Speaker
Description
The contribution study the minimum distance density estimators based on Kolmogorov and Cramér -von Mises distance. Inequalities between Kolmogorov and Cramér -von Mises distances are proven to achieve $n^{-\gamma}$ consistency in (expected) L$_1$ norm of M(CM)E. Further, the generalized Cramér - von Mises distance is defined together with so called Kolmogorov - Cramér distance which includes both Kolmogorov and Cramér - von Mises distance as limiting special cases. We prove $n^{-\gamma}$ consistency in the (expected) L$_1$ norm of both minimum distance estimators based on newly defined distances.
Our numerical simulation illustrates the quality of consistency property covered by theoretical results. The proportionality constants of the consistency order are approximated from simulated data since they are not given by the proofs of theorems. Dependence of consistency in the L$_1$ norm on $\varepsilon$ - contamination neighbourhood of the true model is studied and, further, the robustness of these newly defined estimators is investigated for contaminated Normal family. Numerical simulations are used to compare statistical properties of all studied estimators and to determine the optimal or preferable choice of parameters of newly defined estimators. Finally, we bring comparison of all studied estimators with Rényi and Power divergence estimators.
