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.
"On optimal adaptive estimation of a quadratic functional." Ann. Statist. 24 (3) 1106 - 1125, June 1996. https://doi.org/10.1214/aos/1032526959