NAG Fortran Library Routine Document G08CBF

G08CBF Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details. 1 Purpose G08CBF performs the one-sample Kolmogorov–Smirnov test, using one of the standard distributions provided. The data consists of a single sample of n observations denoted by x 1 ; x 2 ;. .. ; x n. Let S n ðx ðiÞ Þ and F 0 ðx ðiÞ Þ represent the sample cumulative distribution function and the theoretical (null) cumulative distribution function respectively at the point x ðiÞ , where x ðiÞ is the ith smallest sample observation. The Kolmogorov–Smirnov test provides a test of the null hypothesis H 0 : the data are a random sample of observations from a theoretical distribution specified by the user against one of the following alternative hypotheses (i) H 1 : the data cannot be considered to be a random sample from the specified null distribution. (ii) H 2 : the data arise from a distribution which dominates the specified null distribution. In practical terms, this would be demonstrated if the values of the sample cumulative distribution function S n ðxÞ tended to exceed the corresponding values of the theoretical cumulative distribution function F 0 ðxÞ. (iii) H 3 : the data arise from a distribution which is dominated by the specified null distribution. In practical terms, this would be demonstrated if the values of the theoretical cumulative distribution function F 0 ðxÞ tended to exceed the corresponding values of the sample cumulative distribution function S n ðxÞ. One of the following test statistics is computed depending on the particular alternative null hypothesis specified (see the description of the parameter NTYPE in Section 5). For the alternative hypothesis H 1 : D n – the largest absolute deviation between the sample cumulative distribution function and the theoretical cumulative distribution function. Formally D n ¼ maxfD þ n ; D À n g. For the alternative hypothesis H 2 : D þ n – the largest positive deviation between the sample cumulative distribution function and the theoretical cumulative distribution function. Formally D þ n ¼ maxfS n ðx ðiÞ Þ À F 0 ðx ðiÞ Þ; 0g for both discrete and continuous null distributions. For the alternative hypothesis H 3 : D À n – the largest positive deviation between the theoretical cumulative distribution function and the sample …