An Asymptotic Expansion of the Distribution of Hotelling's T²-Statistic Under General Distributions

SYNOPTIC ABSTRACT

Distribution theory in nonnormal populations is important but is not fully exploited particularly in multivariate analysis. In this paper we derive an asymptotic expansion of the distribution of Hotelling's multivariate T²-statistic under general distributions. Our 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.

Key Words and Phrases:

• Commutation matrix
• cumulant
• duplication matrix
• Edgeworth expansion
• elliptical distribution
• Hermite polynomial
• symmetric tensor