Adaptive estimation of distribution density in the basis of algebraic polynomials

This paper is devoted to the problem of adaptive statistical estimation of the distribution density defined on a finite interval. Projective-type estimators in the basis of Jacobi polynomials is considered. An adaptive statistical estimator, which is asymptotically minimax in the case of mean-square losses for all sets from a certain family of contracting sets of functions having different smoothness, is constructed. The smoothness conditions are stated in terms of $L_2$-norms of residuals of distribution densities when approximating them by linear combinations of a finite number of the first Jacobi polynomials. Extension of the result to other orthonormal bases possessing some natural regularity properties is also discussed.