Anderson–Darling test

The Anderson–Darling test is a statistical test used to determine if a given sample of data is drawn from a specified probability distribution. Developed by Theodore W. Anderson and Donald A. Darling in 1952, the test is a modification of the Cramér–von Mises criterion and is used extensively in statistics for hypothesis testing.

Overview

The Anderson–Darling test is particularly sensitive to deviations in the tail ends (extreme values) of the distribution. This sensitivity makes it more effective than other tests, such as the Kolmogorov–Smirnov test, for detecting differences in the tails of the distributions. The test is applicable to a variety of distributions including the normal distribution, exponential distribution, Weibull distribution, and others.

Methodology

The test statistic, A², is calculated by: \[ A^2 = -n - \frac{1}{n} \sum_{i=1}^n (2i - 1) \left[ \ln(F(Y_i)) + \ln(1 - F(Y_{n+1-i})) \right] \] where:

• \( n \) is the sample size,
• \( Y_i \) are the sorted data points,
• \( F \) is the cumulative distribution function of the specified theoretical distribution.

The null hypothesis \( H_0 \) of the Anderson–Darling test states that the data follow the specified distribution. Rejection of the null hypothesis indicates that the data do not follow the distribution.

Applications

The Anderson–Darling test is widely used in quality control, reliability engineering, and other fields that require robust assessment of distribution fit. It is particularly useful in the context of parametric statistical inference, where the form of the distribution might significantly influence conclusions.

Limitations

While the Anderson–Darling test is powerful for detecting departures from a specified distribution, it can be overly sensitive to sample size. Large samples might lead to rejection of the null hypothesis for trivial deviations that are of no practical significance. Additionally, the test requires that the parameters of the distribution be fully specified; if parameters are estimated from the data, the critical values of the test statistic change, complicating its application.

See also

• Shapiro–Wilk test
• Lilliefors test
• Q–Q plot