Estimating a density and its derivatives via the minimum distance method

Summary This paper deals with minimum distance (MD) estimators and minimum penalized distance (MPD) estimators which are based on the L _p distance. Rates of strong consistency of MPD density estimators are established within the family of density functions which have a bounded m-th derivative. For the case p=2, it is also proved that the MPD density estimator achieves the optimum rate of decrease of the mean integrated square error and the L ₁ error. Estimation of derivatives of the density is considered as well.In a class parametrized by entire functions, it is proved that the rate of convergence of the MD density estimator (and its derivatives) to the unknown density (its derivatives) is of order in expected L ₁ and L ₂ distances. In the same class of distributions, MD estimators of unknown density and its derivatives are proved to achieve an extraordinary rate (log log n/n)^1/2 of strong consistency.