Estimation of parameters in diffusion models is usually handled by maximum likelihood and involves the calculation of a Radon–Nikodym derivative. This methodology is often not available when minor changes are made to the model. However, these complications can usually be avoided and results obtained under more general conditions using quasi-likelihood methods. The basic ideas are explained in this paper and are illustrated through discussion of the Cox–Ingersoll–Ross model and a modification of the Langevin model.

A rumour model due to Maki and Thompson (1973) is slightly modified to incorporate a continuous-time random contact process and varying individual behaviours in front of the rumour. Two important measures of the final extent of the rumour are provided by the ultimate number of people who have heard the rumour, and the total personal time units during which the rumour is spread. Our purpose in this note is to derive the exact joint distribution of these two statistics. That will be done by constructing a family of martingales for the rumour process and then using a particular family of Gontcharoff polynomials.

We give formulas for updating both the steady-state probability distribution and the fundamental matrices of a singularly perturbed Markov chain. This formula generalizes Schweitzer's regular perturbation formulas to the case of singular perturbations.

In this note we point out an inherent difficulty in numerical identification of bacteria. The problem is that of uniqueness of the taxonomic structure or, in mathematical terms, the lack of statistical identifiability of finite mixtures of multivariate Bernoulli probability distributions shown here.

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

A family of distribution-free statistics along with related tests is defined and properties of its members are studied. These statistics, one of which is the Smirnov-Wald and Wolfowitz statistic, Dn+ yield tests of the one-sided hypotheses. The minimax and maximin tests in this family against a restricted class of alternatives of minimum power are obtained. The connection to a confidence bound of Stringer for the mean is also remarked.

In this short note, we present a simple characterization of the increasing convex ordering on the set of probability distributions on ℝ. We show its usefulness by providing a very short proof of a comparison result for M/GI/1 queues due to Daley and Rolski, and obtained by completely different means.

We provide the explicit expression for the mean coverage function of a generalized voter model on a regular lattice and establish a characterization of the class of the above processes. As a result, we derive the exact rate of convergence of the considered processes to the steady state. We also prove the existence of different processes with the same mean coverage function on a given lattice.

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

We prove that the expected number of particles of the critical contact process on a homogeneous tree is bounded above. This is the first graph for which the behavior of the expected number of particles of the critical contact process is known. As an easy corollary of our result we get that the critical contact process dies out on any homogeneous tree. This completes the work of Pemantle (1992).

Let {ξn, n ≧1} be a sequence of independent real random variables, F denote the common distribution function of identically distributed random variables ξn, n ≧1 and let ξ1 have an arbitrary distribution. Define Xn+1 = k max(Xn, ξ n+1), Yn+ 1 = max(Yn, ξ n+1) – c, Un+1 = l min(Un, ξ n+1), Vn+1 = min(Vn, ξ n+1) + c, n ≧ 1, 0 < k < 1, l > 1, 0 < c < ∞, and X1 = Υ1= U1 = V1 = ξ1. We establish conditions under which the limit law of max(X1, · ··, Xn) coincides with that of max(ξ2, · ··, ξ n+1) when both are appropriately normed. A similar exercise is carried out for the extreme statistics max(Y1, · ··, Yn), min(U1,· ··, Un) and min(V1, · ··, Vn).

In the paper, we discuss the properties of Cowan and Mecke's Markov chain P in general form. We give some criteria for determining whether the chain P is transient, recurrent null or recurrent non-null, and for the chain P to have invariant measures.

Choi and Park [2] derived an expression for the joint stationary distribution of the number of customers of k types who arrive in batches at an infinite-server system of M/M/∞ type. We propose another method of solving this problem and extend the result to the case of general service times (not necessarily independent). We also get a transient solution. Our main result states that the k- dimensional vector of the number of customers of k types in the system is a certain linear function of a (2k – 1)-dimensional vector whose coordinates are independent Poisson random variables.

We rigorously prove that for a stochastic process, , the existence of a first regeneration time, R1, implies the existence of an infinite sequence of such times, {R1, R2, · ·· }, and hence that the definition of regenerative process need only demand the existence of a first regeneration time. Here we include very general processes up to and including processes where cycles are stationary but not necessarily independent and identically distributed.

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

In this paper we study the behavior of a delayed compound renewal process, S, about some fixed level, L. Normally, a jump process S increases at random times τ1, τ2, …, in random increments until it crosses L. S would then be terminated in a random number v of phases at time τv. In many applications, a more general termination scenario assumes that S may evolve either through v or σ random phases, whichever of the two is smaller (denoted by T). The number T of actual phases is called the termination index, and we evaluate a joint functional of T, the termination time τ T and the termination level ST. We also seek information about the process S a step before its termination, and derive a joint functional for all relevant processes. Examples of these processes and their applications to various stochastic models are discussed.

We give a sample path proof of the well-known Wiener–Hopf identity F = G– + G+ – G– ∗G+ which relates the ladder-height distributions G– and G+ of a simple random walk to the step distribution F. Unlike previous approaches this direct proof is both simple and intuitive.

The Jackson network under study receives batch arrivals at i.i.d. intervals and features Markovian routing and exponentially distributed service times. The system is shown to be stable, in the sense of not being overloaded, if and only if, for each node, the total arrival rate of external and internal customers is less than the service rate. The method of proof is of more general interest.

In this paper, a repairable system consisting of one unit and a single repairman is studied. Assume that the system after repair is not as good as new. Under this assumption, a bivariate replacement policy (T, N), where T is the working age and N is the number of failures of the system is studied. The problem is to determine the optimal replacement policy (T, N)∗such that the long-run average cost per unit time is minimized. The explicit expression of the long-run average cost per unit time is derived, and the corresponding optimal replacement policy can be determined analytically or numerically. Finally, under some conditions, we show that the policy (T, N)∗ is better than policies N∗ or T∗.

An observer watches one of a set of Poisson streams. He may switch from one stream to another instantaneously. If an arrival occurs in a stream while the observer is watching another stream, he does not see the arrival. The experiment terminates when the observer sees an arrival. We derive a formula which states essentially that the expected total time that the observer watches a stream is equal to the probability that he sees the arrival in this stream divided by the intensity of the stream. This formula is valid independently of the observation policy. We also discuss applications of this formula.