On the Complex Analogues of T²- and R²-Tests

Let be a complex Gaussian random variable with mean E () = and Hermitian positive definite complex covariance matrix = E (-) (-) ^, where (-) ^ is the adjoint of (-). Its probability density function is given by equation* (0. 1) p (, ) = ^-p () ^-1 - (-) ^^-1 (-), equation* with E (-) (-) ' = 0. Write equation* =pmatrix₁₁ & ₁₂ \\ ^₁₂ & ₂₂pmatrix, equation* where ₂₂ is the (p - 1) (p - 1) lower right-hand submatrix of. Goodman (1963) found the maximum likelihood estimate of and ² = ₁₂^-1₂₂^₁₂/₁₁ when = 0 and also found the distributions of these estimates. The problems considered here are of (i) testing the hypothesis H₀₁: = 0 that the mean of the vector is 0 against the alternative H₁: ^^-1 > 0 and (ii) of testing the hypothesis H₀₂: ₁₂ = 0 that the first component of is independent of the others against the alternative H₂: 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.