The Exact and Limiting Distributions for the Number of Successes in Success Runs Within a Sequence of Markov-Dependent Two-State Trials
Tóm tắt
The total number of successes in success runs of length greater than or equal to k in a sequence of n two-state trials is a statistic that has been broadly used in statistics and probability. For Bernoulli trials with k equal to one, this statistic has been shown to have binomial and normal distributions as exact and limiting distributions, respectively. For the case of Markov-dependent two-state trials with k greater than one, its exact and limiting distributions have never been considered in the literature. In this article, the finite Markov chain imbedding technique and the invariance principle are used to obtain, in general, the exact and limiting distributions of this statistic under Markov dependence, respectively. Numerical examples are given to illustrate the theoretical results.
Tài liệu tham khảo
Benson, G. (1999). Tandem repeats finder: A program to analyze DNA sequences, Nucleic Acids Research, 27, 573–580.
Billingsley, P. (1968). Convergence of Probability Measures, Wiley, New York.
Fu, J. C. (1986). Reliability of consecutive-k-out-of-n F system with (k-1)-step Markov dependence, IEEE Transactions on Reliability, R 35, 602–606.
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials, Statist. Sinica, 6, 957–974.
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach, J. Amer. Statist. Assoc., 89, 1050–1058.
Huntington's Disease Collaborative Research Group (1993). A novel gene containing a trinucleotide repeat that is expanded and unstable on Huntington's disease chromosomes, Cell, 72, 971–983.
Mcleish, D. L. (1974). Dependent central limit theorem and invariance principles, Ann. Probab., 2, 620–628.
Nagaev, S. V. (1957). Some limit theorems for stationary Markov chains, Theory Probab. Appl., 2, 378–406.
Nagaev, S. V. (1961). More exact statements of limit theorems for homogeneous Markov chains, Theory Probab. Appl., 6, 62–81.
Petrov, V. V. (1975). Sums of Independent Random Variables, Springer, Berlin.