Adaptive estimation of the spectrum of a stationary Gaussian sequence

In this paper, we study the problem of nonparametric adaptive estimation of the spectral density f of a stationary Gaussian sequence. For this purpose, we consider a collection of finite-dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on possibly irregular grids or spaces of trigonometric polynomials). We estimate the spectral density by a projection estimator based on the periodogram and constructed on a data-driven choice of linear space from the collection. This data-driven choice is made via the minimization of a penalized projection contrast. The penalty function depends on parallel to integral parallel to (infinity), but we give results including the estimation of this bound. Moreover, we give extensions to the case of unbounded spectral densities (long-memory processes). In all cases, we state non-asymptotic risk bounds in L-2-norm for our estimator, and we show that it is adaptive in the minimax sense over a large class of Besov balls.