A Double Recursion for Calculating Moments of the Truncated Normal Distribution and its Connection to Change Detection

The integral ∫0∞xme−12(x−a)2dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\int }_{0}^{\infty }x^{m} e^{-\frac {1}{2}(x-a)^{2}}dx$\end{document} appears in likelihood ratios used to detect a change in the parameters of a normal distribution. As part of the mth moment of a truncated normal distribution, this integral is known to satisfy a recursion relation, which has been used to calculate the first four moments of a truncated normal. Use of higher order moments was rare. In more recent times, this integral has found important applications in methods of changepoint detection, with m going up to the thousands. The standard recursion formula entails numbers whose values grow quickly with m, rendering a low cap on computational feasibility. We present various aspects of dealing with the computational issues: asymptotics, recursion and approximation. We provide an example in a changepoint detection setting.