Approximating the mode of the non-central chi-squared distribution

TL;DR

This paper derives an approximation for the mode of the non-central chi-squared distribution, enhancing computational methods and extending the applicability of existing software implementations.

Contribution

It provides a new approximation for the mode of the non-central chi-squared distribution and demonstrates improved performance of Boost's implementation.

Findings

Derived an approximate mode formula for the non-central chi-squared distribution.

Showed improved performance of Boost's non-central chi-squared implementation.

Extended the domain of applicability for computational methods.

Abstract

In this paper we consider the probability density function (PDF) of the non-central $\chi^2$ distribution with arbitrary number of degrees of freedom and non-centrality. For this function we find the approximate location of the maximum and discuss related edge cases of 1 and 2 degrees of freedom. We also use this expression to demonstrate the improved performance of the C++ Boost's implementation of the non-central $\chi^2$ and extend the domain of its applicability.