Notes on consistency of some minimum distance estimators with simulation results

Abstract

We focus on the minimum distance density estimators ${\widehat{f}}_n$ of the true probability density $f_0$ on the real line. The consistency of the order of $n^{-1/2}$ in the (expected) L$_1$-norm of Kolmogorov estimator (MKE) is known if the degree of variations of the nonparametric family $\mathcal {D}$ is finite. Using this result for MKE we prove that minimum Lévy and minimum discrepancy distance estimators are consistent of the order of $n^{-1/2}$ in the (expected) L$_1$-norm under the same assumptions. Computer simulation for these minimum distance estimators, accompanied by Cramér estimator, is performed and the function $s(n)=a_0+a_1\sqrt{n}$ is fitted to the L$_1$-errors of ${\widehat{f}}_n$ leading to the proportionality constant $a_1$ determination. Further, (expected) L$_1$-consistency rate of Kolmogorov estimator under generalized assumptions based on asymptotic domination relation is studied. No usual continuity or differentiability conditions are needed.