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Approximate probabilities for runs and patterns in i.i.d. and Markov-dependent multistate trials

  • James C. Fu (a1) and Brad C. Johnson (a1)

Abstract

Let X n (Λ) be the number of nonoverlapping occurrences of a simple pattern Λ in a sequence of independent and identically distributed (i.i.d.) multistate trials. For fixed k, the exact tail probability P{Xn (∧) < k} is difficult to compute and tends to 0 exponentially as n → ∞. In this paper we use the finite Markov chain imbedding technique and standard matrix theory results to obtain an approximation for this tail probability. The result is extended to compound patterns, Markov-dependent multistate trials, and overlapping occurrences of Λ. Numerical comparisons with Poisson and normal approximations are provided. Results indicate that the proposed approximations perform very well and do significantly better than the Poisson and normal approximations in many cases.

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Copyright

Corresponding author

Postal address: Department of Statistics, University of Manitoba, Winnipeg, Canada R3T 2N2.
∗∗ Email address: brad_johnson@umanitoba.ca

Footnotes

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This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.

Footnotes

References

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Arratia, R., Goldstein, L. and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5, 403434.
Barbour, A. D., Chryssaphinou, O. and Roos, M. (1996). Compound Poisson approximation in systems reliability. Naval Res. Logistics 43, 251264.
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 10501058.
Fu, J. C. and Lou, W. Y. W. (2003). Distribution Theory of Runs and Patterns and Its Applications. World Scientific, River Edge, NJ.
Fu, J. C., Wang, L.and Lou, W. Y. W. (2003). On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials. J. Appl. Prob. 40, 346360.
Godbole, A. P. (1991). Poisson approximations for runs and patterns of rare events. Adv. Appl. Prob. 23, 851865.
Godbole, A. P. and Schaffner, A. A. (1993). Improved Poisson approximations for word patterns. Adv. Appl. Prob. 25, 334347.
Mood, A. M. (1940). The distribution theory of runs. Ann. Math. Statist. 11, 367392.
Riordan, J. (1958). An Introduction to Combinatorial Analysis. John Wiley, New York.
Wald, J. and Wolfowitz, J. (1940). On a test whether two samples are from the same population. Ann. Math. Statist. 11, 147162.
Wishart, J. and Hirschfeld, H. O. (1936). A theorem concerning the distribution of Joins between line segments. J. London Math. Soc. 11, 227235.
Wolfowitz, J. (1943). On the theory of runs with some applications to quality control. Ann. Math. Statist. 14, 280288.

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Approximate probabilities for runs and patterns in i.i.d. and Markov-dependent multistate trials

  • James C. Fu (a1) and Brad C. Johnson (a1)

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