No abstract

For nonparametric density estimation, first we give optimal sampling schemes of continuous time processes. Next we study effects of known or small errors-in-variables on such samplings. Throughout the paper various simulations are also presented.MSC 2000: Primary 62G07 Secondary 62D05, 62M09, 60H35.

In this paper, we adopt a Bayesian point of view for predicting real continuous-time processes. We give two equivalent definitions of a Bayesian predictor and study some properties: admissibility, prediction sufficiency, nonunbiasedness, comparison with efficient predictors. Prediction of Poisson process and prediction of Ornstein-Uhlenbeck process in the continuous and sampled situations are considered. Various simulations illustrate comparison with non-Bayesian predictors.2000 Mathematics Subject Classification. Primary 62M20, 62F15.

We consider a real Gaussian process X with unknown smoothness r 0 ∈ N 0 where the mean-square derivative X (r 0) is supposed to be Hölder continuous in quadratic mean. First from selected sampled observations, we study reconstruction of X(t), t ∈ [0, 1] with X r (t), a piecewise polynomial interpolation of degree r ≥ 1. We show that the mean-square error of the interpolation is a decreasing function of r but becomes stable as soon as r ≥ r 0. Next, from an interpolation-based empirical criterion and n sampled observations of X, we derive an estimator r n of r 0 and prove its strong consistency by giving an exponential inequality for P(r n = r 0). Finally, we establish the strong consistency of X max(rn,1) (t) with an almost optimal rate.

In continuous time, rates of convergence of density estimators fluctuate with the nature of observed sample paths. In this paper, we give a family of rates reached by the kernel estimator and we show that these rates are minimax. Finally, we study applications of these results for specific classes of processes including the Gaussian ones.