SSJ User’s Guide Package gof Goodness-of-fit test Statistics

This package provides facilities for performing and reporting different types of univariate goodness-of-fit statistical tests. Overview This package contains tools for performing univariate goodness-of-fit (GOF) statistical tests. Static methods for computing (or approximating) the distribution function F (x) of certain GOF test statistics, as well as their complementary distribution function ¯ F (x) = 1 − F (x), are implemented in classes FDist and FBar. Tools for computing the GOF test statistics and the corresponding p-values, and for formating the results, are provided in classes GofStat and GofFormat. We are concerned here with GOF test statistics for testing the hypothesis H 0 that a sample of N observations X 1 ,. .. , X N comes from a given univariate probability distribution F. We consider tests such as those of Kolmogorov-Smirnov, Anderson-Darling, Crámer-von Mises, etc. These test statistics generally measure, in different ways, the distance between a continuous distribution function F and the empirical distribution function (EDF) ˆ F N of X 1 ,. .. , X N. They are also called EDF test statistics. The observations X i are usually transformed into U i = F (X i), which satisfy 0 ≤ U i ≤ 1 and which follow the U (0, 1) distribution under H 0. (This is called the probability integral transformation.) Methods for applying this transformation, as well as other types of transformations, to the observations X i or U i are provided in GofStat. Then the GOF tests are applied to the U i sorted by increasing order. The corresponding p-values are easily computed by calling the appropriate static methods in FDist. If a GOF test statistic Y has a continuous distribution under H 0 and takes the value y, its (right) p-value is defined as p = P [Y ≥ y | H 0 ]. The test usually rejects H 0 if p is deemed too close to 0 (for a one-sided test) or too close to 0 or 1 (for a two-sided test). In the case where Y has a discrete distribution under H 0 , we distinguish the right p-value p R = P [Y ≥ y | H 0 ] and the left p-value p L = P [Y ≤ y | H 0 ]. We then define the p-value for a two-sided test as p = (1) Why such a definition? Consider for example a Poisson random variable Y with …