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

  • Moshe PollakEmail author
  • Michal Shauly-Aharonov


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.


Changepoint On-line Shiryaev–Roberts Surveillance 

Mathematics Subject Classification (2010)

62L10 62E15 60E05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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.


  1. Barr D R, Sherrill E T (1999) Mean and variance of truncated normal distributions. Amer Statist 53:357–361Google Scholar
  2. Burkardt J (2014) The truncated normal distribution.
  3. Cook JD (2010) Upper bounds on non-central chi-square tails and truncated normal moments.
  4. Dhrymes P J (2005) Moments of truncated (normal) distributions.
  5. Gordon L, Pollak M (1995) A robust surveillance scheme for stochastically ordered alternatives. Ann Statist 23:1350–1375MathSciNetCrossRefzbMATHGoogle Scholar
  6. Horrace WC (2015) Moments of the truncated normal distribution. J Prod Anal 43:133–138CrossRefGoogle Scholar
  7. Krieger AM, Pollak M, Yakir B (2003) Surveillance of a simple linear regression. J Amer Statist Assoc 98:1–15MathSciNetCrossRefzbMATHGoogle Scholar
  8. Lee A (1914) Table of the Gaussian “tail” tunctions; when the “tail” is larger than the body. Biometrika 10:208–214Google Scholar
  9. Liquet B, Nazarathy Y (2015) A dynamic view to moment matching of truncated distributions. Statist Prob Letts 104(53):87–93MathSciNetCrossRefzbMATHGoogle Scholar
  10. MATLAB (2014) MATLAB and statistics toolbox release 2014b. The Mathworks Inc., NatickGoogle Scholar
  11. Toolbox (2016) MATLAB multiprecision computing toolbox (ADVANPIX Release 2016). The MathWorks, Inc., Natick. http://www.advanpix Google Scholar
  12. O’Connor AN (2011) Probability distributions used in reliability engineering. Reliability information analysis center (RIAC)Google Scholar
  13. Pearson K, Lee A (1908) On the generalized probable error in multiple normal correlation. Biometrika 6:59–68CrossRefGoogle Scholar
  14. Pollak M (1987) Average run lengths of an optimal method of detecting a change in distribution. Ann Statist 15:749–779MathSciNetCrossRefzbMATHGoogle Scholar
  15. Pollak M, Siegmund D (1991) Sequential detection of a change in a normal mean when the initial value is unknown. Ann Statist 19:394–416MathSciNetCrossRefzbMATHGoogle Scholar
  16. Pollak M, Croarkin C, Hagwood C (1993) Surveillance schemes with application to mass calibration. NISTIR 5158 Technical Report, Statistical Division, The National Institute of Standards and Technology, Gaithersburg, MD 20899, USAGoogle Scholar
  17. Quesenberry C (1991) SPC Q processes for start-up processes and short or long runs. J Qual Tech 23:213–224CrossRefGoogle Scholar
  18. van Dobben de Bruyn CS (1968) Cumulative Sum Tests: Theory and Practice. Griffin, LondonGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of StatisticsThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations