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Minimum distance procedures. (English) Zbl 0597.62032
Nonparametric methods, Handb. Stat. 4, 741-754 (1984).
[For the entire collection see Zbl 0587.00012.]
We examine more closely the minimum distance estimates and tests based on the Cramér-von Mises metric \[ d(G,F)=\{\int [G(x)-F(x)]^ 2\}d\mu, \] where F, G are distribution functions and \(\mu\) is a probability measure. The theory for these procedures is relatively simple, yields interesting results, and most importantly, illustrates general features of minimum distance procedures. Arguments that work for the Cramér-von Mises metric often have counterparts for other metrics, even when substantial technical differences occur (as is the case for the Hellinger and Kolmogorov-Smirnov metrics).
Section 2 develops asymptotic distribution theory for minimum Cramér- von Mises distance estimates and establishes their robustness under local perturbations from the parametric model. Asymptotic performance of the minimized Cramér-von Mises distance is studied in Section 3. The discussion includes theoretical justification of the boostrap technique for selecting critical values and asymptotic power calculations for the corresponding goodness-of-fit tests.

62G10 Nonparametric hypothesis testing
62G05 Nonparametric estimation
62G30 Order statistics; empirical distribution functions