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

  • Moshe Pollak
  • Michal Shauly-Aharonov
Article
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Abstract

The integral \({\int }_{0}^{\infty }x^{m} e^{-\frac {1}{2}(x-a)^{2}}dx\) 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.

Keywords

Changepoint On-line Shiryaev–Roberts Surveillance 

Mathematics Subject Classification (2010)

62L10 62E15 60E05 

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Notes

Acknowledgements

This work was supported by grant 1450/13 from the Israel Science Foundation and by the Marcy Bogen Chair of Statistics at the Department of Statistics, The Hebrew University of Jerusalem. The authors would like to thank the referees for their salient comments, which greatly improved the paper.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsThe Hebrew University of JerusalemJerusalemIsrael

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