Refined normal approximations for the Student distribution

TL;DR

This paper develops a local limit theorem for the Student distribution, improving normal approximations of its survival function and deriving bounds for approximation errors, with applications to quantile estimation.

Contribution

It introduces a refined local limit theorem for the Student distribution, enhancing the accuracy of normal approximations and providing asymptotic error bounds.

Findings

Improved normal approximation for the Student survival function.

Derived asymptotic bounds for maximal approximation errors.

Provided new formulas for Student distribution quantiles in terms of normal quantiles.

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.