Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series B, Pt. 3, 407--432

COCHRAN THEOREMS TO VECTOR ELLIPTICALLY CONTOURED DISTRIBUTIONS OF GAMMA TYPE

By

CHI SONG WONG University of Windsor, Windsor
and
HUA CHENG Syracuse University, Syracuse

SUMMARY. For a normal n x p matrix Y with mean m and covariance $\bA \otimes \Sigma$, $(\bY'\bW_i\bY)^k_{i=1}$, with non-negative definite Wi, is independent Wishart $\bW_p(m_i, \Sigma, \eta_i)$ if and only if AWiAWi =AWi, mi = r (AWi),WibAWj = 0 for i!=j, and hi = m' Wi = m' WiAWim. This Cochran theorem is generalized to a vector-elliptical Y induced by a gamma (a, b) random variable r2. Applications are given for normal and vector-elliptical growth curve models Y with $\mu \in \bS$, where S is the linear span of x1 x2' with $x_i \in \bS_i$ and Si is of the regression form
E = { X b : K'b = Mvf for some v}. Tests for T'm =0 are obtained through certain decreasing convex functions; each test statistic involves the projection P of $\Re^n$ onto E :

P = X( X' X)- X' - X( X' X)- K (K'( X' X)-K )- K'( X' X)- X'
+X( X' X)- K( K'( X' X)- K)- M'[ M( K'( X' X)- K)- M']-
x M( K'( X' X)- K)- K'( X' X)- X'.

AMS (1991) subject classification. Primary 62H05, secondary 62H10.

Key words and phrases. Cochran theorem, decreasing convex function, estimable function, gamma random variable, generalized inverse, generalized Wishart distribution, growth curve model, idempotent, multivariate normal and elliptically contoured distributions, Moore-Penrose inverse, quadratic form, spectral theorem, testing hypothesis, zonal polynomial.

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