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 for any , . 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.