# Notes on consistency of some minimum distance estimators with simulation results

Research paper by Jitka Hrabáková, Václav Kůs

Indexed on: 09 Nov '16Published on: 18 Oct '16Published in: Metrika

#### Abstract

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.AbstractWe 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. $${\widehat{f}}_n$$ $${\widehat{f}}_n$$ $$f_0$$ $$f_0$$ $$n^{-1/2}$$ $$n^{-1/2}$$ $$_1$$ $$_1$$ $$\mathcal {D}$$ $$\mathcal {D}$$ $$n^{-1/2}$$ $$n^{-1/2}$$ $$_1$$ $$_1$$ $$s(n)=a_0+a_1\sqrt{n}$$ $$s(n)=a_0+a_1\sqrt{n}$$ $$_1$$ $$_1$$ $${\widehat{f}}_n$$ $${\widehat{f}}_n$$ $$a_1$$ $$a_1$$ $$_1$$ $$_1$$