Problems of detecting the arrival of a ‘disorder' are of central importance in quality control. One such problem is formulated as an optimal stopping problem in which information about the process to be stopped may be bought at any time. A control policy will involve a determination of times at which information should be bought and of the time at which the process should be stopped. It is shown that in all circumstances there is a unique policy satisfying necessary conditions for optimality; the various circumstances and control policies are specified.

This paper considers the discrete time Markov decision model with a denumerable state space and finite action space. Under certain conditions it is proved that the minimal total expected cost for a planning horizon of n epochs minus n times the minimal long-run average expected cost per unit time has a finite limit as n → ∞ for each initial state.

The paper deals with two approaches to the estimation of the parameters β and σ2 in the General Gauss-Markoff (GGM) model represented by the triplet (Y, Xß, σ2V), where E(Y)=Xβ and D(Y) =σ2V, when no assumptions are made about the ranks of X and V. One is called Inverse Partition Matrix (IPM) method, which depends on the numerical evaluation of the g-inverse of a partitioned matrix. The second is an analogue of least squares theory applicable even when V is singular, unlike Atiken's method which is applicable only for non-singular V, and is called Unified Least Square (ULS) method.

In many Markov chain models, the immediate characteristic of importance is the positive recurrence of the chain. In this note we investigate whether positivity, and also recurrence, are robust properties of Markov chains when the transition laws are perturbed. The chains we consider are on a fairly general state space : when specialised to a countable space, our results are essentially that, if the transition matrices of two irreducible chains coincide on all but a finite number of columns, then positivity of one implies positivity of both; whilst if they coincide on all but a finite number of rows and columns, recurrence of one implies recurrence of both. Examples are given to show that these results (and their general analogues) cannot in general be strengthened.

A model for the transition probability matrices (t.p.m.'s) of finite discrete Markov chains is suggested which may help those who wish to use a larger number of states than would seem reasonable with the data available in the current estimation situation. The model is especially useful in that a finite t.p.m. of arbitrary size can be specified by as few as two parameters. An example of the model's estimation and use is presented, showing it in a fair light in comparison with the conventional method of t.p.m. specification.

The coefficient of correlation between suitably defined statistics provides a means of testing various parametric hypotheses. The distribution of such coefficients, when the null hypothesis is not true, takes various forms in different applications. In particular, the correlation coefficients used to test hypotheses about the structural parameters in linear functional relations have doubly non-central distributions.

For tests of correlation in some typical situations, the curvature of the function defining the power in terms of the parameter in the neighbourhood of the null hypothesis is determined. This curvature is shown to be simply related to that of the function defining the squared correlation coefficient in terms of the parameter, in the neighbourhood of its minimum.

This paper is concerned with the solution of an initial and boundary value problem for a parabolic differential equation driven by a stochastic point process.

Given two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n–1f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).

The Markov-Dobrushin condition for (weak) ergodicity of non-homogeneous discrete-time Markov chains, and an analogous criterion for continuous chains, are derived by means of coupling techniques.

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

In this paper a problem arising in queueing and dam theory is studied. We shall consider a G/G*/1 queueing model, i.e., a G/G/1 queueing model of which the service process is a separable centered process with stationary independent increments. This is a generalisation of the well-known G/G/1 model with constant service rate.

Several results concerning the amount of work done by the server, the busy cycles etc., are derived, mainly using the well-known method of Pollaczek. Emphasis is laid on the similarities and dissimilarities between the results of the ‘classical’ G/G/1 model and the G/G*/1 model.

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.

The NIPALS approach is applied to the ‘soft’ type of model that has come to the fore in sociology and other social sciences in the last five or ten years, namely path models that involve latent variables which serve as proxies for blocks of directly observed variables. Such models are seen as hybrids of the ‘hard’ models of econometrics where all variables are directly observed (path models in the form of simultaneous equations systems) and the ‘soft’ models of psychology where the human mind is described in terms of latent variables and their directly observed indicators. For hybrid models that involve one or two latent variables the NIPALS approach has been developed in [38], [41] and [42]. The present paper extends the NIPALS approach to path models with three or more latent variables. Each new latent variable brings a rapid increase in the pluralism of possible model designs, and new problems arise in the parameter estimation of the models. Iterative procedures are given for the point estimation of the parameters. With a view to cases when the iterative estimation does not converge, a device of range estimation is developed, where high profile versus low profile estimates give ranges for the parameter estimates.

This paper gives the asymptotic distributions, as the time period grows infinite, of the first exit times above a fixed constant and from upper and lower constant boundaries for the Ornstein-Uhlenbeck stochastic process. The results of a large amount of numerical analysis illustrate the asymptotic forms.

The purpose of this paper is to show that the empirical characteristic function, when suitably normalised, converges weakly to a stationary Gaussian process whose autocovariance function is the theoretical characteristic function.

This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution.

The G/M/m queue with only s waiting places is studied. We start by studying the joint distribution of the number of customers present at time t and the time elapsing until the next arrival after t. This gives the asymptotic distribution of the number of customers at the moment of an arrival and at an arbitrary moment. Then waiting time and virtual waiting time distributions are easily obtained. For the G/M/1 queue also the transient solution is given. Finally the case s = ∞ is considered.

In this study some of the basic ideas needed for the application of the Wiener-Hopf Technique in solving problems occurring in applied probability theory are discussed; the paper aims to give a short introduction. The method is illustrated by applying it to two problems; one, although basic in probability theory, is rather simple to handle by this method. The second is much more intricate, but shows clearly the power of the method.

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk > c‥kp], c ≥ 0, 0 ≤ p < 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 < α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].

A general birth-and-death queueing model is considered with N waiting positions (0 ≦ N ≦ ∞), s servers (1 ≦ s ≦ ∞) and a first-come-first-served queueing discipline. The first and second order moments of the steady-state waiting-time (excluding service) for a non-lost arriving customer are given. By setting the busy period equal to the time where at least one service is in progress, we obtain the first and second order moments of the length of a busy period and also the distribution of the number served during it, given an arbitrary number of customers present originally. Using a direct approach all expressions are given in explicit forms which, although being far from elegant, are suitable for evaluation on a computer.