Minimum Distance Estimators for Location and Goodness of Fit

A weighted Cramér-von Mises distance between the empirical distribution function and the assumed model F0(x - θ) is minimized to produce estimators θ n that are asymptotically normal. If the weight function is taken proportional to (- ln f 0)″/f 0 , then θ n is asymptotically efficient and the minimized distance has the appropriate loss of one degree of freedom. Special attention is focused on the limiting distribution of this latter goodness-of-fit statistic in both null and alternative situations.