Abstract:
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.
Fulltext:
PDF file (1605 kB)