Refined normal approximations for the Student distribution

Abstract

In this paper, we develop a local limit theorem for the Student distribution. We use it to improve the normal approximation of the Student survival function given in Shafiei & Saberali (2015) and to derive asymptotic bounds for the corresponding maximal errors at four levels of approximation. As a corollary, approximations for the percentage points (or quantiles) of the Student distribution are obtained in terms of the percentage points of the standard normal distribution.

Local normal approximations and probability metric bounds for the matrix-variate $T$ distribution and its application to Hotelling's $T$ statistic

“…In Theorem 1 below, we prove an asymptotic expansion for the ratio of the centered matrixvariate T density to the centered matrix-variate normal (MN) density with the same covariances. The case d = m = 1 was proven recently in Ouimet (2022). The result extends significantly the convergence in distribution result from Theorem 4.3.4 in Gupta & Nagar (1999).…”