The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions $e_k$, k=0,1,2,..., for the observation model $y_i=f\left(x_{in}\right)+{\eta }_i$, i=1,...,n, is considered, where ${\eta }_i$ are uncorrelated random variables with zero mean value and finite variance, and the observation points $x_{in}\in [0,2\pi ]$, i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $(1/n){\sum }_{i=1}^{n}E{(f\left(x_{in}\right)-{\widehat{f}}_{N\left(n\right)}\left(x_{in}\right))}^2$, the integrated mean-square error $E\Vert f-{\widehat{f}}_{N\left(n\right)}{\Vert }^2$ and the pointwise mean-square error $E{\left(f\left(x\right){-}_{N\left(n\right)}\left(x\right)\right)}^2$ of the estimator ${\widehat{f}}_{N\left(n\right)}\left(x\right)={\sum }_{k=0}^{N\left(n\right)}{\widehat{c}}_k{e}_k\left(x\right)$ for f ∈ C[0,2π] and...

In this paper, we investigate the problem of the conditional cumulative of a scalar response variable given a random variable taking values in a semi-metric space. The uniform almost complete consistency of this estimate is stated under some conditions. Moreover, as an application, we use the obtained results to derive some asymptotic properties for the local linear estimator of the conditional quantile.

We study the estimation of the mean function of a continuous-time stochastic process and its derivatives. The covariance function of the process is assumed to be nonparametric and to satisfy mild smoothness conditions. Assuming that n independent realizations of the process are observed at a sampling design of size N generated by a positive density, we derive the asymptotic bias and variance of the local polynomial estimator as n,N increase to infinity. We deduce optimal sampling densities, optimal...

We construct a data-driven projection density estimator for continuous time processes. This estimator reaches superoptimal rates over a class F0 of densities that is dense in the family of all possible densities, and a «reasonable» rate elsewhere. The class F0 may be chosen previously by the analyst. Results apply to Rd-valued processes and to N-valued processes. In the particular case where square-integrable local time does exist, it is shown that our estimator is strictly better than the local...