W. Y. Wendy Lou

Distribution Theory of Runs and Patterns and Its Applications: A Finite Markov Chain Imbedding Approach

* Finite Markov Chain Imbedding * Runs and Patterns in a Sequence of Two-State Trials * Runs and Patterns in Multi-State Trials * Waiting-Time Distributions * Random Permutations * Applications

The exact distribution of the k-tuple statistic for sequence homology

The distribution theory of runs and patterns has become increasingly useful in the field of biological sequence homology. One important application in detecting tandem duplications among DNA sequence segments is the k-tuple statistic Sn,k, the sum of matches in matching-runs of length k or longer in a sequence of n i.i.d. Bernoulli trials with success/matching probability p. Current approaches to this distribution problem are based on various approximations, due mainly to the numerical complexity of computing the exact distribution using a straightforward combinatorial approach. In this paper, we obtain a simple and efficient expression for the exact distribution of Sn,k using the principle of finite Markov chain imbedding. Our numerical results illustrate most importantly that for pattern lengths in the range n=10 to 100, a range commonly used in detecting DNA tandem repeats, the distribution, in general, is highly skewed and far from normal.

On reliabilities of certain large linearly connected engineering systems

Many engineering systems such as series system, standby redundant system, k-out-of-n system, consecutive k-out-of-n: F system, deterioration system, and repairable system, all of them can be viewed as linearly connected systems. This paper mainly studies the reliabilities and survival times of large linearly connected systems with deterioration-type transition matrices.

The exact and limiting distributions for the number of successes in success runs within a sequence of Markov-dependent two-state trials

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.

On Probabilities for Complex Switching Rules in Sampling Inspection

Switching rules between different levels of sampling are widely used in quality control, as in the well-known Military Standard 105E (MIL STD 105E) and similar acceptance-sampling schemes. These switching rules are typically defined by specific patterns of inspection outcomes within a sequence of previous inspections. The probability distributions of the rules are usually hard to find, and many of them remain unknown. In this chapter, we will provide a general and simple method, the finite Markov chain imbedding technique, to obtain the distributions of switching rules. We demonstrate the utility of this method primarily by (i) deriving the generating function of a basic switching rule (k consecutive acceptances) for the practically important case of a two-state, first-order autoregressive AR(1) sequence, (ii) treating jointly the normal and tightened inspection regimes of MIL STD 105E including the overall probability of discontinuing inspection, and (iii) considering a stratified sampling scheme with three possible inspection outcomes.