Automatic Distribution Fit Analyzer

The Anderson-Darling statistic measures the distance between the empirical distribution function of a sample and a candidate theoretical CDF, but unlike the Kolmogorov-Smirnov statistic, which takes the largest vertical gap anywhere, it integrates the squared difference using the weight 1 divided by F(x)(1 - F(x)). That weight grows without bound as F approaches 0 or 1, making the test unusually sensitive to disagreement in the tails - exactly where reliability, risk and extreme-value work needs accuracy. In computational form it becomes a sum over the sorted sample of (2i - 1) times the log of F at the i-th order statistic plus the log of one minus F at the mirrored order statistic, subtracted from minus n. Interpretation depends on how the parameters were obtained. When the mean and standard deviation are estimated from the same data being tested, as they are here, the statistic must be adjusted by the factor (1 + 0.75/n + 2.25/n squared) and compared against case-specific critical values: approximately 0.752 at the 5 percent level and 1.035 at the 1 percent level for the normal case. Ranking candidate distributions by raw AD value is a screening heuristic rather than a formal test, since a uniform fit using the sample minimum and maximum is being judged against the very points that defined it, and families with more free parameters carry an inherent advantage - information criteria such as AIC or BIC handle that comparison more honestly. Below roughly 20 observations the test has little power to separate similar shapes, and 50 to 100 points are preferable.

Goodness-of-fit testing begins with Karl Pearson, who introduced the chi-squared test in 1900 to judge whether observed frequencies matched a hypothesized distribution. The chi-squared approach required binning continuous data, and conclusions shifted depending on how the bins were drawn, so a second family of tests was built directly on the empirical distribution function. Andrey Kolmogorov proposed the supremum-distance statistic in 1933, Nikolai Smirnov extended the idea to the two-sample case later in that decade, and Harald Cramer and Richard von Mises independently developed integrated squared-difference criteria in 1928. Theodore Anderson and Donald Darling published their weighted refinement in the Annals of Mathematical Statistics in 1952, deriving the asymptotic theory, and followed it in 1954 in the Journal of the American Statistical Association with the practical test. Their contribution was the weighting function that magnifies discrepancies in the tails, addressing the known insensitivity of the Cramer-von Mises and Kolmogorov-Smirnov statistics at the extremes. The test became usable in routine work only after Michael Stephens published tables of critical values and small-sample corrections for the case where distribution parameters are estimated from the data, in a 1974 paper in the Journal of the American Statistical Association and later in the 1986 volume Goodness-of-Fit Techniques edited with Ralph D'Agostino. Those corrected values are what statistical packages and the NIST Engineering Statistics Handbook still publish, and they are the reason the Anderson-Darling test became the default normality check in reliability and quality engineering software.