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Description
This paper investigates the consistency and efficiency of generalized Cramér–von Mises (GCM) minimum distance estimators in the context of statistical estimation, focusing particularly on the L$_1$ norm and the expected L$_1$ norm. It presents new inequality between Kolmogorov and generalized Cramér–von Mises distances, leading to the proof of consistency of Cramér–von Mises estimator with the convergence rate of $n^{-1/3}$, and consistency of GCM of the order of $n^{-p/2(p+q)}$ in the (expected) L$_1$ norm. Through theoretical analysis and computer simulations, the study explores practical application of these estimators across various distribution types, contributing to the understanding of minimum distance estimation in statistical analysis. The computer simulation also suggests further possible improvements of proven L$_1$ convergence rate of GCM up to $n^{-1/2}$.
