A new asymptotic representation and inversion method for the Student's t distribution

TL;DR

This paper introduces a new asymptotic representation and inversion method for the Student's t distribution, enhancing computational efficiency and accuracy in statistical applications like hypothesis testing and random sample generation.

Contribution

It presents a novel asymptotic representation of the Student's t distribution using the complementary error function, improving distribution function computation and inversion techniques.

Findings

The new asymptotic representation improves approximation accuracy.

Numerical examples demonstrate enhanced computational performance.

The method facilitates better random sample generation and hypothesis testing.

Abstract

Some special functions are particularly relevant in applied probability and statistics. For example, the incomplete beta function is the cumulative central beta distribution. In this paper, we consider the inversion of the central Student's-$t$ distribution which is a particular case of the central beta distribution. The inversion of this distribution functions is useful in hypothesis testing as well as for generating random samples distributed according to the corresponding probability density function. A new asymptotic representation in terms of the complementary error function, will be one of the important ingredients in our analysis. As we will show, this asymptotic representation is also useful in the computation of the distribution function. We illustrate the performance of all the obtained approximations with numerical examples.