Approximating the Distribution of the Maximum Partial Sum of Normal Deviates
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.
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- Andrews, Donald W K, 1993.
"Tests for Parameter Instability and Structural Change with Unknown Change Point,"
Econometric Society, vol. 61(4), pages 821-56, July.
- Donald W.K. Andrews, 1990. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Cowles Foundation Discussion Papers 943, Cowles Foundation for Research in Economics, Yale University.
- 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.
- Ploberger, Werner & Kramer, Walter, 1992. "The CUSUM Test with OLS Residuals," Econometrica, Econometric Society, vol. 60(2), pages 271-85, March.
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