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Approximating the Distribution of the Maximum Partial Sum of Normal Deviates

Author

Listed:
  • Denis Conniffe

    (Economic and Social Research Institute (ESRI))

  • John E. Spencer

    (The Queen's University of Belfast)

Abstract

The largest partial sum of deviations from the mean is a statistic of importance in several areas of application, including hydrology and in testing for a change-point. Approximations to its distribution for the simple normal case have appeared in the literature, based either on functionals of Brownian motion asymptotics or on a methodology developed for crossing problems in sequential analysis. The former approximation is inaccurate except for very large samples, while the latter is based on rather difficult theory. In this paper, we first review some early findings about exact moments and extend them somewhat. We then use moments to fit simple Chi-squared and Beta approximations and show that they work very well.

Suggested Citation

  • Denis Conniffe & John E. Spencer, 1999. "Approximating the Distribution of the Maximum Partial Sum of Normal Deviates," Papers WP102, Economic and Social Research Institute (ESRI).
    • Handle: RePEc:esr:wpaper:wp102
    as

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    File URL: https://www.esri.ie/pubs/WP102.pdf
    File Function: First version, 1999
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    References listed on IDEAS

    as
    1. Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-856, July.
    2. Ploberger, Werner & Kramer, Walter, 1992. "The CUSUM Test with OLS Residuals," Econometrica, Econometric Society, vol. 60(2), pages 271-285, March.
    3. Benoit B. Mandelbrot, 1972. "Statistical Methodology for Nonperiodic Cycles: From the Covariance To R/S Analysis," NBER Chapters, in: Annals of Economic and Social Measurement, Volume 1, number 3, pages 259-290, National Bureau of Economic Research, Inc.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Reza Habibi, 2010. "Distribution Approximations for Cusum and Cusumsq Statistics," Statistics in Transition new series, Główny Urząd Statystyczny (Polska), vol. 11(3), pages 585-596, December.
    2. Denis Conniffe & John E. Spencer, 2000. "Approximating the Distribution of the R/s Statistic," The Economic and Social Review, Economic and Social Studies, vol. 31(3), pages 237-248.
    3. Conniffe, Denis & Kelly, Robert, 2011. "Structural Breaks - An Instrumental Variable Approach," Research Technical Papers 4/RT/11, Central Bank of Ireland.
    4. Habibi Reza, 2011. "A note on approximating distribution functions of cusum and cusumsq tests," Monte Carlo Methods and Applications, De Gruyter, vol. 17(1), pages 1-10, January.

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