C. G. Khatri

Testing some covariance structures under a growth curve model

This paper considers three types of problems: (i) the problem of independence of two sets, (ii) the problem of sphericity of the covariance matrix [Sigma], and (iii) the problem of intraclass model for the covariance matrix [Sigma], when the column vectors of X are independently distributed as multivariate normal with covariance matrix [Sigma] and E(X) = B[xi]A,A and B being given matrices and [xi] and [Sigma] being unknown. These problems are solved by the likelihood ratio test procedures under some restrictions on the models, and the null distributions of the test statistics are established.

Test for a specified signal when the noise convariance matrix is unknown

In the univariate case it is well known that the one sided t test is uniformly most powerful for the null hypothesis against all one sided alternatives. Such a property does not easily extend to the multivariate case. In this paper, a test derived for the hypothesis that the mean of a vector random variable is zero against specified alternatives, when the covariance matrix is unknown. This test depends on the given alternatives and is more powerful than Hotellings T2. The results are derived both for real and complex vector observations and under normal and spherical distributions. The properties of the proposed tests are investigated in detail when a single alternative is specified.

Some asymptotic inferential problems connected with elliptical distributions

Asymptotic confidence bounds on the location parameters of the linear growth curve, asymptotic distribution of the canonical correlations and asymptotic confidence bounds on the discriminatory value for the linear discriminant function are established when a set of independent observations are taken from an elliptical distribution (or from a distribution possessing some properties on the moments).

Robustness study for a linear growth model

For the linear growth curve model introduced by Potthoff and Roy (Biometrika 51 (1964), 313-326), various likelihood ratio tests and some ad hoc tests are available for the location and scale parameters on the basis of normally distributed error components. We study these tests under the assumption of elliptical (or spherical) distributions of the error components and show that these tests are null robust; and the tests for the location parameters are shown to be unbiased. These results are extended to the linear growth model in complex variables having elliptical (or spherical) complex distributions.

On Monotonicity of the Modified Likelihood Ratio Test for the Equality of Two Covariances

For testing the hypothesis of equality of two covariances ([Sigma]1 and [Sigma]2) of two p-dimensional multivariate normal populations, it is shown that the power function of the modified likelihood ratio test increases as [lambda]1 increases from one and [lambda]r decreases from one where [lambda]1 > ... > [lambda]r > 0 are the distinct characteristic roots of [Sigma]1[Sigma]2-1, r

On characterization of gamma and multivariate normal distributions by solving some functional equations in vector variables

The main aim of this paper is to solve the functional equations and k > 1, in vector variables t1,...,tk satisfying the condition ti = ([Sigma]j = 1m tij2)1/2