On the Complex Analogues of $T^2$- and $R^2$-Tests

Abstract

Let $\xi$ be a complex Gaussian random variable with mean $E(\xi) = \alpha$ and Hermitian positive definite complex covariance matrix $\Sigma = E(\xi - \alpha)(\xi - \alpha)^\ast$, where $(\xi - \alpha)^\ast$ is the adjoint of $(\xi - \alpha)$. Its probability density function is given by \begin{equation*}\tag{(0.1)}p(\xi \mid \alpha, \Sigma) = \pi^{-p}(\det \Sigma)^{-1} \exp \lbrack -(\xi - \alpha)^\ast\Sigma^{-1}(\xi - \alpha)\rbrack,\end{equation*} with $E(\xi - \alpha)(\xi - \alpha)' = 0$. Write \begin{equation*} \Sigma =\begin{pmatrix}\Sigma_{11} & \Sigma_{12} \\ \Sigma^\ast_{12} & \Sigma_{22}\end{pmatrix},\end{equation*} where $\Sigma_{22}$ is the $(p - 1) \times (p - 1)$ lower right-hand submatrix of $\Sigma$. Goodman (1963) found the maximum likelihood estimate of $\Sigma$ and $\rho^2 = \Sigma_{12}\Sigma^{-1}_{22}\Sigma^\ast_{12}/\Sigma_{11}$ when $\alpha = 0$ and also found the distributions of these estimates. The problems considered here are of (i) testing the hypothesis $H_{01} : \alpha = 0$ that the mean of the vector $\xi$ is 0 against the alternative $H_1 : \alpha^\ast\Sigma^{-1}\alpha > 0$ and (ii) of testing the hypothesis $H_{02} : \Sigma_{12} = 0$ that the first component of $\xi$ is independent of the others against the alternative $H_2 : \rho^2 > 0$. Since likelihood ratio test has some optimum properties and has been found satisfactory for similar problems in the real case, we find the likelihood ratio tests of these problems and show that these tests possess certain optimum properties which are counterparts of the real case. These results will be presented in Sections 3 and 4. Section 1 deals with some known results of complex matrix algebra. In Section 2, we will prove some preliminary results which are useful for complex Gaussian statistical analysis. For an application of these results the reader is referred to Goodman (1963). It may be remarked here that the likelihood ratio test is invariant under all transformations which leave the problem invariant and may be obtained from the densities of maximal invariant under the null hypothesis and the alternative.