Anderson–Darling statistic and its “inverse”

Resumo

For more than 60 years, the Anderson–Darling test is most frequently used among all Cramér–von Mises (omega-square) tests. This statistic modifies a classical empirical process defined within the [0, 1] interval by multiplying it by weighting function ψ(t) = (t(1–t))^–1/2. The weighting function redistributes the test sensitivity to deviations of the distribution function of the observed stochastic quantity from a hypothetical distribution function in different its segments. However, the tests with other weighting functions may also be of interest in practice. New formulas for the eigenvalues of the Anderson–Darling statistic are proposed. The statistic “inverse” to the Anderson–Darling statistic with weighting function ψ(t) = (t(1–t))^1/2 is considered. Tests with other weighting functions may also be of interest when weighted Cramér–von Mises statistics are used. The table of quantiles of statistics with weighting functions ψ(t) = t^α(1–t)^β, α >–1, β >–1 is presented. The quantiles are given for 36 different combinations of parameters α >–1 and β >–1. The table was calculated using accurate numerical methods and without application of modeling techniques.