Cochran theorems for a multivariate elliptically contoured model

• Published in: Journal of statistical planning and inference 1995, Vol.43 (1), p.257-270
• Main Authors: Tonghui, Wang, Chi Song, Wong
• Format: Article
• Language: English
• Subjects: Characteristic function; Exact sciences and technology; Generalized Wishart distribution; Mathematics; Multivariate analysis; Multivariate elliptically contoured distribution; Probability and statistics; Sciences and techniques of general use; Statistics; Stochastic spherical representation
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/0378-3758(94)00025-Q

Let Y be a multivariate elliptically contoured n × p random matrix with an MEC n × p ( μ,∑ Y , ø) distribution and P( Y = μ) < 1. Let i = 1,2, …, L, W i be an n × n nonnegative definite (n.n.d.) matrix, and m i ϵ {1,2,…}, Q i(Y) = (Y − μ)′ W i(Y − μ) and ∑( ≠ 0) be a p × p n.n.d. matrix. Then (a) and (b) are equivalent: 1. (a) ( Q 1( Y), Q 2( Y), …, Q L ( Y)) ∼ GW p ( m 1, m 2, …, m L ; n − ∑ L i = 1 m i ; ∑; ø). 2. (b) For some n.n.d. n × n matrix A and for any distinct i, j = 1,2, …, L, 2.1. (i) ( W i ⊗ I p ) (∑ Y − A ⊗ ∑) ( W i ⊗ I p ) = 0, 2.2. (ii) AW i AW i = AW i , r( AW i ) = m i , 2.3. (iii) W i AW j = 0 and 2.4. (iv) ( W i ⊗ I p )∑ Y ( W j ⊗ I p ) = 0. Moreover if r( W i ) = m i for each i, then (a′) and (b′) are equivalent: 1. (a′) ( Q 1( Y), Q 2( Y), …, Q L ( Y)) ∼ GW p ( r( W 1), r( W 2), …, r( W L ); n − ∑ L i = 1 r( W i ); ∑; ø). 2. (b′) For any distinct i, j = 1, 2, …, L, 2.1. (i) ( W i ⊗ I p ) ∑ Y ( W i ⊗ I p ) = W i ⊗ ∑ and 2.2. (ii) ( W i ⊗ I p ) ∑ Y ( W i ⊗ I p ) = 0. Applications are given for certain MANOVA models and multivariate components of variance models. The above results are general in that ∑ Y is no longer required to have the form A ⊗ ∑.