Anderson-Darling testΒΆ
The Anderson-Darling test is a statistical test of whether a given sample of data is drawn from a given probability distribution. The library only provides the Anderson-Darling test for normal distributions.
Let be a sample of dimension 1 drawn from the (unknown) cumulative distribution function assumed to be continuous. We want to test whether the sample is drawn from a normal distribution ie whether , where is the cumulative distribution function of the normal distribution.
This test involves the calculation of the test statistic which is the distance between the empirical cumulative distribution function and . Letting be independent random variables respectively distributed according to , we define the the order statistics by:
The test statistic is defined by:
This distance is a quadratic type, as in the Cramer-Von Mises test, but gives more weight to deviations of tail values. The empirical value of the test statistic denoted by is evaluated from the sample sorted in ascending order:
Under the null hypothesis , the asymptotic distribution of the test statistic is known i.e. when . If is sufficiently large, we can use the asymptotic distribution to apply the test as follows. We fix a risk (error type I) and we evaluate the associated critical value which is the quantile of order of .
Then a decision is made, either by comparing the test statistic to the theoretical threshold (or equivalently by evaluating the p-value of the sample defined as and by comparing it to ):
if (or equivalently ), then we reject the normal distribution,
if (or equivalently ), then the normal distribution is considered acceptable.