Distributions of Goodness-of-Fit Statistics

This book is devoted to the theory of goodness-of-fit tests based on weighted empirical processes. Much attention has been given to the limit distributions of statistics of these tests as well as to convergence problems. Cramer-von Mises statistics are studied throughout most of the book, but attention is also given to the Kolmogorov-Smirnov test, the chi-square goodness-of-fit test as well as some others. The authors describe statistics to test simple and complex parametric hypotheses. Other hypotheses are also be considered, namely the hypothesis of distributional symmetry, the hypothesis of uniformity for random variables on a circle, the hypothesis of uniformity of distribution on a multidimensional cube, and the hypothesis of independence of the components of multidimensional vectors. Tests based on the transformed empirical process are also discussed. The expressions for eigenvalues and eigenfunctions are derived for many covariance operators corresponding to various empirical processes. The resulting eigenfunctions are often expressed in terms of known special functions. Methods for computing the distribution of various types of quadratic form from normal random variables are also described. These methods are not well known in the literature. The theory is accompanied by a collection of small tables of distributions, and in some cases by small programs in the MATHEMATICA language. All tables of quantiles for the Cramer-von Mises tests have been computed by exact numerical methods. Some tables of distributions of the similar Kolmogorov-Smirnov statistics, once again found using simulations, are also included in this book.