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