New asymptotic representations of the noncentral $t$-distribution

TL;DR

This paper introduces new asymptotic approximations for the noncentral t-distribution, improving computational efficiency for large parameters using elementary and special functions, validated by numerical tests.

Contribution

It provides novel asymptotic expansions for the noncentral t-distribution applicable to large noncentrality and degrees of freedom, including cases with multiple large parameters.

Findings

Asymptotic formulas in elementary functions and special functions.

Numerical tests confirm high accuracy of approximations.

Applicable to large parameter regimes in statistical analysis.

Abstract

New asymptotic approximations of the non-central $t$ distribution are given, a generalization of the Student's $t$ distribution. Using new integral representations, we give new asymptotic expansions for large values of the noncentrality parameter but also for large values of the degrees of freedom parameter. In some case we accept more than one large parameter. These results are in terms of elementary functions, but also in terms of the complementary error function and the incomplete gamma function. A number of numerical tests demonstrate the performance of the asymptotic approximations.