Pathologies of some Minimum Distance Estimators

Minimum distance estimates are studied at the $N(\theta, 1)$ model. Estimates based on a non-Hilbertian distance $\mu (\mu = \text{Kolmogorov-Smirnov, Levy, Kuiper, variation and Prohorov})$ can exhibit very large variances, or even outright inconsistency, at distributions arbitrarily close to the model in terms of $\mu$-distance. For Hilbertian distances $(\mu = \text{Cramer-von Mises, Hellinger})$ this problem does not seem to occur. Geometric motivation for these results is provided.