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Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test

Published online by Cambridge University Press:  14 July 2016

J. Durbin
J. Durbin*
Affiliation:
London School of Economics and Political Science
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Let w(t), 0 ≦ t ≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1, E{w(t1)w(t2)}= min (t1, t2), and let a, b denote the boundaries defined by y = a(t), y = b(t), where b(0) < 0 < a(0) and b(t) < a(t), 0 ≦ t ≦ T ≦ ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path of w(t) crosses a or b before t = T. This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the condition w(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 3 , September 1971 , pp. 431 - 453
Copyright
Copyright © Applied Probability Trust 1971 

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References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. National Bureau of Standards, Washington.Google Scholar
Anderson, T. W. (1960) A modification of the sequential probability ratio test to reduce the sample size. Ann. Math. Statist. 31, 165–197.CrossRefGoogle Scholar
Anderson, T. W. and Darling, D. A. (1952) Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. Ann. Math. Statist. 23, 193–212.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Breiman, L. (1967) First exit times from a square root boundary. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 2, 9–16.Google Scholar
Cooper, B. (1968) Algorithm AS2. The normal integral. Appl. Statist. 17, 186–188.CrossRefGoogle Scholar
Daniels, H. E. (1945) The statistical theory of the strength of bundles of threads. I. Proc. Roy. Soc. Ser. A 183, 405–435.Google Scholar
Daniels, H. E. (1964) The Poisson process with a curved absorbing boundary. Bull. Int. Statist. Inst. 40, 994–1011.Google Scholar
Daniels, H. E. (1969) The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.Google Scholar
Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.Google Scholar
Dinges, H. (1962) Ein verallgemeinertes Spiegelungsprinzip für den Prozess der Brownschen Bewegung. Z. Wahrscheinlichkeitsth. 1, 177–196.Google Scholar
Doob, J. L. (1949) Heuristic approach to the Kolmogorov-Smimov theorem. Ann. Math. Statist. 20, 393–403.Google Scholar
Epanechnikov, V. A. (1968) The significance level and power of the two-sided Kolmogorov test in the case of small sample sizes. Theor. Probability Appl. 13, 689–690.CrossRefGoogle Scholar
Fortet, R. (1943) Les fonctions aléatoires du type de Markoff associées à certaines équations linéaires aux derivées partielles du type parabolique. J. Math. Pures Appl. 22, 177–243.Google Scholar
Hill, I.D. (1969) Remark ASR2. A remark on Algorithm AS2 “The normal integral”. Appl. Statist. 18, 299–300.Google Scholar
Hill, I. D. and Joyce, S. A. (1967) Algorithm 304. Normal curve integral. Comm. Ass. Comp. Mach. 10, 374–375.Google Scholar
I. B. M. (1968) System 360 Scientific Sub-routines Package. 360A–CM–03X, Version III.Google Scholar
Knott, M. (1970) The small sample power of one-sided Kolmogorov tests for a shift in location of the normal distribution. J. Amer. Statist. Assoc. 65, 1384–1391.CrossRefGoogle Scholar
Milton, R. C. and Hotchkiss, R. (1969) Computer evaluation of the normal and inverse normal distribution functions. Technometrics 11, 817–822.CrossRefGoogle Scholar
Noé, M. and Vandewiele, G. (1968) The calculation of distributions of Kolmogorov-Smirnov type statistics including a table of significance points for a particular case. Ann. Math. Statist. 39, 233–241.Google Scholar
Robbins, H. and Siegmund, D. (1970) Boundary crossing probabilities for the Wiener process and sample sums. Ann. Math. Statist. 41, 1410–1429.Google Scholar
Steck, G. P. (1971) Rectangle probabilities for uniform order statistics and the probability that the empirical distribution function lies between two distribution functions. Ann. Math. Statist. 42.Google Scholar
Suzuki, G. (1967) On exact probabilities of some generalised Kolmogorov's D-statistics. Ann. Inst. Statist. Math. Tokyo 19, 373–388.CrossRefGoogle Scholar
Suzuki, G. (1968) Kolmogorov-Smimov tests of fit based on some general bounds. J. Amer. Statist. Assoc. 63, 919–924.Google Scholar
Sweet, A. L. and Hardin, J. C. (1970) Solutions for some diffusion processes with two barriers. J. Appl. Prob. 7, 423–431.Google Scholar
Wald, A. and Wolfowitz, J. (1939) Confidence limits for continuous distribution functions. Ann. Math. Statist. 10, 105–118,CrossRefGoogle Scholar