ON BICKEL AND RITOVS CONJECTURE ABOUT ADAPTIVE ESTIMATION OF THE INTEGRAL OF THE SQUARE OF DENSITY DERIVATIVE

Bickel and Ritov suggested an optimal estimator for the integral of th e square of the hth derivative of a density when the unknown density b elongs to a Lipschitz class of a given order beta. In this context opt imality means that the estimate is asymptotically efficient, that is, it has the best constant and rate of risk convergence, whenever beta > 2k + 1/4, and it is rate optimal otherwise. The suggested optimal est imator crucially depends on the value of beta which is obviously unkno wn. Bickel and Ritov conjectured that the method of cross validation l eads to a corresponding adaptive estimator which has the same optimal statistical properties as the optimal estimator based on prior knowled ge of beta. We show for probability densities supported over a finite interval that when beta > 2k + 1/4 adaptation is not necessary for the construction of an asymptotically efficient estimator. On the other h and, it is not possible to construct an adaptive estimator which has t he same rate of convergence as the optimal nonadaptive estimator as so on as k < beta less than or equal to 2k + 1/4.