of UNIARIM distributions.

Let A_i~ (a_i, b_i), B_i~ (c_i, d_i), i = 1, 2, A ~ (a, b), B ~ (c, d) and define

and

Z₄ = B^1/2A(B^1/2)^H

(A and B are independent).

Then, from Theorem 1.7, it follows that Z_i ∈ _m, i = 1, 2, 3, 4. From Cui, Gupta and Nagar [3], the p.d.f. of Z₁ is available as

where ₂

₁ is the Gauss hypergeometric function of Hermitian matrix argument (James [8]). The density of Z₂ and Z₃ can be shown to be (Gupta, Nagar and Vélez-Carvajal [7]),

and

respectively. Note that the distribution of Z₄ is same as that of Z₃.

Next, let X_i~ W_m(ν_i, I_m), i = 1, 2, Y_i ~ I

W_m(µ_i, I_m), i = 1, 2, X ~ W_m(ν, I_m), and Y ~ I

W_m(µ, I_m), where X₁ and X₂ are independent, Y₁ and Y₂ are independent, and X and Y are independent. Let

and

Z₈ = Y^1/2X(Y^1/2)^H.

Then, the p.d.f. of Z₅ is (Gupta and Nagar [6]),

where

_δ(·) is the Herz's type II Bessel function of Hermitian matrix argument. Since ~ W_m(µ₁, I_m), ~ W_m(µ₂, I_m), the p.d.f. of Z₆, obtained from the p.d.f. of Z₅, is given as

Note that Z₇ ~ (µ, ν) and Z₈ ~ (µ, ν) which follows from Tan [18] and Cui, Gupta and Nagar [3].

Further, define the following complex random matrices which again belong to the class

_m:

Z₉ = A^1/2X(A^1/2)^H,

Z₁₀ = X^1/2A(X^1/2)^H,

Z₁₁ = B^1/2X(B^1/2)^H,

Z₁₂ = X^1/2B(X^1/2)^H,

Z₁₃ = A^1/2Y(A^1/2)^H,

Z₁₄ = Y^1/2A(Y^1/2)^H,

Z₁₅ = B^1/2Y(B^1/2)^H,

and

Z₁₆ = Y^1/2B(Y^1/2)^H.

The random matrices Z₉ and Z₁₀ have the same density given by

where is the confluent hypergeometric function of Hermitian matrix argument. The random matrices Z₁₁ and Z₁₂ have the same density derived as

Similarly, the random matrices Z₁₃ and Z₁₄ have the same density derived in Theorem 3.1 and the random matrices Z₁₅ and Z₁₆ have the same density obtained in Theorem 3.3.

3 Properties of UNIARIM Distributions

In the previous sections we have discussed a number of properties of UNIARIM distributions and generated them by noting that if X ∈ _m and Y ∈ _m are independent, then for any square root T of Y, the distribution of Z = T X T^H belongs to _m.

Several properties, distributional results, expected values, etc. of random matrices defined in Section 2 are available in the literature. The distributional results and properties of Z₁, Z₂, Z₃ and Z₄ were studied by Gupta, Nagar and Vélez-Carvajal [7]. Results related to Z₅ and Z₆ were derived by Gupta and Nagar [6] and properties of Z₉, Z₁₀, Z₁₁ and Z₁₂ were obtained by Khatri, Khattree and Gupta [12].

In this section we derive distributions of Z₁₃ and Z₁₅ and study their properties. First we obtain the densities of Z₁₃ and Z₁₅.

Theorem 3.1. Let A and Y be independent, A ~ (a, b) and Y ~ IW_m(µ, I_m ). Then, the density of Z₁₃ = A^1/2Y (A^1/2)^H is derived as

where ₁

₁ is the confluent hypergeometric function of Hermitian matrix argument.

Proof. The joint density of A and Y is given by

where 0 < A = A^H < I_m and Y = Y^H > 0. Making the transformation Z₁₃ = A^1/2Y (A^1/2)^H with the Jacobian J (A, Y → A, Z₁₃) = det(A)^-m and integrating A we obtain

where Z₁₃ = > 0. Now, integration using (A.1) yields the result.

The distribution of Z₁₃ is designated by (µ, a, b).

Corollary 3.2. If x and y are mutually independent, x ~ B^I(a, b ) and y ~ IG(µ, 1), then xy ~ (µ, a, b). Further, the p.d.f. of z₁₃ = xy is given by

where ₁F₁ is the confluent hypergeometric function of scalar argument (Luke [13]).

Theorem 3.3. Let B and Y be independent, B ~ (c, d) and Y ~ IW_m(µ, I_m). Then, the density of Z₁₅ = B ^1/2Y (B^1/2)^H is derived as

Proof. The joint density of B and Y is given by

where B = B^H > 0 and Y = Y^H > 0. Transforming Z₁₅ = B^1/2Y (B^1/2)^H with the Jacobian J(B, Y → B, Z₁₃) = det(B)^-m and integrating B we obtain

where Z₁₅ = > 0. The desired result is obtained by using (A.2).

The above distribution will be denoted by (µ, c, d).

Corollary 3.4. If x and y are independent, x ~ B ^II(c, d) and y ~ IG(µ, 1), then xy ~

(µ, c, d). Further, the p.d.f. of z₁₃ = xy is given by

where ψ is the confluent hypergeometric function of scalar argument (Luke [13]).

Our next two results are of importance in deriving marginal distributions of certain submatrices of Z₁₃ and Z₁₅.

Theorem 3.5. Let A and Y be independent, A ~ (a, b) and Y ~ IW_m1+m2 (µ, I_m). Then,

are independent. Further,

are independent where A_11·2, Y_11·2, A_22·1 and Y_22·1are Schur complements of A₂₂, Y₂₂, A₁₁and Y₁₁, respectively.

Proof. From Theorem A.8 and Theorem A.9, A₁₁ , Y₁₁, A_22·1 and Y_22·1 are independent, A₁₁ ~ (a, b), Y₁₁ ~ I(µ - m₂,I_m1), A_22·1 ~ (a - m₁, b) and Y_22·1 ~ I(µ, I_m2). Now, application of Theorem 3.1 yields the desired result. The proof of the second part is similar.

Theorem 3.6. Let B and Y be independent, Y ~ I

( µ , I_m) and B ~

(c , d). Then,

are independent. Further,

are independent where Y_11·2, B_11·2, Y_22·1and B_22·1are Schur complements of Y₂₂, B₂₂, Y₁₁and B₁₁, respectively.

Proof. From Theorem A.8 and Theorem A.10, Y₁₁ , B₁₁, Y_22·1 and B_22·1 are independent, Y₁₁ ~ I(µ - m₂, I_m1), B₁₁ ~ (c, d - m₂) , Y_22·1 ~ I(µ, I_m2) and B_22·1 ~ (c - m₁, d). Now, application of Theorem 3.3 yields the desired result. The proof of the second part is similar.

3.1 Properties of Z ₁₃

In this section some properties of the random matrix Z₁₃ are derived using the fact that Z₁₃ belongs to the class of UNIARIM distributions.

(i) Let Z₁₃ = , Z₁₃₁₁ (m₁× m₁) , m₁ + m₂ = m. Then, using Theorem 1.6, Theorem 1.7 and Theorem 3.5, Z₁₃₁₁ and its Schur complement Z_1322·1 are independent, Z₁₃₁₁ ~ (µ - m₂, a, b) and Z_1322·1 ~ (µ, a - m₁, b). Further, Z₁₃₂₂ and its Schur complement Z_1311·2 are independent, Z₁₃₂₂ ~ (µ - m₁, a, b) and Z_1311·2 ~ (µ, a - m₂, b).

(ii) For a q × m complex non-random matrix C of rank q ( < m),

(CC^H)^-1/2CZ₁₃C^H (CC^H)^-1/2 ~ (µ - m², a, b)

and

(iii) For c ∈ ^m, c ≠ 0,

and

Note that the distributions of

do not depend on c. Thus, if y (m × 1) is a complex random vector independent of Z₁₃, and P(y ≠ 0) = 1, then it follows that

and

(iv) Let Z₁₃ = (z_13ij) and . Then, z_13ii ~ (µ - m + 1, a, b), i = 1,..., m and 1/ ~ (µ, a - m + 1, b), i = 1,..., m.

(v) Let = (z_13jk), 1 < j, k < i. Define

Then, the random variables v₁, ..., v_m are mutually independent and using Theorem A.11, Theorem A.7 and Corollary 3.1, v_i ~ (µ - i + 1, a, b), i = 1,..., m. Further the distribution of det(Z₁₃) is the same as that of .

3.2 Moments of functions of Z ₁₃

In this section we derive several expected values of scalar and complex matrix valued functions of the complex random matrix Z₁₃.

Using the representation Z₁₃ = A^1/2Y(A ^1/2)^H and (A.10)-(A.17), one obtains

Further, using (A.4), (A.5) and above expressions, the expected values of (tr Z₁₃)² and ² are evaluated as

and

Now, applying the results [n]₍₂₎ = n(n + 1), [n] = n(n - 1), [-n + m]₍₂₎ = (n - m)(n - m - 1), [-n + m] = (n - m)(n - m + 1), ₍₂₎(I_m) = m(m + 1)/2 and (I_m) = m(m - 1)/2 in the above expressions and simplifying, we obtain

and

Similarly, using (A.6)-(A.8), the expected values of (tr Z₁₃)³ and ³ are obtained as

and

respectively. Now, using the results [n]₍₃₎ = n(n + 1)(n + 2), [n]_(2,1) = n(n + 1)(n - 1), = n(n - 1)(n - 2), [-n + m]₍₃₎ = - (n - m) (n - m - 1)(n - m - 2), [- n + m]_(2,1) = -(n - m) (n - m - 1)(n - m + 1), = - (n - m) (n - m + 1)(n - m + 2), ₍₃₎(I_m) = m(m + 1)(m + 2)/6, _(2,1)(I_m) = 2 m(m² - 1)/3 and (I_m) = m(m - 1)(m - 2)/6 in the above expressions and simplifying, we obtain

where µ > m + 1, and

where a > m + 2. Similarly, the expected values of tr(Z₁₃) tr and tr are obtained as

and

respectively. Since, E = _αI_m, we have E[tr] = _αm. Thus, the coefficient of m in E[tr] is _α. Therefore, evaluating E[tr], E[tr], E[tr] and E[tr using the technique describe above, and computing the coefficients of m in the resulting expressions, one obtains

and

3.3 Properties of Z ₁₅

In this section we give several properties of Z₁₅.

(i) Let Z₁₅ = , Z₁₅₁₁ (m₁× m₁) , m₁ + m₂ = m. Then, using Theorem 1.6, Theorem 1.7 and Theorem 3.6, Z₁₅₁₁ and Z_1522·1 are independent, Z₁₅₁₁ ~ (µ - m₂, c, d - m₂) and Z_1522·1 ~ (µ, c - m₁, d). Further, Z₁₅₂₂ and Z_1511·2 are independent, Z₁₅₂₂ ~ (µ - m₁, c, d - m₁) and Z_1511·2 ~ (µ, c - m₂, d).

(ii) For a q × m complex non-random matrix C of rank q (< m),

(CC^H)^-1/2CZ₁₅C^H (CC^H)^-1/2 ~ (µ - m₂, c, d - m₂)

and

(CC^H)^-1/2 (C

C^H)^-1 (CC^H)^-1/2 ~ (µ , c - m₁, d)

(iii) If y (m × 1) is a non-random complex vector with y ≠ 0, or a complex random vector independent of Z₁₅ with P(y ≠ 0) = 1, then it follows that

and

(iv) Let Z₁₅ = (z_15ij) and = . Then, for i = 1,..., m, z_15ii ~ (µ - m + 1, c, d - m + 1), and 1/ ~ (µ, c - m + 1, d).

(v) Let = (z_15jk), 1 < j, k < i. Define

Then, the random variables v₁,..., v_m are mutually independent and using Theorem A.11, Theorem A.12 and Corollary 3.4, v_i ~ (µ - i + 1, c, d - i + 1), i = 1,..., m. Further, the distribution of det(Z₁₅) is the same as that of .

3.4 Moments of functions of Z ₁₅

This section deals with expected values of scalar and complex matrix valued functions of the complex random matrix Z₁₅.

Using the representation Z₁₅ = Y^1/2B(Y^1/2)^H ~ (µ, c, d), (A.10)-(A.13), (A.18), (A.19), and the technique of computing expected values given in Subsection 3.2, we obtain

and

Acknowledgements. The authors would like to thank the worthy referee for his comments and suggestions which improved the presentation of this article.

Received: 27/V/08. Accepted: 03/XII/08.

#768/08.

Appendix

The confluent hypergeometric function ₁

₁ of Hermitian matrix argument has the integral representation (James [8] and Chikuse [2]),

valid for all Hermitian X, Re(α) > m - 1 and Re(γ - α) > m - 1. The confluent hypergeometric function of m × m Hermitian matrix R is defined by (Chikuse [2]),

where Re(R) > 0 and Re(α) > m - 1.

Let

_κ(X) be a zonal polynomial of an m × m Hermitian matrix X corresponding to the partition κ = (k₁, ..., k_m), k₁ + ... + k_m = k, k₁> ... > k_m > 0. Then, for small values of k, explicit formulas for _κ(X) are available as (James [8], Khatri [11]),

Also, substituting X = I_m in (A.3)-(A.8), it is easy to see that ₍₁₎(I_m) = m, ₍₂₎(I_m) = m(m + 1)/2, (I_m) = m(m - 1)/2, ₍₃₎(I_m) = m(m + 1)(m + 2)/6, _(2,1)(I_m) = 2m(m² - 1)/3, (I_m) = m(m - 1)(m - 2)/6. The complex generalized hypergeometric coefficient [a]_κ is defined as

with (a)_r = a(a + 1)...(a + r - 1) = (a)_r-1(a + r - 1) for r = 1,2, ..., and (a)₀ = 1. Further, substituting appropriately in (A.9), it is easy to observe that [a]₍₂₎ = a(a + 1), = a(a - 1), [a]₍₃₎ = a(a + 1)(a + 2), [a]_(2,1) = a(a + 1)(a - 1) and = a(a - 1)(a - 2).

If Y ~ I

W_m(µ, Ψ), A ~ (a, b) and B ~ (c, d) then (Bedoya, Nagar and Gupta [1], James [8], Khatri [10], Nagar and Gupta [15], Shaman [16]),

and

where R is an m × m Hermitian matrix.

Theorem A.7. Let A ~ W_m(ν, I_m) and A = TT^H, where T = (t_ij) is a complex lower triangular matrix with t_ii > 0 and t_ij = t_1ij +

t_2ij, j < i. Then, t_ij, 1< j < i < m are independently distributed,

~ G(ν - i + 1), 1 < i < m and t_ij ~ N(0,1), 1< j < i < m.

Proof. See Goodman [4].

The univariate gamma distribution denoted by G(a) is defined by the p.d.f.

{Γ (a)}^-1 exp(-x)x^a-1, x > 0, a > 0.

Theorem A.8. Let Y ~ IW_m(µ,Ψ) and partition Y and Ψ as

where Y₁₁ and Ψ₁₁ are q × q matrices. Then, Y₁₁ and its Schur complement Y_22·1 are independent, Y₁₁ ~ IW_q(µ - m + q, Ψ₁₁) and Y_22·1 ~ IW_m-q(µ, Ψ_22·1). Further, Y₂₂ and its Schur complement Y_11·2 are independent, Y₂₂ ~ IW_m-q(µ - q, Ψ₂₂) and Y_11·2 ~ I W_q(µ, Ψ_11·2).

Proof. See Tan [17].

Theorem A.9. Let A ~ (a, b) and partition A as

Then,

(i) A₁₁ and its Schur complement A_22·1 are independent, A₁₁ ~ (a, b) and A_22·1 ~ (a - q, b) and

(ii) A₂₂ and its Schur complement A_11·2 are independent, A₂₂ ~ (a, b) and A_11·2 ~ (a - m + q, b).

Proof. See Tan [18].

Theorem A.10. Let B ~ (c, d) and partition B as

Then, (i) B₁₁and its Schur complement B_22·1are independent, B₁₁ ~ (c, d - m + q) and B_22·1 ~ (c - q, d) and (ii) B₂₂and its Schur complement B_11·2are independent, B₂₂ ~ (c, d - q) and B_11·2 ~ (c - m + q, d).

Proof. See Tan [18].

Theorem A.11. Let A ~ (a, b) and A = TT^H, where T = (t_ij) is a complex lower triangular matrix with positive diagonal elements. Then, are independently distributed, ~ B^I(a - i + 1, b), i = 1,..., m.

The univariate beta type I distribution, denoted by B^I(a, b), is defined by the p.d.f.

{B(a, b)}^-1x^a-1 (1 - x)^b-1, 0 < x < 1,

where B(a, b) = .

Theorem A.12. Let B ~ (c, d) and B = TT^H, where T = (t_ij) is a complex lower triangular matrix with positive diagonal elements. Then, are independently distributed, ~ B^II(c - i + 1, d - m + i), i = 1,..., m.

The univariate beta type II distribution, denoted by B^II(a, b), is defined by the p.d.f.

{B(a, b)}^-1x^a-1 (1 + x)^-(a+b), x > 0.

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*

Supported by the Comité para el Desarrollo de la Investigación, Universidad de Antioquia research grant no. IN515CE.

Publication Dates

• Publication in this collection
  30 Mar 2009
• Date of issue
  2009

History

• Received
  27 May 2008
• Accepted
  03 Dec 2008