Goodness-of-fit tests based on quadratic functionals of transformed empirical processes

A family of quadratic functionals of the sample, each based on a Transforme d Empirical Process (TEP) is introduced, thus generalizing the Modified And erson-Darling statistics studied in [4], by letting a weight function be ar bitrarily selected. These statistics are used to construct goodness-of-fit tests consistent aga inst any alternative. It is shown that, in addition, the power against a gi ven sequence of contiguous alternatives can be increased by properly select ing the score function of the TEP. For such tests, the asymptotic power does not depend neither on the null hy pothesis distribution nor on the sequence of alternatives for which the tes t is designed. The asymptotic distribution of the test statistic under the null hypothesis, and under the priviledged alternatives, depends only on th e weight function, and can be simulated for any practical purpose. It is al so shown that the shape of the weight function has a little effect on the r esulting power. The paper finally adds some theoretical comments on the distribution of the test statistic, particularly for the case of a constant weight function.