The integral of the mean squared error of an estimator of a regression function is used as a criterion for defining an optimal design measure in the context of local linear regression, when the bandwidth is chosen in a locally optimal manner. An algorithm is proposed that constructs a sequence of piecewise uniform designs with the help of current estimates of the integral of the mean squared error. These estimates do not require direct estimation of the second derivative of the regression function. Asymptotic properties of the algorithm are established and numerical results illustrate the gains that can be made, relative to a uniform design, by using the optimal design or sub-optimal, piecewise uniform designs. The behaviour of the algorithm in practice is also illustrated.

Asymptotically minimax nonparametric estimation of a regression function observed in white Gaussian noise over a bounded interval is considered, with respect to a L₂-loss function. The unknown function f is assumed to be m times differentiable except for an unknown although finite number of jumps, with piecewise mth derivative bounded in L₂ norm. An estimator is constructed, attaining the same optimal risk bound, known as Pinsker's constant, as in the case of smooth functions (without jumps).

The large deviation properties of p^*, the approximation to the conditional density of the maximum likelihood estimator, and r^*, the modified directed likelihood, are studied. Attention is restricted to curved exponential models. Various specifications of an approximate ancillary, which are required in the construction of p^*and r^*, are considered, including: a modified directed likelihood ancillary, a^*, and an unmodified directed likelihood ancillary, a^o. It is shown that if a^* is used then p^* and r^* achieve saddlepoint accuracy on both normal and large deviation regions; if, on the other hand, a^o is used in the construction of p^*, then saddlepoint accuracy is not achieved, though the relative error still stays bounded on large deviation regions. It is also shown that if a^o rather than a^* is held fixed in the sample space differentiations needed to calculate r^*, then saddlepoint accuracy is still attained in both normal and large deviation regions. On a first impression, the last result is a little surprising because, in a repeated sampling framework, a^o is only ancillary to order O(n^-1/2). However, this finding is also of direct practical interest because, from the point of view of calculation, a^o is often substantially easier to work with than a^*. An important aspect of our approach is the development of guidelines, referred to as Laplace-spa calculus, for the construction of invariant saddlepoint-style approximations to marginal and conditional densities. Finally, connections between recent work by Jensen and the results of this paper are discussed and clarified.

Cox showed that the likelihood of regression models for discrete-time processes factors into a partial likelihood and a product of conditional laws for the covariates, given the history. Jacod constructed a partial likelihood for continuous-time regression models in terms of the predictable characteristics of the response process. Here we prove a factorization of the likelihood, analogous to Cox's, assuming both the response and the covariates to be semimartingales. The result is useful for counting process regression modelling and inference, and also for regression involving continuous processes and diffusions with jumps.

An anticipating stochastic integral is proposed for 'normal martingales'. It agrees with the Skorohod integral in the Brownian case. A variational derivative of Malliavin type is also defined. An integration by parts formula is given which has some subtle and important differences from the formula in the Brownian case. The existence and uniqueness of solutions of linear stochastic differential equations with anticipating exogenous driving terms are also established.

Let $\left\{L\right(t):t\ge 0\}$ be Lévy's, let $\gamma :R_{+}⤅R$, and let $\{Z_t:t\ge 3\}$ be the stochastic process defined by $Z_t\left(s\right)=L\left(\mathrm{ts}\right)/\left(2t\log \log t\right),0\le s\le 1$. Conditions on $\gamma$ are given such that the set of all limit points of $\left\{\gamma \right(t)Z_t:t\ge 3\}$ as $t\to \mathrm{\infty }$ is a.s. equal to the set of all continuous functions defined on $[0,1]$ which vanish at 0.

For a self-similar α-stable process with stationary increments {X(t), 0 ≤t ≤1} we study the asymptotic behaviour of the probability that the process stays within the interval [-ε,ε], as ε becomes small. This behaviour turns out to be only partially determined by the index of stability α and parameter of self-similarity H.