AndersonDarlingTest

The Anderson-Darling test is a statistical method used to assess whether a sample comes from a specified probability distribution. It is a type of goodness-of-fit test that places more emphasis on the tails of the distribution than some other tests, making it particularly sensitive to deviations in the tails. The test was introduced by Anderson and Darling in 1954 and has since become a widely used tool for evaluating distributional assumptions in fields such as statistics, engineering, and the natural and social sciences.

In its general form, the test analyzes a sample x(1) ≤ x(2) ≤ ... ≤ x(n) and a hypothesized cumulative distribution function F. The Anderson-Darling statistic is computed as A^2 = -n - ∑_{i=1}^n (2i - 1)/n [ ln F(x(i)) + ln(1 - F(x(n+1-i))) ]. Large values of A^2 suggest the data do not come from F. When F depends on unknown parameters (for example, testing for normality with unknown mean and variance), those parameters are estimated from the data, and the null distribution of A^2 is adjusted accordingly, typically via tabulated critical values or Monte Carlo simulation to obtain p-values.

A common specialization is the Anderson-Darling test for normality, where F is the normal CDF with parameters estimated from the sample. More generally, the AD test can be applied to any specified distribution by choosing the corresponding F. It is often preferred over the Kolmogorov-Smirnov test in situations where tail behavior is important. Limitations include dependence of p-values on the chosen F and potential complexity in obtaining accurate critical values when parameters are estimated.