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

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  • Moshe Pollak

    (The Hebrew University of Jerusalem)

  • Michal Shauly-Aharonov

    (The Hebrew University of Jerusalem)

Abstract

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

Suggested Citation

  • Moshe Pollak & Michal Shauly-Aharonov, 2019. "A Double Recursion for Calculating Moments of the Truncated Normal Distribution and its Connection to Change Detection," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 889-906, September.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:3:d:10.1007_s11009-018-9622-7
    DOI: 10.1007/s11009-018-9622-7
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    References listed on IDEAS

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    1. Liquet, Benoit & Nazarathy, Yoni, 2015. "A dynamic view to moment matching of truncated distributions," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 87-93.
    2. Krieger A.M. & Pollak M. & Yakir B., 2003. "Surveillance of a Simple Linear Regression," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 456-469, January.
    3. William Horrace, 2015. "Moments of the truncated normal distribution," Journal of Productivity Analysis, Springer, vol. 43(2), pages 133-138, April.
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