On the Moments and Density Function of Hotelling's Trace and Other Test Criteria Concerning Test of Independence under Violation in the Complex Case

N. Saweris (1987). On the Moments and Density Function of Hotelling's Trace and Other Test Criteria Concerning Test of Independence under Violation in the Complex Case . Journal of Statistical Research, Vol. 21, No. 1-2, pp. ??-??.

Abstract

This paper deals with the moments and density functions of Hotelling's trace $T^{(p)}=tr(S_1S_1^{-1})=\sum r_i$ , in connection with the test of independence under violation. It may be noted that $R=\mbox{diag}\{r_1,\ldots, r_p\};$ $0\leq r_1\leq\cdots\leq r_p$ , are the latent roots of $R=S_1^{1/2}S_2S_1^{1/2}$ , where $S_l$ and $S_2$ are independently distributed and having respectively the complex noncentral Wishart $W_0(P,n_1,\Sigma_1,\Omega),\Omega=MM\Sigma_1^{-1},$ and central Wishart $W_0(P,n_2,\Sigma_2,\Omega)$ . The joint density function of $(r_1,\ldots,r_p)$ in the canonical correlation case under violation is also given. Based on this function, the moments and the density function of Wilke and Wilks-Lawely test criteria are also derived in connection with the same test of independence. Throughout the context, an aasumption is made on $\Omega$ , accordingly, the normality is violated and that of common covariance matrix is also disturbed.