Estimating a density and its derivatives via the minimum distance method
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 1/\(\sqrt{n}\) 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.
scientific article; zbMATH DE number 4050763
scientific article