Limit Distributions of Kolmogorov-Smirnov Type Statistics Under a Fixed Alternative with Estimated Location and Scale Parameters.

摘要

Much attention has been devoted to Monte Carlo simulations of the power of Kolmogorov-Smirnov type goodness-of-fit statistics when nuisance parameters of the hypothesized distribution are estimated. Here we consider the asymptotic behavior of such statistics at fixed alternatives when location and scale parameters are estimated. It is shown that suitably normalized Kolmogorov-Smirnov statistics converge in distribution to Gaussian-related random variables depending on the alternative distribution and the maximum deviation between the null and alternative distribution functions. The work of Raghavachari is thus extended from simple hypotheses to the case of composite hypotheses with estimated nuisance parameters. (Author)