Anderson–Darling test

Tests for families of distributions

Essentially the same test statistic can be used in the test of fit of a family of distributions, but then it must be compared against the critical values appropriate to that family of theoretical distributions and dependent also on the method used for parameter estimation.

Test for normality

Empirical testing has found that the Anderson–Darling test is not quite as good as the Shapiro–Wilk test, but is better than other tests. Stephens found ${\displaystyle A^2}$ to be one of the best empirical distribution function statistics for detecting most departures from normality.

The computation differs based on what is known about the distribution:

• Case 0: The mean ${\displaystyle \mu }$ and the variance ${\displaystyle {\sigma }^2}$ are both known.
• Case 1: The variance ${\displaystyle {\sigma }^2}$ is known, but the mean ${\displaystyle \mu }$ is unknown.
• Case 2: The mean ${\displaystyle \mu }$ is known, but the variance ${\displaystyle {\sigma }^2}$ is unknown.
• Case 3: Both the mean ${\displaystyle \mu }$ and the variance ${\displaystyle {\sigma }^2}$ are unknown.

The n observations, ${\displaystyle X_i}$, for ${\displaystyle i=1,\dots n}$, of the variable ${\displaystyle X}$ must be sorted such that ${\displaystyle X_1\le X_2\le \cdots \le X_n}$ and the notation in the following assumes that X_i represent the ordered observations. Let $${\displaystyle \hat{\mu }=\{\begin{array}{ll}\mu ,& \text{if the mean is known.}\\ \overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i,& \text{otherwise.}\end{array}}$$

$${\displaystyle {\hat{\sigma }}^2=\{\begin{array}{ll}{\sigma }^2,& \text{if the variance is known.}\\ \frac{1}{n}\sum _{i=1}^n(X_i-\mu {)}^2,& \text{if the variance is not known, but the mean is.}\\ \frac{1}{n-1}\sum _{i=1}^n(X_i-\overline{X}{)}^2,& \text{otherwise.}\end{array}}$$ The values ${\displaystyle X_i}$ are standardized to create new values ${\displaystyle Y_i}$, given by $${\displaystyle Y_i=\frac{X_i-\hat{\mu }}{\hat{\sigma }}.}$$ With the standard normal CDF ${\displaystyle \mathrm{\Phi }}$, ${\displaystyle A^2}$ is calculated using $${\displaystyle A^2=-n-\frac{1}{n}\sum _{i=1}^n\left(2i-1\right)\left[\ln \mathrm{\Phi }(Y_i)+\ln (1-\mathrm{\Phi }(Y_{n+1-i}))\right].}$$ An alternative expression in which only a single observation is dealt with at each step of the summation is: $${\displaystyle A^2=-n-\frac{1}{n}\sum _{i=1}^n\left[(2i-1)\ln \mathrm{\Phi }(Y_i)+(2(n-i)+1)\ln (1-\mathrm{\Phi }(Y_i))\right].}$$ A modified statistic can be calculated using $${\displaystyle A^{\ast 2}=\{\begin{array}{ll}{A}^2\left(1+\frac{0.75}{n}+\frac{2.25}{n^2}\right),& \text{if the variance and the mean are both unknown.}\\ A^2,& \text{otherwise.}\end{array}}$$

. If ${\displaystyle A^2}$ or ${\displaystyle A^{\ast 2}}$ exceeds a given critical value, then the hypothesis of normality is rejected with some significance level. The critical values are given in the table below for values of ${\displaystyle A^{\ast 2}}$.

Note 1: If ${\displaystyle \hat{\sigma }}$ = 0 or any ${\displaystyle \mathrm{\Phi }(Y_i)=}$(0 or 1) then ${\displaystyle A^2}$ cannot be calculated and is undefined.

Note 2: The above formula for the modified statistic ${\displaystyle A^{\ast 2}}$ is taken from D'Agostino (1986, p. 123). Care is required in comparisons across different sources as often the specific adjustment formula is not stated.

Note 3: Stephens notes that the test becomes better when the parameters are computed from the data, even if they are known.

Note 4: Marsaglia & Marsaglia provide a more accurate result for Case 0 at 85% and 99%.

Tests for other distributions

Above, it was assumed that the variable ${\displaystyle X_i}$ was being tested for normal distribution. Any other family of distributions can be tested, but the test for each family is implemented by using a different modification of the basic test statistic, with reference to critical values specific to that family of distributions. The modifications of the statistic and tables of critical values are given by Stephens (1986) for the exponential, extreme-value, Weibull, gamma, logistic, Cauchy, and von Mises distributions. Tests for the (two-parameter) log-normal distribution can be implemented by transforming the data using a logarithm and using the above test for normality. Details for the required modifications to the test statistic and for the critical values for the normal distribution and the exponential distribution have been published by Pearson & Hartley (1972, Table 54). Details for these distributions, with the addition of the Gumbel distribution, are also given by Shorack & Wellner (1986, p239). Details for the logistic distribution are given by Stephens (1979). A test for the (two parameter) Weibull distribution can be obtained by making use of the fact that the logarithm of a Weibull variate has a Gumbel distribution.

Non-parametric k-sample tests

Fritz Scholz and Michael A. Stephens (1987) discuss a test, based on the Anderson–Darling measure of agreement between distributions, for whether a number of random samples with possibly different sample sizes may have arisen from the same distribution, where this distribution is unspecified. The R package kSamples and the Python package Scipy implements this rank test for comparing k samples among several other such rank tests.

For ${\displaystyle k}$ samples the statistic can be computed as follows under the assumption that the distribution function ${\displaystyle F_i}$ of ${\displaystyle i}$-th sample is continuous

$${\displaystyle A_{kN}^2=\frac{1}{N}\sum _{i=1}^k\frac{1}{n_i}\sum _{j=1}^{N-1}\frac{(N{M}_{ij}-j{n}_i{)}^2}{j(N-j)}}$$

where

• ${\displaystyle n_i}$ is the number of observations in the ${\displaystyle i}$-th sample
• ${\displaystyle N}$ is the total number of observations in all samples
• ${\displaystyle Z_1<\cdots <Z_N}$ is the pooled ordered sample
• ${\displaystyle M_{ij}}$ is the number of observations in the ${\displaystyle i}$-th sample that are not greater than ${\displaystyle Z_j}$.