AndersonDarling

The Anderson-Darling test is a statistical test used to decide whether a sample of data comes from a specified continuous distribution. It is named after Theodore W. Anderson and Donald A. Darling, who introduced the method in 1954. The test is a modification of the Kolmogorov-Smirnov test that gives more weight to the tails of the distribution, making it more sensitive to deviations in the tails.

There are two main variants: a one-sample goodness-of-fit test for a given distribution, and, less commonly, a k-sample extension that compares multiple samples to determine if they come from the same distribution. In the one-sample form, the test requires the cumulative distribution function (CDF) F of the reference distribution.

The standard statistic for the one-sample test, based on an ordered sample X(1) ≤ ... ≤ X(n), is A^2 = -n - (1/n) sum_{i=1}^n [ (2i-1) ( ln F(X(i)) + ln(1 - F(X(n+1-i))) ) ]. F(X(i)) denotes the CDF value of the i-th order statistic. Finite-sample corrections and tabulated critical values or p-value approximations are used in practice. Under the null hypothesis that the data follow the specified distribution, A^2 has a known asymptotic distribution, with distributional specifics depending on F and sample size.

The test is widely implemented in statistical software, with common usage including testing normality (a frequent default case) and fitting data to other parametric families. It is valued for its sensitivity to tail discrepancies, though its distribution must be handled carefully when parameters are estimated from the data.