An asymptotic expansion of the distribution of Student's t type statistic under spherical distributions

An asymptotic expansion of the distribution of Student's t type statistic based on the multivariate standardized or studentized sample mean vector is obtained by making use of an Edgeworth expansion up to the order O(N−2), where N is sample size and Student's t type transformation is defined by Tα(X)=(p−1)12α′X(‖X‖2−(α′X)2)−12 for any α∈Rp, α′α=1. It turns out that a t-approximation to Student's t type statistic based on the studentized sample mean vector has the error o(N−ℓ), if a certain spherical population has at least 4(ℓ+1)th moment, where ℓ=0,1,2. Some numerical experiments are also conducted to evaluate the accuracy of the result.