A Note on Computing Extreme Tail Probabilities of the Noncentral T Distribution with Large Noncentrality Parameter

TL;DR

This paper introduces a new numerical integration method for accurately and efficiently computing the extreme tail probabilities of the noncentral t-distribution, especially for large noncentrality parameters and degrees of freedom.

Contribution

It proposes an alternative approach based on direct numerical integration for computing the noncentral t-distribution's CDF, improving precision and speed for challenging parameter values.

Findings

High-precision tail probability calculations achieved

Fast evaluation for large noncentrality parameters

Implementation available in MATLAB

Abstract

The noncentral $t$-distribution is a generalization of the Student's $t$-distribution. In this paper we suggest an alternative approach for computing the cumulative distribution function (CDF) of the noncentral $t$-distribution which is based on a direct numerical integration of a well behaved function. With a double-precision arithmetic, the algorithm provides highly precise and fast evaluation of the extreme tail probabilities of the noncentral $t$-distribution, even for large values of the noncentrality parameter $\delta$ and the degrees of freedom $\nu$. The implementation of the algorithm is available at the MATLAB Central, File Exchange: http://www.mathworks.com/matlabcentral/fileexchange/41790-nctcdfvw.