Markos V. Koutras

Distribution Theory of Runs: A Markov Chain Approach

Abstract The statistics of the number of success runs in a sequence of Bernoulli trials have been used in many statistical areas. For almost a century, even in the simplest case of independent and identically distributed Bernoulli trials, the exact distributions of many run statistics still remain unknown. Departing from the traditional combinatorial approach, in this article we present a simple unified approach for the distribution theory of runs based on a finite Markov chain imbedding technique. Our results cover not only the identical Bernoulli trials, but also the nonidentical Bernoulli trials. As a byproduct, our results also yield the exact distribution of the waiting time for the mth occurrence of a specific run.

Runs, scans and URN model distributions: A unified Markov chain approach

This paper presents a unified approach for the study of the exact distribution (probability mass function, mean, generating functions) of three types of random variables: (a) variables related to success runs in a sequence of Bernoulli trials (b) scan statistics, i.e. variables enumerating the moving windows in a linearly ordered sequence of binary outcomes (success or failure) which contain prescribed number of successes and (c) success run statistics related to several well known urn models. Our approach is based on a Markov chain imbedding which permits the construction of probability vectors satisfying triangular recurrence relations. The results presented here cover not only the case of identical and independently distributed Bernoulli variables, but the non-identical case as well. An extension to models exhibiting Markov dependence among the successive trials is also discussed in brief.

On a Markov chain approach for the study of reliability structures

In this paper we consider a class of reliability structures which can be efficiently described through (imbedded in) finite Markov chains. Some general results are provided for the reliability evaluation and generating functions of such systems. Finally, it is shown that a great variety of well known reliability structures can be accommodated in this general framework, and certain properties of those structures are obtained on using their Markov chain imbedding description.

Non-central stirling numbers and some applications

Non-central Stirling numbers of the first and second kind are introduced and corresponding representations and recurrences are given along with some applications in occupancy problems and discrete distribution theory.

Waiting Time Distributions Associated with Runs of Fixed Length in Two-State Markov Chains

In the present article a general technique is developed for the evaluation of the exact distribution in a wide class of waiting time problems. As an application the waiting time for the r-th appearance of success runs of specified length in a sequence of outcomes evolving from a first order two-state Markov chain is systematically investigated and asymptotic results are established. Several extensions and generalisations are also discussed.

On the signature of coherent systems and applications

In the present article we provide a formula that facilitates the evaluation of the signature of a reliability structure by a generating function approach. A simple sufficient condition is also derived for proving the nonpreservation of the IFR property for the systems lifetime (when the components are IFR) by exploiting the signature of the system. As an application of the general results, we deduce recurrence relations for the signature of a linear consecutive k-out-of-n: F system. We establish a simple relation between the signature of a linear and a circular system and investigate the IFR preservation property under the formulation of such systems.

On a waiting time distribution in a sequence of Bernoulli trials

In the present article we investigate the exact distribution of the waiting time for the r-th non-overlapping appearance of a pair of successes separated by at mosk k−2 failures (k≥2) in a sequence of independent and identically distributed (iid) Bernoulli trials. Formulae are provided for the probability distribution function, probability generating function and moments and some asymptotic results are discussed. Expressions in terms of certain generalised Fibonacci numbers and polynomials are also included.

Reliability of 2-dimensional consecutive-k-out-of-n:F systems

The authors derive lower and upper reliability bounds for the two-dimensional consecutive k-out-of-n:F system (Salvia Lasher, 1990) with independent, but not necessarily identically distributed, components. A Weibull limit theory is proven for system time-to-failure for i.i.d. components. >

Runs on a circle

We study the number of occurrences of non-overlapping success runs of length k in a sequence of (not necessarily identical) Bernoulli trials arranged on a circle. An exact formula is given for the probability function, along with some sharp bounds which turn out to be very useful in establishing limiting (Poisson convergence) results. Certain applications to statistical run tests and reliability theory are also discussed.

Non-parametric randomness tests based on success runs of fixed length

Let Xn be a random variable enumerating the number of appearances of a specific pattern in a sequence of n Bernoulli trials. A new method is presented for obtaining the conditional distribution of Xn given the number of successes in the n trials. The method is applied to three fixed-length run statistics and the results are used for establishing and investigating certain non-parametric tests of randomness.