Asymptotic expansions of the null distributions of some test statistics for profile analysis in general distributions

This paper discusses asymptotic expansions for the null distributions of some test statistics for profile analysis under non-normality. It is known that the null distributions of these statistics converge to chi-square distribution under normality Siotani, M., 1956. On the distributions of the Hotelling's T²$T^2$-statistics. Ann. Inst. Statist. Math. Tokyo 8, 1–14; Siotani, M., 1971. An asymptotic expansion of the non-null distributions of Hotelling's generalized T²$T^2$-statistic. Ann. Math. Statist. 42, 560–571]. We extend this result by obtaining asymptotic expansions under general distributions. Moreover, the effect of non-normality is also considered. In order to obtain all the results, we make use of matrix manipulations such as direct products and symmetric tensor, rather than usual elementwise tensor notation.