Notes on consistency of some minimum distance estimators with simulation results

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• 作者：Jitka Hrabáková ; Václav Kůs
• 关键词：Minimum distance estimators ; Consistency ; Kolmogorov distance ; Degree of variations
• 刊名：Metrika
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：80
• 期：2
• 页码：243-257
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Statistics, general; Statistics for Business/Economics/Mathematical Finance/Insurance; Probability Theory and Stochastic Processes; Economic Theory/Quantitative Economics/Mathematical Methods;
• 出版者：Springer Berlin Heidelberg
• ISSN：1435-926X
• 卷排序：80

文摘

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.