Distributions of numbers of failures and successes until the first consecutivek successes

Exact distributions of the numbers of failures, successes and successes with indices no less thanl (1lk–1) until the first consecutivek successes are obtained for some {0, 1}-valued random sequences such as a sequence of independent and identically distributed (iid) trials, a homogeneous Markov chain and a binary sequence of orderk. The number of failures until the first consecutivek successes follows the geometric distribution with an appropriate parameter for each of the above three cases. When the {0, 1}-sequence is an iid sequence or a Markov chain, the distribution of the number of successes with indices no less thanl is shown to be a shifted geometric distribution of orderk - l. When the {0, 1}-sequence is a binary sequence of orderk, the corresponding number follows a shifted version of an extended geometric distribution of orderk - l.This research was partially supported by the ISM Cooperative Research Program (92-ISM-CRP-16) of the Institute of Statistical Mathematics.     

Joint distributions associated with patterns,successes and failures in a sequence of multi-state trials

Let {Z _t ,t≥1} be a sequence of trials taking values in a given setA={0, 1, 2,...,m}, where we regard the value 0 as failure and the remainingm values as successes. Let ε be a (single or compound) pattern. In this paper, we provide a unified approach for the study of two joint distributions, i.e., the joint distribution of the numberX _n of occurrences of ε, the numbers of successes and failures inn trials and the joint distribution of the waiting timeT _r until ther-th occurrence of ε, the numbers of successes and failures appeared at that time. We also investigate some distributions as by-products of the two joint distributions. Our methodology is based on two types of the random variablesX _n (a Markov chain imbeddable variable of binomial type and a Markov chain imbeddable variable of returnable type). The present work develops several variations of the Markov chain imbedding method and enables us to deal with the variety of applications in different fields. Finally, we discuss several practical examples of our results. This research was partially supported by the ISM Cooperative Research Program (2002-ISM·CRP-2007).     

Joint Distributions of Numbers of Success-Runs and Failures Until the First Consecutive k Successes in a Binary Sequence

Joint distributions of the numbers of failures, successes andsuccess-runs of length less than k until the first consecutive k successesin a binary sequence were derived recently by Aki and Hirano (1995, Ann.Inst. Statist. Math., 47, 225-235). In this paper, we present an alternatederivation of these results and also use this approach to establish someadditional results. Extensions of these results to binary sequences of orderh are also presented.     

Joint distributions of numbers of success runs of specified lengths in linear and circular sequences

In this paper, we study two joint distributions of the numbers of success runs of several lengths in a sequence ofn Bernoulli trials arranged on a line (linear sequence) or on a circle (circular sequence) based on four different enumeration schemes. We present formulae for the evaluation of the joint probability functions, the joint probability generating functions and the higher order moments of these distributions. Besides, the present work throws light on the relation between the joint distributions of the numbers of success runs in the circular and linear binomial model. We give further insights into the run-related problems arisen from the circular sequence. Some examples are given in order to illustrate our theoretical results. Our results have potential applications to other problems such as statistical run tests for randomness and reliability theory. This research was partially supported by the ISM Cooperative Research Program (2003-ISM.CRP-2007).     

Joint Distributions of Runs in a Sequence of Multi-State Trials

In this paper we introduce a Markov chain imbeddable vector of multinomial type and a Markov chain imbeddable variable of returnable type and discuss some of their properties. These concepts are extensions of the Markov chain imbeddable random variable of binomial type which was introduced and developed by Koutras and Alexandrou (1995, Ann. Inst. Statist. Math., 47, 743–766). By using the results, we obtain the distributions and the probability generating functions of numbers of occurrences of runs of a specified length based on four different ways of counting in a sequence of multi-state trials. Our results also yield the distribution of the waiting time problems.     

Start-Up Demonstration Tests Under Markov Dependence Model with Corrective Actions

A general probability model for a start-up demonstration test is studied. The joint probability generating function of some random variables appearing in the Markov dependence model of the start-up demonstration test with corrective actions is derived by the method of probability generating function. By using the probability generating function, several characteristics relating to the distribution are obtained.     

Joint Distributions of Numbers of Success-runs Until the First Consecutive k Successes in a Higher-Order Two-State Markov Chain

Let X-m+1, X-m+2,.., X0, X1, X2,.., be a time-homogeneous {0, 1}-valued m-th order Markov chain. Joint distributions of the numbers of trials, failures and successes, of the numbers of trials and success-runs of length l (m l k) and of the numbers of trials and success-runs of length l (l m k) until the first consecutive k successes are obtained in the sequence X1, X2,.., There are some ways of counting numbers of runs of length l. This paper studies the joint distributions based on four ways of counting numbers of runs, i.e., the number of non-overlapping runs of length l, the number of runs of length greater than or equal to l, the number of overlapping runs of length l and the number of runs of length exactly l. Marginal distributions of them can be obtained immediately, and surprisingly their distributions are very simple.     

Waiting Time Problems in a Two-State Markov Chain

Let F ₀ be the event that l ₀ 0-runs of length k ₀ occur and F ₁ be the event that l ₁ 1-runs of length k ₁ occur in a two-state Markov chain. In this paper using a combinatorial method and the Markov chain imbedding method, we obtained explicit formulas of the probability generating functions of the sooner and later waiting time between F ₀ and F ₁ by the non-overlapping, overlapping and "greater than or equal" enumeration scheme. These formulas are convenient for evaluating the distributions of the sooner and later waiting time problems.     

Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k

A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer less than the specified length of the runs, an enumeration scheme called -overlapping way of counting is defined. Let k and be positive integers satisfying < k. Based on independent Bernoulli trials, it is shown that the number of (– 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length follows the generalized binomial distribution of order (k–). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k– 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.     

Waiting time problems for a sequence of discrete random variables

Let X ₁, X ₂,... be a sequence of nonnegative integer valued random variables.For each nonnegative integer i, we are given a positive integer k _i . For every i = 0, 1, 2,..., E _i denotes the event that a run of i of length k _i occurs in the sequence X ₁, X ₂,.... For the sequence X ₁, X ₂,..., the generalized pgf's of the distributions of the waiting times until the r-th occurrence among the events % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaiWabeaacaWGfbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzF% aaWaa0baaSqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIukaaaaa!43D8!\[\left\{ {E_i } \right\}_{i = 0}^\infty\]are obtained. Though our situations are general, the results are very simple. For the special cases that X's are i.i.d. and {0, 1}-valued, the corresponding results are consistent with previously published results.This research was partially supported by the ISM Cooperative Research Program (90-ISM-CRP-11) of the Institute of Statistical Mathematics.     

On the Waiting Time for the First Success Run

Let k and m are positive integers with k ≥ m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k−1 is obtained. This research was partially supported by the ISM Cooperative Research Program(2004-ISM-CRP-2006) and by a Grant-in-Aid for Scientific Research (C) of the JSPI (Grant Number 16500183)     

Generalized binomial and negative binomial distributions of order<Emphasis Type="Italic">k</Emphasis> by the<Emphasis Type="Italic">l</Emphasis>-overlapping enumeration scheme

In this paper, we investigate the exact distribution of the waiting time for ther-th ℓ-overlapping occurrence of success-runs of a specified length in a sequence of two state Markov dependent trials. The probability generating functions are derived explicitly, and as asymptotic results, relationships of a negative binomial distribution of orderk and an extended Poisson distribution of orderk are discussed. We provide further insights into the run-related problems from the viewpoint of the ℓ-overlapping enumeration scheme. We also study the exact distribution of the number of ℓ-overlapping occurrences of success-runs in a fixed number of trials and derive the probability generating functions. The present work extends several properties of distributions of orderk and leads us a new type of geneses of the discrete distributions.     

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2^.m conditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

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.     

Sooner and Later Waiting Time Problems Based on a Dependent Sequence

This paper introduces a new concept: a binary sequence of order (k,r), which is an extension of a binary sequence of order k and a Markov dependent sequence. The probability functions of the sooner and later waiting time random variables are derived in the binary sequence of order (k,r). The probability generating functions of the sooner and later waiting time distributions are also obtained. Extensions of these results to binary sequence of order (g,h) are also presented.     

On Number of Occurrences of Success Runs of Specified Length in a Higher-Order Two-State Markov Chain

Let X^-m+1, X^-m+2,..., X⁰, X¹, X²,..., Xⁿ be a time-homogeneous {0, 1}-valued m-th order Markov chain. The probability distributions of numbers of runs of "1" of length k (k m) and of "1" of length k (k < m) in the sequence of a {0, 1}-valued m-th order Markov chain are studied. There are some ways of counting numbers of runs with length k. This paper studies the distributions based on four ways of counting numbers of runs, i.e., the number of non-overlapping runs of length k, the number of runs with length greater than or equal to k, the number of overlapping runs of length k and the number of runs of length exactly k.     

Joint distributions of runs in a sequence of higher-order two-state Markov trials

Consider a time homogeneous {0, 1}-valued m-dependent Markov chain . In this paper, we study the joint probability distribution of number of 0-runs of length and number of 1-runs of length in n trials. We study the joint distributions based on five popular counting schemes of runs. The main tool used to obtain the probability generating function of the joint distribution is the conditional probability generating function method. Further a compact method for the evaluation of exact joint distribution is developed. For higher-order two-state Markov chain, these joint distributions are new in the literature of distributions of run statistics. We use these distributions to derive some waiting time distributions.     

Estimation of parameters in the discrete distributions of order k

This paper considers estimating parameters in the discrete distributions of order k such as the binomial, the geometric, the Poisson and the logarithmic series distributions of order k. It is discussed how to calculate maximum likelihood estimates of parameters of the distributions based on independent observations. Further, asymptotic properties of estimators by the method of moments are investigated. In some cases, it is found that the values of asymptotic efficiency of the moment estimators are surprisingly close to one.     

Joint Distributions of Numbers of Runs of Specified Lengths in a Sequence of Markov Dependent Multistate Trials

Let Z ₀, Z ₁,...,Z _n be a sequence of Markov dependent trials with state space Ω = {F ₁,...,F _λ, S ₁,...,S _ν}, where we regard F ₁,...,F _λ as failures and S ₁,...,S _ν as successes. In this paper, we study the joint distribution of the numbers of S _i -runs of lengths k _ij (i = 1,2,...,ν, j = 1,2,...,r _i ) based on four different enumeration schemes. We present formulae for the evaluation of the probability generating functions and the higher order moments of this distribution. In addition, when the underlying sequence is i.i.d. trials, the conditional distribution of the same run statistics, given the numbers of success and failure is investigated. We give further insights into the multivariate run-related problems arising from a sequence of the multistate trials. Besides, our results have potential applications to problems of various research areas and will come to prominence in the future. This research was partially supported by the ISM Cooperative Research Program (2004-ISM·CRP-2007).     

On multiparameter distributions of order k

A multiparameter negative binomial distribution of order k is obtained by compounding the extended (or multiparameter) Poisson distribution of order k by the gamma distribution. A multiparameter logarithmic series distribution of order k is derived next, as the zero truncated limit of the first distribution. Finally a few genesis schemes and interrelationships are established for these three multiparameter distributions of order k. The present work extends several properties of distributions of order k.