Sankhya: The Indian Journal of Statistics

1995, Volume 57, Series A, Pt. 3, pp. 342--359

LAPLACE-WISHART DISTRIBUTIONS AND COCHRAN THEOREMS

By

  CHI SONG WONG, University of Windsor

and

TON GHUI WANG, New Mexico State University

SUMMARY.   Let E, V be n-, p-dimensional inner product spaces over the real field, $\mathcal{L}(V, E)$ be the set of all linear maps of V into $E, A \in \mathcal{L}(E, E)$ and $\sum \in \mathcal{L}(V, V)$ be nonnegative definite, Y be a normal random operator in $\mathcal{L}(V, E)$ with mean $\mu_Y=\mu$ covariance $\sum_Y=A \otimes \sum \ne 0, r(\sum) 1, i \in {1, 2, ..., \mathcal{l}}, B_i, C_i \in \mathcal{L}(V, E), D_i \in \mathcal{L}(V, V)$, and $W_i$ be self adjoint in $\mathcal{L}(E, E)$. The distribution of Q1-Q2 with Q1, Q2 being independent Wishart operators $W_p(m_1, \sum, \lambda_1), W_p(m_2, \sum, \lambda_2)$ in $\mathcal{L}(V, V)$ is referred to as $DW_p(m_1, m_2, \sum, \lambda_1, \lambda_2). It is proved that $(*) : " {Q_{i}(Y)}$ with $Q_{i}(Y)= Y\prime W_{i} Y+ B_{i}\prime Y+Y\prime C_i +D_i$ is an independent family of $DW_{p}(m_{1i}, m_{2i}, \sum, \lambda{1i}, \lambda_{2i}) random operators $Q_{i}(Y)"$ if and only if for any distinct i, j $(i) AW_{i} AW_{i} AW_{i} A=AW_{i}A, (ii) tr(AW_{i})=m_{1i}- m_{2i}, tr(AW_{i})_{2}=m_{1i}+ m_{2i}, (iii) AB_{i}=AC_{i}, (iv) \lambda{1i}- \lambda_{2i}=Q_{i}(\mu)=(B_i+W_{i\mu}), (v) \lambda{1i}+ \lambda_{2i}=(B_i + W_{i \mu})\prime A(B_i + W_{i \mu})=(B_i + W_{i \mu})\prime AW_{i}AW_{i}A(B_{i} + W_{i \mu}), (vi) AW_{i}AW_{j}A=0 (vii)  (B_i + W_{i \mu})\prime A(B_j + W_{j \mu})=0." A necessary and sufficient condition for (*) is also obtained for the general case where $\sum_Y$ need not be of the form $A \otimes \sum$. This combines certain results of Tan (1975, 1976), Mathai (1993), and the authors into more general one.

AMS (1980) subject classification.   Primary 62H05, secondary 62H10.

Key words and phrases. Cochran theorem, inner product, Kronecker product, Laplacian distribution, moment generating function, multivariate normal operator,  outer product, quadratic random operator, spectral decomposition, Wishart distribution.

Full paper (PDF)

This article in mathematical reviews.