Distribution theory in nonnormal populations is important but is not fully exploited particularly in multivariate analysis. This paper derives an asymptotic expansion of the distribution of Hotelling’s multivarite T2-statistic under general distributions. The present general expansion specializes the existing expansions under elliptical and normal distributions. The previous research on robustness of T2 to violation of normal assumption, based on Monte Carlo study, concludes that nonnormality of an underlined population influences substantially upon the distribution of T2 for small or medium samples and that the third-order cumulants of the underlined distribution affects T2 much more seriously than do the fourth-order cumulants. The derived formula is used to provide theoretical grounds for the experimental results. Matrix manipulations such as Kronecker products and symmetric tensors are utilized to derive all the results, rather than usual elementwise tensors with Einstein’s convention.
