Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramér–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

We describe a computational procedure for evaluating the quasi-stationary distributions of a continuous-time Markov chain. Our method, which is an ‘iterative version' of Arnoldi's algorithm, is appropriate for dealing with cases where the matrix of transition rates is large and sparse, but does not exhibit a banded structure which might otherwise be usefully exploited. We illustrate the method with reference to an epidemic model and we compare the computed quasi-stationary distribution with an appropriate diffusion approximation.

This paper considers the histogram of unit cell size built up from m independent observations on a Poisson (μ) distribution. The following question is addressed: what is the limiting probability of the event that there are no unoccupied cells lying to the left of occupied cells of the histogram? It is shown that the probability of there being no such isolated empty cells (or isolated finite groups of empty cells) tends to unity as the number m of observations tends to infinity, but that the corresponding almost sure convergence fails. Moreover this probability does not tend to unity when the Poisson distribution is replaced by the negative binomial distribution arising when μ is randomized by a gamma distribution. The relevance to empirical Bayes statistical methods is discussed.

We consider the likelihood ratio tests to detect an epidemic alternative in the following two cases of normal observations: (1) the alternative specifies a square wave drift in the mean value of an i.i.d. sequence; (2) the alternative permits a square wave drift in the intercept of a simple linear regression model. To develop the approximations for the significance levels leads us to consider boundary-crossing problems of some two-dimensional discrete-time Gaussian fields. By the method which was proposed originally by Woodroofe (1976) and adapted to study maxima of some random fields by Siegmund (1988), some large deviations for the conditional non-linear boundary-crossing probabilities are developed. Some results of Monte Carlo experiments confirm the accuracy of these approximations.

Let {Xn, n ≧ 1} be an i.i.d. sequence of positive random variables with a continuous distribution function having a regularly varying upper tail. Denote by {X(n), n ≧ 1} the corresponding sequence of record values. We introduce two statistics based on the sequence of successive record values and investigate their asymptotic behaviour. We also give some numerical results.

The paper deals with a problem which arises in the Monte Carlo optimization of steady state or ergodic systems which can be modelled by Markov chains. The transition probability depends on a parameter, and one wishes to find the parameter value at which some performance function is minimum. The only available data are obtained from either simulation or actual operating information. For such a problem one needs good statistical estimates of the derivatives. Conditions are given for the existence of the derivative of the stationary measure with respect to the parameter, in the sense that the derivative is a signed measure, and is the limit of the natural approximating sequence. Some properties and a useful characterization of the derivative are obtained. It is also shown that, under appropriate conditions, the derivative of the n-step transition function converges to the derivative of the stationary measure as n tends to ∞. This latter result is of particular importance whether one is simply estimating or is actually optimizing via some sequential Monte Carlo procedure, since the basic observations are always taken over a finite time interval.