Pathologies of some minimum distance estimators
Asymptotic properties of minimum distance (MD) estimators are studied at the normal N(\(\theta\),1) model. In contrast with good stability properties of MD functionals the estimators are shown to exhibit sometimes pathological behaviour at distributions arbitrarily close to the model. The asymptotic variance of minimum Kolmogorov-Smirnov and Lévy distance estimators can be as large as desired while inconsistency occurs with Prokhorov and variation distances. Regularity of MD estimators is however present when Hilbertian distances are applied, like Cramér-von Mises and Hellinger. The functionals induced by these metrics are shown to be also Fréchet differentiable.
Fréchet differentiability of statistical functionals
scientific article; zbMATH DE number 4121219