13.11 : The Anderson-Darling Test

The Anderson-Darling test is a statistical method used to determine whether a data sample is likely drawn from a specific theoretical distribution. Unlike parametric tests, it does not require assumptions about specific parameters of the distribution. Instead, it compares the sample's empirical cumulative distribution function (ECDF) with the cumulative distribution function (CDF) of the hypothesized distribution. Critical values for the test are specific to the chosen distribution rather than universal, making it adaptable to various distributions.

Developed by Theodore Wilbur Anderson and Donald Allan Darling in 1952, the test is widely used to check for normality, though it is a common misconception that it applies only to normal distributions. In fact, it can also test goodness-of-fit for distributions like exponential, Weibull, or logistic, as long as the relevant CDF is known.

A key consideration when using the Anderson-Darling test is whether a parametric or nonparametric approach is appropriate, depending on the information about the population distribution. Although it is frequently employed to test for normality, the test can assess fit across a broad range of distributions. It is considered an improvement over the Kolmogorov-Smirnov (K-S) test due to its greater sensitivity to deviations in the tails of the distribution, making it more effective for detecting outliers and extreme values. Finally, while calculating the Anderson-Darling test statistic manually can be complex, computer-based tools and software packages have simplified the process, providing both the test statistic and the critical values needed to interpret results efficiently.