The minimum distance method of testing

A method is developed for generalising tests of Kolmogorov-Smirnov and Cramér-von Mises type to cases where parameters jave to be estimated, and where the underlying distribution is replaced by a sequence of alternatives.

Abstract

In this paper a method is developed for generalising tests of Kolmogorov-Smirnov and Cramér-von Mises type to cases where parameters jave to be estimated. The procedures are based on comparing the empirical distribution functionF n , as a random point in a normed linear space, with a parametric surface $$\{ F(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } ):\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } \in \Theta \} $$ which represents the family of possible underlying distributions. Asymptotic results are proved for the distribution of the minimum distance $$\sqrt n \mathop {\inf }\limits_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } } \parallel F_n - F(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } )\parallel $$ and for the corresponding minimizing value of $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } $$ . The results are extended to cases where ‖·‖ is replaced by a parameter dependent norm $$\parallel \cdot \parallel _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } } $$ , and where the underlying distribution is replaced by a sequence of alternatives. The basic assumptions require convergence in distribution of $$\sqrt n [F_n - F(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } _0 )]$$ and differentiability in norm of the map $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } \to F(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } )$$ .