We discuss the estimation of the asymptotic covariance matrix of semi-parametric maximum likelihood estimators by the observed profile information. We show that a discretized version of the second derivative of the profile likelihood function yields consistent estimators of minus the efficient information matrix.

We study a bootstrap method for stationary real-valued time series, which is based on the sieve of autoregressive processes. Given a sample $X_1,...,X_n$ from a linear process $\{X_t{\}}_{t\in \mathbb{Z}}$, we approximate the underlying process by an autoregressive model with order $p=p\left(n\right)$, where $p\left(n\right)\to \mathrm{\infty },p\left(n\right)=o\left(n\right)$ as the sample size $n\to \mathrm{\infty }$. Based on such a model, a bootstrap process $\{X_t^{*}{\}}_{t\in \mathbb{Z}}$ is constructed from which one can draw samples of any size.

We show that, with high probability, such a sieve bootstrap process $\{X_t^{*}{\}}_{t\in \mathbb{Z}}$ satisfies a new type of mixing condition. This implies that many results for stationary mixing sequences carry over to the sieve bootstrap process. As an example we derive a functional central limit theorem under a bracketing condition.

We study the functional estimation of the space-dependent diffusion coefficient in a one-dimensional framework. The sample path is observed at discrete times. We study global $L_p$-loss errors $(1\le p<+\mathrm{\infty })$ over Besov spaces $B_{\mathrm{sp}\mathrm{\infty }}$. We show that, under suitable conditions, the minimax rate of convergence is the usual $n^{-s/(1+2s)}$. Linking our model to nonparametric regression, we provide an estimating procedure based on a linear wavelet method which is optimal in the minimax sense.

Multivariate versions of the law of large numbers and the central limit theorem for martingales are given in a generality that is often necessary when studying statistical inference for stochastic process models. To illustrate the usefulness of the results, we consider estimation for a multidimensional Gaussian diffusion, where results on consistency and asymptotic normality of the maximum likelihood estimator are obtained in cases that were not covered by previously published limit theorems. The results are also applied to martingales of a different nature, which are typical of the problems occurring in connection with statistical inference for stochastic delay equations.

This paper introduces in detail a new systematic method to construct approximate finite-dimensional solutions for the nonlinear filtering problem. Once a finite-dimensional family is selected, the nonlinear filtering equation is projected in Fisher metric on the corresponding manifold of densities, yielding the projection filter for the chosen family. The general definition of the projection filter is given, and its structure is explored in detail for exponential families. Particular exponential families which optimize the correction step in the case of discrete-time observations are given, and an a posteriori estimate of the local error resulting from the projection is defined. Simulation results comparing the projection filter and the optimal filter for the cubic sensor problem are presented. The classical concept of assumed density filter (ADF) is compared with the projection filter. It is shown that the concept of ADF is inconsistent in the sense that the resulting filters depend on the choice of a stochastic calculus, i.e. the Itô or the Stratonovich calculus. It is shown that in the context of exponential families, the projection filter coincides with the Stratonovich-based ADF. An example is provided, which shows that this does not hold in general, for non-exponential families of densities.

Let $\{Z_n,n\ge 0\}$ be an aperiodic irreducible recurrent (not necessarily positive recurrent) Markov chain taking values on a countable unbounded subset $S$ of $R^d$, $\pi (\cdot )$ its invariant measure and $f$ is a non-negative function defined on $S$. We first find sufficient conditions under which ${\int }_S^{}f\left(z\right)\pi (dz)=\mathrm{\infty }$ (the corresponding result for the finiteness of ${\int }_S^{}f\left(z\right)\pi (dz)$ was obtained by Tweedie). Then we obtain lower and upper bounds for the values of the invariant measure $\pi$ on the subsets $B$ of $S$, that is, $\pi \left(B\right)$. These bounds are expressed in terms of first passage probabilities and the first exit time from $B$. We also show how to estimate the latter quantities using sub- or supermartingale techniques. The results are finally illustrated for driftless reflected random walks in ${\mathbb{Z}}_{+}^2$ and for Markov chains on non-negative reals with asymptotically small drift of Lamperti type. In both cases we obtain very precise information on the asymptotic behaviour of their stationary measures.