The Consistency and Robustness of Modified Cramér–Von Mises and Kolmogorov–Cramér Estimators

Abstract

This article focuses on the minimum distance estimators under two newly introduced modifications of Cramér–von Mises distance. The generalized power form of Cramér–von Mises distance is defined together with the so-called Kolmogorov–Cramér distance which includes both standard Kolmogorov and Cramér–von Mises distances as limiting special cases. We prove the consistency of Kolmogorov-Cramér estimators in the (expected) L₁-norm by direct technique employing domination relations between statistical distances. In our numerical simulation we illustrate the quality of consistency property for sample sizes of the most practical range from n = 10 to n = 500. We study dependence of consistency in L₁-norm on ϵ-contamination neighborhood of the true model and further the robustness of these two newly defined estimators for normal families and contaminated samples. Numerical simulations are used to compare statistical properties of the minimum Kolmogorov–Cramér, generalized Cramér–von Mises, standard Kolmogorov, and Cramér–von Mises distance estimators of the normal family scale parameter. We deal with the corresponding order of consistency and robustness. The resulting graphs are presented and discussed for the cases of the contaminated and uncontaminated pseudo-random samples.

Keywords:

• Consistency
• Cramér–von Mises distance
• Minimum distance estimates
• Robustness

Mathematics Subject Classification:

• 62F12
• 62F35

Acknowledgment

This work was supported by the grants SGS12/197/OHK4/3T/14 and MSMT INGO-II LG12020. We express many thanks to both referees for valuable suggestions and substantial improvements to this article.