Cramer-von Mises distance: probabilistic interpretation, confidence intervals, and neighbourhood-of-model validation

摘要

We give a probabilistic interpretation of the Cramer-von Mises distance Delta(F, F-0) = integral (F - F-0)(2) dF(0) between continuous distribution functions F and F0. If F is unknown, we construct an asymptotic confidence interval for Delta (F, F-0) based on a random sample from F. Moreover, for given F-0 and some value Delta(0) > 0, we propose an asymptotic equivalence test of the hypothesis that Delta(F, F-0) >= Delta(0) against the alternative Delta(F, F-0) < Delta(0). If such a 'neighbourhood-of-F-0 validation test', carried out at a small asymptotic level, rejects the hypothesis, there is evidence that F is within a distance Delta(0) of F-0. As a neighbourhood-of-exponentiality test shows, the method may be extended to the case that H-0 is composite.