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Boundary crossing for moving sums

Published online by Cambridge University Press:  14 July 2016

J. Glaz  and
B. Johnson
J. Glaz*
Affiliation:
University of Connecticut
B. Johnson
Affiliation:
University of Connecticut
*
∗ Postal address: Department of Statistics, U-120. 196 Auditorium Road. University of Connecticut, Storrs, CT 06268. USA.
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Abstract

Let Xi, i ≧ 1, be a sequence of independent N(0, 1) random variables and Sj,m = Xj + · ·· + Xj+m–1, the jth moving sum. Let τ m = inf{j ≧ 1 : Sj,m > a} + m – 1, the boundary crossing time. Approximation in the spirit of Glaz and Johnson (1984), (1986) and Samuel-Cahn (1983) are given for Pr(τm > n), E(τ m), and σ (τ m),the standard deviation of τm.

Keywords

APPROXIMATIONSBOUNDARY CROSSING PROBABILITIESSHIFT DETECTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 81 - 88
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

†

Regrettably, Professor Johnson died on 4 November 1986.

References

Bauer, P. and Hackl, P. (1980) Extension of the MOSUM technique for quality control. Technometrics 22, 1–7.CrossRefGoogle Scholar
Glaz, J. and Johnson, B. Mck. (1984) Probability inequalities for multivariate distributions with dependence structures. J. Amer. Statist. Assoc. 79, 436–441.CrossRefGoogle Scholar
Glaz, J. and Johnson, B. Mck. (1986) Approximating boundary crossing probabilities with applications to sequential tests. Sequential Anal. 5, 37–72.CrossRefGoogle Scholar
Lai, T. L. (1974) Control charts based on weighted sums. Ann. Statist. 2, 134–147.CrossRefGoogle Scholar
Orey, S. (1971) Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand, London.Google Scholar
Owen, D. B. (1956) Tables for computing bivariate normal probabilities, Ann. Math. Statist. 29, 1075–1090.CrossRefGoogle Scholar
Samuel-Cahn, E. (1983) Simple approximations to the expected waiting time for a cluster of any given size for point processes. Adv. Appl. Prob. 15, 21–38.CrossRefGoogle Scholar
Steck, P. S. (1958) A table for computing trivariate normal probabilities. Ann. Math. Statist. 29, 780–800.CrossRefGoogle Scholar
Tweedie, R. L. (1974a) R-theory for Markov chains on a general state space I. Ann. Prob. 2, 840–864.Google Scholar
Tweedie, R. L. (1974b) Quasi-stationary distributions for Markov chains on a general state space. J. Appl. Prob. 11, 726–741.CrossRefGoogle Scholar