A LOCAL MINIMAX LOWER BOUNDFOR NONPARAMETRIC DECONVOLUTIONSecond

Suppose we have i.i.d. observations with a distribution equal to the convolution of an unknown distribution function F and a known distribution function K. We derive a local minimax lower bound for the problem of estimating F at a xed point x 0. We focus on the dependence of the lower bound on the smoothness of the distribution K. The performance with respect to the lower bound of nonparametric maximum likelihood estimators and of kernel type estimators is reviewed. Suppose we have observations X 1 ; : : : ; X n which are independent. The distribution function of these observations is assumed to be equal to the convolution of an unknown distribution function F and a known distribution function K. Examples of such observations are for instance random measurements plus independent measurement errors with a known distribution, or ages of disease diagnosis as the sum of ages of infection plus independent incubation periods with a known distribution. In the rst example K will typically correspond to a symmetric smooth distribution , while in the second example less{smooth distributions like the exponential and gamma distributions are more plausible. We will investigate the dependence of the hardness of the problem of estimating F on this smoothness , to give yet another quantiication of the phrase \the smoother the known distribution K the harder the decon-volution problem". We derive a local minimax lower bound for the problem of estimating F at a xed point x 0 , i.e. a lower bound for the minimal value over all estimators of the maximal expected absolute error in two cases of underlying distributions. One is a xed distribution F and the other is a local perturbation of F, which as n increases approaches the xed F. Related global lower bounds, where the expected absolute error is maximized over a xed set of distributions with a certain minimal smoothness, i.e. having at least a continuous density, have been derived by Carroll and Hall (1988) and Fan (1991). Our perturbations will have bounded densities with discontinuities so our result corresponds to a situation where no smoothness of F further than the existence of a bounded density can be assumed. First we construct the perturbations of F. Deene the higher order diierences of a function u as follows (((0) t u)(x) = u(x); (1) (((l+1) t u)(x) = (((l) t u)(x + t) ? (((l) t u)(x ? …