Katuomi Hirano

On discrete distributions of orderk

SummaryThis paper gives some results on calculation of probabilities and moments of the discrete distributions of orderk. Further, a new distribution of orderk, which is called the logarithmic series distribution of orderk, is investigated. Finally, we discuss the meaning of theorder of the distributions.

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 (1≤l≤k−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.

On Ling's binomial and negative binomial distributions of order k

Type II binomial distribution of order k was introduced by Ling (1988). In this article the probability function, the probability generating function and asymptotic properties of the distribution are discussed. Some properties of the corresponding negative binomial distribution of order k, which was introduced by Ling (1989) are also examined.

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 μ less than the specified length of the runs, an enumeration scheme called μ-overlapping way of counting is defined. Let k and ℓ be positive integers satisfying ℓ < k. Based on independent Bernoulli trials, it is shown that the number of (ℓ− 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length ℓ follows the generalized binomial distribution of order (k−ℓ). 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.

Joint distributions of numbers of success-runs and failures until the first consecutivek successes

Joint distributions of the numbers of failures, successes and success-runs of length less thank until the first consecutivek successes are obtained for some random sequences such as a sequence of independent and identically distributed integer valued random variables, a {0, 1}-valued Markov chain and a binary sequence of orderk. There are some ways of counting numbers of runs with a specified length. This paper studies the joint distributions based on three ways of counting numbers of runs, i.e., the number of overlapping runs with a specified length, the number of non-overlapping runs with a specified length and the number of runs with a specified length or more. Marginal distributions of them can be derived immediately, and most of them are surprisingly simple.

Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials

In this paper we study exact distributions of sooner and later waiting times for runs in Markov dependent bivariate trials. We give systems of linear equations with respect to conditional probability generating functions of the waiting times. By considering bivariate trials, we can treat very general and practical waiting time problems for runs of two events which are not necessarily mutually exclusive. Numerical examples are also given in order to illustrate the feasibility of our results.

LIFETIME DISTRIBUTION AND ESTIMATION PROBLEMS OF CONSECUTIVE-k-OUT-OF-n:F SYSTEMS*

Explicit formula is given for the lifetime distribution of a consecutive-k-out-of-n:F system. It is given as a linear combination of distributions of order statistics of the lifetimes of n components. We assume that the lifetimes are independent and identically distributed. The results should make it possible to treat the parametric estimation problems based on the observations of the lifetimes of the system. In fact, we take up, as some examples, the cases where the lifetimes of the components follow the exponential, the Weibull, and the Pareto distributions, and obtain feasible estimators by moment method. In particular, it is shown that the moment estimator is quite good for the exponential case in the sense that the asymptotic efficiency is close to one.

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.

Distributions of Numbers of Success-Runs Until the First Consecutive k Successes in Higher Order Markov Dependent Trials

The distributions of numbers of overlapping and non-overlapping occurrences of success-runs of length l, and the distributions of numbers of occurrences of success-runs of exact length l and of length l or more until the first occurrence of success-run of length k in the m-th order Markov dependent trials are studied. When m ≤ l < k the derived distributions do not depend on the initial distribution of the m-th order Markov chain and they are shown to be equal to the corresponding distributions considered in independent trials with a success probability whose value is given by the transition probability that a success occurs after a success-run of length m in the m-th order Markov chain. When l < m the distributions depend on the initial distribution of the m-th order Markov chain and are not necessarily so simple as the above case. A method for deriving the probability generating function of the conditional distribution of the number of overlapping occurrences of success-runs of length l until the first occurrence of success-run of length k under each initial condition is given.

Waiting time problems for a two-dimensional pattern

We consider waiting time problems for a two-dimensional pattern in a sequence of i.i.d. random vectors each of whose entries is 0 or 1. We deal with a two-dimensional pattern with a general shape in the two-dimensional lattice which is generated by the above sequence of random vectors. A general method for obtaining the exact distribution of the waiting time for the first occurrence of the pattern in the sequence is presented. The method is an extension of the method of conditional probability generating functions and it is very suitable for computations with computer algebra systems as well as usual numerical computations. Computational results applied to computation of exact system reliability are also given.