Hybrid projection estimation for a wide class of functional parameters

In the framework of functional parameters estimation (such as e.g. density estimation), we consider a wide class characterized by the fact that its elements can be written as limits of sums of the expected values of random variables. We propose an ``hybrid'' projection estimator of such a general functional parameter when we observe n realizations of a discrete time stochastic process (X_t). The estimator is said ``hybrid'' because the dimension of the projection subspace is chosen differently according to the sample size, very large or not. From the asymptotic point of view, this estimator locally reaches a superoptimal rate for the mean integrated square error (MISE) on a dense subset of the space to which the considered functional parameter is supposed to belong, and we state under which hypotheses there is a near-optimal rate of convergence elsewhere in L^2. Note that some hypothesis have been relaxed with respect to previous literature. The finite sample performance is clearly improved with respect to other estimators of the same kind; indeed performance is evaluated through a simulation study, where the parameter to estimate is the spectral density: the proposed estimator is shown to reduce often drastically the MISE in comparison with that of the classical projection estimator and the kernel estimator. Finally, from the practioner point of view this new estimator can be completely data-driven with only a smoothing parameter to choose.