On optimal adaptive estimation of a quadratic functional

Abstract

Minimax mean-squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes. In contrast to many nonparametric function estimation problems there are both regular and irregular cases. In the regular cases the minimax mean-squared error converges at a rate proportional to the inverse of the sample size, whereas in the irregular case much slower rates are the rule.We investigate the problem of adaptive estimation of a quadratic functional of a smooth function when the degree of smoothness of the underlying function is not known. It is shown that estimators cannot achieve the minimax rates of convergence simultaneously over two parameter spaces when at least one of these spaces corresponds to the irregular case. A lower bound for the mean squared error is given which shows that any adaptive estimator which is rate optimal for the regular case must lose a logarithmic factor in the irregular case. On the other hand, we give a rather simple adaptive estimator which is sharp for the regular case and attains this lower bound in the irregular case. Moreover, we explicitly describe a subset of functions where our adaptive estimator loses the logarithmic factor and show that this subset is relatively small.

Adaptive estimation of a quadratic functional of a density by model selection

“…Efroimovich and Low [8] were the first to propose adaptive estimators of quadratic functionals in the framework of the Gaussian sequence model. The estimator θ n proposed by Efroimovich and Low is inspired by Lepskii's method of adaptation and it has the following adaptive properties: for any positive R and α, provided that the sequence (β λ ) λ≥1 satisfies to the condition β 2 λ λ 2α+1 ≤ R 2 for all λ (which means that (β λ ) λ≥1 belongs to some hyperrectangle) one has…”

On adaptive wavelet estimation of a quadratic functional from a deconvolution problem

“…). See, for instance, Bickel and Ritov (1988), Donoho and Nussbaum (1990), Kerkyacharian and Picard (1996), Efromovich and Low (1996), Gayraud and Tribouley (1999), Johnstone (2001a,b), Laurent (2005), Low (2005, 2006) and Rivoirard and Tribouley (2008).…”

Sharp adaptive estimation of quadratic functionals

“…Efroimovich (1994) gave a sharp adaptive procedure for the regular case. Efroimovich and Low (1996) gave a rate optimal adaptive procedure for the irregular case which was simultaneously sharp adaptive for the regular case. They considered l 2 body and Hölder body (p = ∞).…”

“…The sharp adaptive procedure given in this article is of similar type as the method of Efroimovich and Low (1996), that is, Lepski method is applied to choose between diagonal quadratic estimators. A difference to Efroimovich and Low (1996) is that the parameters of the procedure: the threshold and the coefficients of the quadratic estimators, are chosen in a more careful way.…”

“…A difference to Efroimovich and Low (1996) is that the parameters of the procedure: the threshold and the coefficients of the quadratic estimators, are chosen in a more careful way. The more careful choice of the parameters leads to sharper optimality of the procedure.…”