Let $\{X_n\}_{n\in{\mathbb{N}}}$ be an ${\mathbb{X}}$-valued iterated function system (IFS) of Lipschitz maps defined as $X_0 \in {\mathbb{X}}$ and for $n\geq 1$, $X_n\;:\!=\;F(X_{n-1},\vartheta_n)$, where $\{\vartheta_n\}_{n \ge 1}$ are independent and identically distributed random variables with common probability distribution $\mathfrak{p}$, $F(\cdot,\cdot)$ is Lipschitz continuous in the first variable, and $X_0$ is independent of $\{\vartheta_n\}_{n \ge 1}$. Under parametric perturbation of both F and $\mathfrak{p}$, we are interested in the robustness of the V-geometrical ergodicity property of $\{X_n\}_{n\in{\mathbb{N}}}$, of its invariant probability measure, and finally of the probability distribution of $X_n$. Specifically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with autoregressive conditional heteroscedastic errors, and for IFS under roundoff error or under thresholding/truncation. Moreover, we provide a general set of assumptions covering the classical Feller-type hypotheses for an IFS to be a V-geometrical ergodic process. An accurate bound for the rate of convergence is also provided.

This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of non-Gaussian positive-definite matrix-valued random fields, adapted to the statistical inverse problems in high stochastic dimension for their experimental identification, is introduced and its properties are analysed. A minimal parameterisation of discretised random fields belonging to this general class is proposed. Using this parameterisation of the general class, a complete identification procedure is proposed. New results of the mathematical and numerical analyses of the parameterised stochastic elliptic boundary value problem are presented. The numerical solution of this parametric stochastic problem provides an explicit approximation of the application that maps the parameterised general class of random fields to the corresponding set of random solutions. This approximation can be used during the identification procedure in order to avoid the solution of multiple forward stochastic problems. Since the proposed general class of random fields possibly contains random fields which are not uniformly bounded, a particular mathematical analysis is developed and dedicated approximation methods are introduced. In order to obtain an algorithm for constructing the approximation of a very high-dimensional map, complexity reduction methods are introduced and are based on the use of sparse or low-rank approximation methods that exploit the tensor structure of the solution which results from the parameterisation of the general class of random fields.

This paper applies the theory of the quasi-likelihood method to model-based inference for sample surveys. Currently, much of the theory related to sample surveys is based on the theory of maximum likelihood. The maximum likelihood approach is available only when the full probability structure of the survey data is known. However, this knowledge is rarely available in practice. Based on central limit theory, statisticians are often willing to accept the assumption that data have, say, a normal probability structure. However, such an assumption may not be reasonable in many situations in which sample surveys are used. We establish a framework for sample surveys which is less dependent on the exact underlying probability structure using the quasi-likelihood method.

In this paper, we consider the question of which convergence properties of Markov chains are preserved under small perturbations. Properties considered include geometric ergodicity and rates of convergence. Perturbations considered include roundoff error from computer simulation. We are motivated primarily by interest in Markov chain Monte Carlo algorithms.

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.

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.