Some New Properties of Wishart Distribution

Transcription

1 Applied Mathematical Sciences, Vol., 008, no. 54, Some New Properties of Wishart Distribution Evelina Veleva Rousse University A. Kanchev Department of Numerical Methods and Statistics 8 Studentska str., room 1.44, 7017 Rouse, Bulgaria eveleva@abv.bg Abstract We obtain the exact distributions of determinants and quotient of determinants of some submatrices of a Wishart distributed random matrix. We show an application of the obtained representations in testing hypotheses concerning the covariance matrix of multivariate normal distribution. Mathematics Subject Classification: 6H10 Keywords: Wishart distribution, covariance matrix, correlation matrix, testing hypotheses 1 Introduction Let W be a random matrix with Wishart distribution W n m, I n, where m>n and I n is the identity matrix of order n. The matrix W can be represented as a product see [3] W = DVD, 1 where D is a diagonal random matrix, D = diag τ 1,..., τ n and V =ν i,j is a symmetric random matrix with units on the main diagonal. The random variables τ i, i =1,...,n are mutually independent, independent of ν i,j,1 i<j n and have chi - square distribution, τ i χ m, i =1,...,n. The joint density function of ν i,j,1 i<j n has the form fy i,j, 1 i<j n = [ Γ m ] n Γ m det Y m n 1, n

2 674 E. Veleva 1 y 1n if Y is a positive definite matrix, Y = By Γ n α is denoted y 1n 1 the multivariate Gamma function, Γ n α =π nn 1 4 ΓαΓ α 1...Γ α n 1. Ignatov and Nikolova in [3], denote by ψm, n the joint distribution of ν i,j, 1 i<j nwith density function of the form. For n = the distribution ψm, n is a univariate distribution with density function fy = Γ m 1 Γ m Γ 1 1 y m 3, y 1, 1. Definition 1.1 A random matrix V is said to have distribution ψ n m with parameters n, m, n<m, written as V ψ n m, ifv is a symmetric matrix of order n with units on the main diagonal and the joint distribution of the elements above the main diagonal is ψm, n. Let R be the sample correlation matrix for a sample of size m + 1 from n - variate normal distribution N n μ, Σ with unknown mean vector μ. Suppose that Σ is a diagonal matrix with unknown positive diagonal elements. Then the distribution of the sample correlation matrix R is ψ n m see [7]. In the present paper we obtain some properties of the distribution ψ n m of the matrix V in 1. By equality 1, we get the corresponding properties of Wishart distribution. In an example, we show an application of the obtained representations in testing hypotheses concerning the covariance matrix of multivariate normal distribution. Preliminary Notes Let P n, R be the set of all real, symmetric, positive definite matrices of order n. Let us denote by Dn, R the set of all real, symmetric matrices of order n, with positive diagonal elements, which off-diagonal elements are in the interval -1,1. There exist a bijection h : Dn, R P n, R, considered in [4], [5] and [6]. The image of an arbitrary matrix X =x i,j from Dn, R by the bijection h, is a matrix Y =y i,j from P n, R, such that y j,j = x j,j, j =1,...,n, 3 y 1,j = x 1,j x1,1 x j,j, j =,...,n, 4

3 Some new properties of Wishart distribution 675 [ y i,j = i 1 r 1 x i,i x j,j x r,i x r,j r=1 q=1 i 1 + x i,j q=1 1 x q,i 1 x q,j The next Proposition can be found in [4] and [6]. 1 x q,i 1 x q,j ], i<j n. 5 Proposition.1 Let ξ =ξ i,j be a random symmetric matrix of order n with units on the main diagonal. Suppose that ξ i,j are independent and ξ i,j ψm i +1,, 1 i<j n. LetV be the matrix V = hξ, where h is the bijection, defined by 3-5. Then the matrix V has distribution ψ n m. Let A = a i,j be a real square matrix of order n. Let α and β be nonempty subsets of the set {1,...,n}. ByA[α, β] we denote the submatrix of A, composed of the rows with numbers from α and the columns with numbers from β. Denote by α c the complement of the set α in {1,...,n}, i.e. α c = {1,...,n}\α. For the matrix A[α c,β c ] we use the notation Aα, β. When β α, A[α, α] is denoted simply by A[α] and Aα, α byaα. Let X Dn, R and Y = hx, where h is the bijection, defined by 3-5. We get interesting relations between the elements of the matrices X and Y see [4], [5] and [6]: x i,j = det Y [{1,...,i}, {1,...,i 1,j}] det Y [{1,...,i}] det Y [{1,...,i 1,j}], i<j n; 6 1 x 1,j 1 x,j...1 det Y [{1,...,i,j}] x i,j = y j,j det Y [{1,...,i}], 1 i<j n; 7 i det Y [{1,...,i,j}] =x 1,1...x i,i x j,j 1 x k,s 1 x k,j, 1 k<s i 8 3 Main Results 1 i<j n. Theorem 3.1 Let V ψ n m. Then for all integer i and j, i<j n det V[{1,...,i}, {1,...,i 1,j}] ζ 1...ζ i 1 ζi ζ i+1,

4 676 E. Veleva where the random variables ζ s, s =1,...,i+1 are independent, ζ 1 ψm i +1, and ζ s, s =,...,i+1 are beta distributed, ζ s β m s+1, s 1, s =,...,i, ζ i+1 β m i+1, i 1. Proof. Using Proposition.1 and the representations 6 and 8, for i<j n we have det V[{1,...,i}, {1,...,i 1,j}] = ξ i,j det V[{1,...,i}] det V[{1,...,i 1,j}] i 1 i 1 = ξ i,j 1 ξk,s 1 ξk,i 1 ξk,j i 1 = ξ i,j 1 k<s i 1 s 1 s= i 1 i 1 1 ξk,s 1 ξk,i 1 ξk,j. 9 Denote by ζ s, s =1,...,i+ 1 the random variables s 1 i 1 ζ 1 = ξ i,j, ζ s = 1 ξk,s, s=,...,i, ζ i+1 = 1 ξk,j. The random variables ξ i,j, 1 i < j n are independent, therefore ζ s, s =1,...,i+1 are independent, too. Since ξ i,j ψm i+1,, 1 i<j n, it can be shown that 1 ξi,j β m i, 1,1 i<j n. It is known, that if π 1 and π are independent random variables, π 1 βα, γ, π βα + γ,δ, then π 1 π βα, γ + δ. Consequently, m i 1 ξi 1,j 1 ξ i,j β, 1,..., m i 1 ξ1,j...1 ξi,j β, i. 10 Corollary 3.1 Let V ψ n m. Then for all integer i and j, i<j n, det V[{1,...,i}, {1,...,i 1,j}] ζ 1 ζ ζ 3 det V[{1,...,i 1}] and det V[{1,...,i}, {1,...,i 1,j}] det V[{1,...,i}] ζ 1 ζ ζ 3, where the random variables ζ 1, ζ and ζ 3 are independent, ζ 1 ψm i+1,, ζ,ζ 3 β m i+1, i 1.

5 Some new properties of Wishart distribution 677 Proof. 8. The Corollary follows from Theorem 3.1 and the representation Lemma 3.1 Let V ψ n m. Leti, j be arbitrary integers, 1 i<j n. Suppose that we interchange the places of the i th and j th rows in V and then interchange the places of the i th and j th columns. Then the obtained matrix V is distributed again ψ n m. Proof. The Lemma follows from the properties of determinants and positive definite matrices. Theorem 3. Let V ψ n m, n>. Then for 1 i<j n det V{i}, {j} 1 j i 1 ζ 1 ζ...ζ n ζn 1 ζ n, where ζ s, s =1,...,nare independent, ζ 1 ψm n+,, ζ s β m s+1, s 1, s =,...,n 1 and ζ n β m n+, n. Proof. Let us put the i th and j th rows of V after its n th row; the i th and j th columns after the n th column. We get a new matrix V, V = V{i, j} V{i, j}, {i} c V{i, j}, {j} c V{i} c, {i, j} 1 ν i,j V{j} c, {i, j} ν i,j 1, 11 where α c denotes the set {1,...,n}\α. Applying Lemma 3.1 several times we get that V ψ n m. It is not difficult to see that On the other hand, det V{i}, {j} = 1 j i 1 det V {n 1}, {n}. 1 det V {n 1}, {n} = det V {n}, {n 1} = det V [{1,...,n 1}, {1,...,n,n}]. 13 Now applying Theorem 3.1, we complete the proof. Corollary 3. Let V ψ n m, n>. Then the elements ν i,j, 1 i<j n of the inverse matrix V 1 are identically distributed ν i,j ζ 1 1, 1 ζ1 ζ ζ 3 where the random variables ζ 1,ζ,ζ 3 are independent, ζ 1 ψm n+,,ζ,ζ 3 β m n+, n.

6 678 E. Veleva Proof. Let V be the matrix in 11. Then det V = det V. From 1 and 13 we get ν i,j j i det V{i}, {j} = 1 = det V {n 1}, {n} det V det V = det V [{1,...,n 1}, {1,...,n,n}]. det V The matrix V is distributed ψ n m. Hence, from Proposition.1 and equalities 6 and 8 it follows that det V [{1,...,n 1}, {1,...,n,n}] det V = ξ n 1,n det V [{1,...,n 1}] det V [{1,...,n,n}] det V n ξ n 1,n 1 ξk,s n 1 ξk,n 1 1 ξk,n = 1 k<s n 1 k<s n 1 = n 1 1 ξk,s 1 ξ k,n ξ n 1,n 1 1 ξn 1,n n n 1 ξk,n 1, 1 ξk,n where ξ i,j,1 i<j n are independent and ξ i,j ψm i +1,. Let us denote by ζ 1, ζ and ζ 3 the random variables n n ζ 1 = ξ n 1,n, ζ = 1 ξk,n 1, ζ 3 = 1 ξk,n. Then ζ 1,ζ,ζ 3 are independent and ζ 1 ψm n +,. From 10 we have that ζ,ζ 3 β m n+, n. Using the representation 1, the statements, proved for the distribution ψ n m ofv, can be easily reformulated for the distribution W n m, I n. We shall use several times the following known property of the Gamma and Beta distributions. Proposition 3.1 Let π 1 and π be independent random variables, π 1 is Gamma distributed Gα, λ and π β α δ, δ. Then π 1 π Gα δ, λ. Theorem 3.3 Let W W n m, I n. Then for i<j n det W[{1,...,i}, {1,...,i 1,j}] ζ 1...ζ i ζi+1 ζ i+,

7 Some new properties of Wishart distribution 679 where the random variables ζ s, s =1,...,i+ are independent, ζ 1 ψm i +1,, ζ s χ m s +, s =,...,i+1, ζ i+ χ m i +1. Proof. From the representation 1 it can be seen that det W[{1,...,i}, {1,...,i 1,j}] = τ 1...τ i 1 τi τ j det V[{1,...,i}, {1,...,i 1,j}]. The random variables τ 1,...,τ n are independent and τ i χ m. From Theorem 3.1 we have that det V[{1,...,i}, {1,...,i 1,j}] ζ 1...ζ i 1 ζi ζ i+1, where ζ s, s = 1,...,i + 1 are independent, ζ 1 ψm i +1,, ζ s β m s+1, s 1, s =,...,i, ζi+1 β m i+1, i 1. Now applying Proposition 3.1, we complete the proof. Corollary 3.3 Let W W n m, I n. Then for all integer i and j, i< j n and det W[{1,...,i}, {1,...,i 1,j}] det W[{1,...,i 1}] det W[{1,...,i}, {1,...,i 1,j}] det W[{1,...,i}] ζ 1 ζ ζ 3 ζ 1 ζ ζ 3, where ζ 1,ζ,ζ 3 are independent, ζ 1 ψm i +1,, ζ,ζ 3 χ m i +1. Proof. From the representation 1 it can be seen that det W[{1,...,i}, {1,...,i 1,j}] det W[{1,...,i 1}] = det V[{1,...,i}, {1,...,i 1,j}] τ i τ j, det V[{1,...,i 1}] det W[{1,...,i}, {1,...,i 1,j}] det W[{1,...,i}] = τj det V[{1,...,i}, {1,...,i 1,j}] τ i det V[{1,...,i}] Now using Corollary 3.1 and Proposition 3.1, the Theorem follows. Theorem 3.4 Let W W n m, I n. Then for all integer i and j, 1 i< j n det W{i}, {j} 1 j i 1 ζ 1 ζ...ζ n 1 ζn ζ n+1, where the random variables ζ k, k =1,...,n+1 are independent, ζ 1 ψm n +,, ζ k χ m k +, k =,...,n, ζ n+1 χ m n +..

8 680 E. Veleva Proof. From the representation 1 it can be seen that det W{i}, {j} =τ 1...τ i 1 τ i+1...τ j 1 τ j+1...τ n τi τ j det V{i}, {j}. According to Theorem 3., det V{i}, {j} 1 j i 1 ζ 1 ζ...ζ n ζn 1 ζ n, where ζ s, s =1,...,nare independent, ζ 1 ψm n+,, ζ s β m s+1, s 1, s =,...,n 1 and ζ n β m n+, n. Now applying Proposition 3.1, we complete the proof. Corollary 3.4 Let W W n m, I n. Then the element w i,j on i th row and j th column of the inverse matrix W 1, 1 i<j n, is distributed w i,j ζ 1 1, 1 ζ1 ζ ζ 3 where the random variables ζ 1, ζ, ζ 3 are independent, ζ 1 ψm n +,, ζ,ζ 3 χ m n +. Proof. From the representation 1 it can be seen that w i,j = 1j i det W{i}, {j} det W = 1j i τi τ j det V{i}, {j} det V = νi,j τi τ j, where ν i,j is the i, j element of the matrix V 1. Now applying Corollary 3. and Proposition 3.1, we complete the proof. The next Proposition see [4], [6] follows from Proposition.1, the representation 1 and equalities 3-5. Proposition 3. Let ξ =ξ i,j be a random symmetric matrix of order n. Suppose that ξ i,j, 1 i j n are independent, ξ i,j ψm i +1, for 1 i<j n and ξ i,i χ m, i =1,...,n. Let W be the matrix W = hξ, where h is the bijection, defined by 3-5. Then the matrix W has distribution W n m, I n. Let W W n m, I n. From Proposition 3. we have that W = hξ, where ξ =ξ i,j is a random symmetric matrix, ξ i,j ψm i+1, for 1 i<j n and ξ i,i χ m, i =1,...,n. From equation 8 for i = n 1 and j = n, we get that det W = ξ 1,1...ξ n,n 1 ξk,s k<s n

9 Some new properties of Wishart distribution 681 Using 3 we obtain that trw = ξ 1,1 + + ξ n,n. 15 The Wishart distribution arises frequently in multivariate statistical analysis. In an example below we show an application of the obtained representations. Example 3.1 Let x i =x i,1,...x i,n t, i =1,...,m be a random sample of size m m >nfromn - variate normal distribution with unknown mean vector μ and unknown positive definite covariance matrix Σ. We are interested in testing the null hypothesis H 0 : Σ = Σ 0 against the alternatives H 1 : Σ Σ 0, where Σ 0 is a fixed positive definite matrix. By a linear transformation of the observations see [], we can reduce the task to testing the hypotheses H 0 : Σ = I n against H 1 : Σ I n. Let us denote the transformed observations by y i, i =1,...,m. The likelihood ratio criterion for testing H 0 : Σ = I n is given by see [] λ = e mmn det S m e 1 trs, 16 where S = n m y i ȳy i ȳ t, ȳ = 1 y m i. Under H 0, the distribution of i=1 i=1 the sample covariance matrix S is W n m 1, I n. Using 14 and 15, λ can be written in the form λ = e mmn 1 k<s n 1 ξ k,s m n ξ m k,k e 1 ξ k,k, where ξ k,s, 1 k s n are independent, ξ k,s ψm k, for k s and ξ k,k χ m 1. From Proposition.1 and equality 8 we have that if V ψ n m 1 then det V = 1 ξk,s. From 10 it follows that 1 k<s n 1 k<s n 1 ξ k,s ζ 1...ζ n 1, where ζ i, i =1,...,n 1 are independent and ζ i β m i 1, i. Consequently, e λ ζ 1...ζ n 1 mmn m ν1...ν n, 17

10 68 E. Veleva where ν i, i = 1,...,n are mutually independent, independent of ζ i, i = 1,...,n 1 and are identically distributed, ν i ξ m e 1 ξ, ξ χ m 1. The relation 17 is an exact representation of λ as a product of independent random variables. Using 16, for simulating of each value of λ we have to generate a n n covariance matrix and calculate its determinant. With the representation 17, we get each value of λ by n 1 independent realizations of chi-square and beta random variables. References [1] J.E. Gentle, Matrix Algebra. Theory, Computations, and Applications in Statistics, Springer Science+Business Media, LLC, New York, 007. [] N.C. Giri, Multivariate Statistical Analysis, Marcel Dekker Inc., New York, 004. [3] T.G. Ignatov, A.D. Nikolova, About Wishart s Distribution, Annuaire de l Universite de Sofia St.Kliment Ohridski, Faculte des Sciences Economiques et de Gestion 3 004, [4] E.I. Veleva, Positive definite random matrices, Comptes Rendus de L Academie Bulgare des Sciences, , [5] E.I. Veleva, Uniform random positive definite matrix generating, Math. and Education in Math., , [6] E.I. Veleva, A representation of the Wishart distribution by functions of independent random variables, Annuaire de l Universite de Sofia St.Kliment Ohridski, Faculte des Sciences Economiques et de Gestion, 6 007, [7] E.I. Veleva, Test for independence of the variables with missing elements in the same column of the empirical correlation matrix, Serdica Math. J., , Received: June, 008