Asymptotic Formulae for the Distribution of Hotelling's Generalized $T^2_0$ Statistic

In this paper the asymptotic expansion of a percentage point of Hotelling's generalized $T^2_0$ distribution is derived in terms of the corresponding percentage point of a $chi^2$ distribution. Our result generalizes Hotelling's and Frankel's asymptotic expansion for the generalized Student $T$ [3], [4]. The technique used in this paper for obtaining the asymptotic expansion of $T^2_0$ is an extension of the previous methods of Welch [8] and of James [5], [6], who used them to solve the distribution problem of various statistics in connection with the Behrens-Fisher problem. An asymptotic formula for the cumulative distribution function (c.d.f.) of $T^2_0$ is also given together with an upper bound for the error committed when all but the first few terms are omitted in the series. This formula is a sort of multivariate analogue of Hartley's formula of "Studentization" [2].