GoodnessofFit tests for Statistical Distributions

Of the many quantitative goodnessoffit techniques (e.g.: KomolgorovSmirnov, AndersonDarling, ShipiroWilk, von Mises), I prefer the AndersonDarling test because it is more sensitive to deviations in the tails of the distribution than is the older KomolgorovSmirnov test. Note: The AndersonDarling test (or KomolgorovSmirnov or ShipiroWilk) does not tell you that you do have a Normal density. It only tells you when the data make it unlikely that you do not.^{(1)} AndersonDarling can be applied to any distribution, but finding tables of critical values isn't so easy. Included here are two of the most useful tables, for the normal and lognormal, and for the Weibull, exponential, and Gumbel. For the normal and lognormal distributions, the test statistic, A^{2} is calculated from where n is the sample size, and w is the standard normal cdf, F[(xm)/s]. This formula needs to be modified for small samples, and then compared to an appropriate critical value from the table below.
(Reference: D'Agostino and Stephens, GoodnessOfFit Techniques, MarcelDekker, New York, 1986, Table 4.7, p.123. All of Chapter 4, pp.97193, deals with goodnessoffit tests based on empirical distribution function (EDF) statistics.) The other popular family of distributions includes the Weibull for distributions of minima, and Gumbel for distributions of maxima. The Gumbel variable X, and Weibull variable Y are related by X=ln(1/Y) . A Weibull distribution with the shape parameter equal to one produces the exponential distribution as a special case. For the Weibull ^{(2)} (and Gumbel) distributions, the test statistic, A^{2} is again calculated from just as for the normal, but w is the cdf for the distribution under consideration. For the Weibull this is and h, b, are the model scale and shape parameters. This formula needs to be modified for small samples, and then compared to an appropriate critical value from the table below.
(Ref: D'Agostino and Stephens, 1986, Table 4.17, p.146) _____________

