Approximating the distributions of runs and patterns

Brad C Johnson,James C Fu

doi:10.1186/2195-5832-1-5

Abstract

The distribution theory of runs and patterns has been successfully used in a variety of applications including, for example, nonparametric hypothesis testing, reliability theory, quality control, DNA sequence analysis, general applied probability and computer science. The exact distributions of the number of runs and patterns are often very hard to obtain or computationally problematic, especially when the pattern is complex and n is very large. Normal, Poisson and compound Poisson approximations are frequently used to approximate these distributions. In this manuscript, we (i) study the asymptotic relative error of the normal, Poisson, compound Poisson and finite Markov chain imbedding and large deviation approximations; and (ii) provide some numerical studies to comparing these approximations with the exact probabilities for moderately sized n. Both theoretical and numerical results show that, in the relative sense, the finite Markov chain imbedding approximation performs the best in the left tail and the large deviation approximation performs best in the right tail. Primary 60E05; Secondary 60J10

Highlights

Introduction and notationLet {Xi}ni=1 be a sequence of m-state trials (m ≥ 2) taking values in the set S = {s1, . . . , sm} of m symbols
It is important to know how these approximations perform with respect to each other and the exact distribution from both a theoretical and numerical standpoint. The aims of this manuscript are two-fold: (i) we first study the asymptotic relative error of the normal, Poisson, and Finite Markov chain imbedding approximation (FMCI) approximations with respect to the exact distribution; and (ii) we provide a numerical study of these three approximations with the exact probabilities in cases where x is fixed and n → ∞ and when n is fixed and x varies
The primary tool used to obtain μn and the bound εn is the Stein-Chen method (Chen 1975), and this method has been refined by various authors Arratia et al (1990), Barbour and Eagleson (1983), Barbour and Eagleson (1984), Barbour and Eagleson (1987), Barbour and Hall (1984), Godbole (1990a), Godbole (1990b), Godbole (1991), Godbole and Schaffner (1993), and Holst et al (1988). This method has been extended to compound Poisson approximations for the distributions of runs and patterns and Barbour and Chryssaphinou (2001) provides an excellent theoretical review of these approximations

Summary

Normal approximation

The normal approximation is one of the most popular for approximating the distribution of the number of runs or patterns Xn( ) in Statistics. When occurrences of correspond to a delayed renewal process, which can occur for Markov dependent trials and/or overlapping counting, we could use the mean and variance of W2( ) for the normalizing constants, which are obtained by modifying ξ 0 in (7) and (8). The primary tool used to obtain μn and the bound εn is the Stein-Chen method (Chen 1975), and this method has been refined by various authors Arratia et al (1990), Barbour and Eagleson (1983), Barbour and Eagleson (1984), Barbour and Eagleson (1987), Barbour and Hall (1984), Godbole (1990a), Godbole (1990b), Godbole (1991), Godbole and Schaffner (1993), and Holst et al (1988) This method has been extended to compound Poisson approximations for the distributions of runs and patterns and Barbour and Chryssaphinou (2001) provides an excellent theoretical review of these approximations. Let Pc(λ, ν) denote the compound Poisson distribution, that is, the distribution of the random variable

Yj where the random variable

FMCI approximations

If g

Discussion and conclusions