2000 Mathematics Subject Classification: 62G07, 60F10.In this paper we prove large and moderate deviations principles for the recursive kernel estimator of a probability density function and its partial derivatives. Unlike the density estimator, the derivatives estimators exhibit a quadratic behaviour not only for the moderate deviations scale but also for the large deviations one. We provide results both for the pointwise and the uniform deviations.

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...

We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by two parameters. The first parameter governs the lacunarity of the wavelet coefficients while the second one governs its intensity. In this paper, we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build optimal...

We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by two parameters. The first parameter governs the lacunarity of the wavelet coefficients while the second one governs its intensity. In this paper, we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build 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...