# Computation of the expected value of a function of a chi-distributed random variable

## TL;DR

This paper introduces an efficient numerical method using a transformation and trapezoidal rule for accurately computing the expected value of functions of chi-distributed variables, relevant in statistical inference and confidence region analysis.

## Contribution

It presents a novel application of Mori's transformation combined with exponentially convergent trapezoidal rule and an error bound for evaluating expectations involving chi-distributions.

## Key findings

- Method achieves exponential convergence for suitable integrands.
- Provides a new upper bound for the infinite sum approximation error.
- Demonstrates suitability for computing coverage probabilities and volumes of confidence regions.

## Abstract

We consider the problem of numerically evaluating the expected value of a smooth bounded function of a chi-distributed random variable, divided by the square root of the number of degrees of freedom. This problem arises in the contexts of simultaneous inference, the selection and ranking of populations and in the evaluation of multivariate t probabilities. It also arises in the assessment of the coverage probability and expected volume properties of the some non-standard confidence regions. We use a transformation put forward by Mori, followed by the application of the trapezoidal rule. This rule has the remarkable property that, for suitable integrands, it is exponentially convergent. We use it to create a nested sequence of quadrature rules, for the estimation of the approximation error, so that previous evaluations of the integrand are not wasted. The application of the trapezoidal rule requires the approximation of an infinite sum by a finite sum. We provide a new easily computed upper bound on the error of this approximation. Our overall conclusion is that this method is a very suitable candidate for the computation of the coverage and expected volume properties of non-standard confidence regions.