Anderson-Darling statistic and another weighted Cramer-von Mises statistics

For more than sixty years, Anderson-Darling statistic is one of the most popular in applications among the Cramer-von-Mises goodness-of-t tests.This statistic modies the classical empirical process in the interval $[0; 1]$ by multiplying it by a weighting function $\psi(t) = (t(1-t))^{1/2}$. The weighting function redistributes the test sensitivity to deviations of the alternative distribution function from the hypothetical on different subsets of $[0,1]$. In practice, the tests can be of interest having other weight functions. The paper proposes new formulas for eigenfunctions of the Anderson-Darling statistics. Also, it was analyzed a statistic "inverse" to the Anderson-Darling statistic with the weighting function $\psi(t)= (t(1-t))1=2$. There was considered also another weighting functions. The proposed theory is based on the use of various special functions. In practice, could be useful the Cramer- von-Mises tests with weighting functions from the family $\psi(t) = t^\alpha(1-t)^\beta; \alpha>-1; \beta> -1$. The paper contains a table of distribution for these statistics with different values of the degrees $\alpha>-1$ and $\beta>-1$. The table was calculated by different methods with good precision without using the statistical simulation. Shortly, must be considered the problem of the testing the complex hypothesis and connections of the proposed theory with the Riccaty equation.
The work was carried out at IITP RAS and supported by Russian Science Foundation (grant RSF No. 14-50-00150).
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