ON OPTIMAL ADAPTIVE ESTIMATION OF A QUADRATIC FUNCTIONAL

Minimax mean-squared error estimates of quadratic functionals of smoot h 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 quadr atic functional of a smooth function when the degree of smoothness of the underlying function is not known. It is shown that estimators cann ot achieve the minimax rates of convergence simultaneously over two pa rameter spaces when at least one of these spaces corresponds to the ir regular case. A lower bound for the mean squared error is given which shows that any adaptive estimator which is rate optimal for the regula r case must lose a logarithmic factor in the irregular case. On the ot her hand, we give a rather simple adaptive estimator which is sharp fo r the regular case and attains this lower bound in the irregular case, Moreover, we explicitly describe a subset of functions where our adap tive estimator loses the logarithmic factor and show that this subset is relatively small.