Integrated empirical measures and generalizations of classical goodness-of-fit statistics

Based on multiply-integrated empirical measures, we introduce three new classes of goodness-of-fit tests that generalize the Anderson-Darling, Cramér-von Mises, and Watson statistics, respectively. These classes of test statistics are designed to detect various distributional features, making them a versatile and valuable toolbox for goodness-of-fit testing. By analyzing the corresponding limiting stochastic processes, we derive the limiting null distributions of these statistics and obtain explicit, closed-form expressions for the associated Karhunen-Loève expansions and their covariance kernels. In particular, we obtain explicit formulas for the eigenvalues for the corresponding integral operators. Furthermore, we represent the moment-generating functions of the limiting distributions as infinite products, after which we simplify those infinite products subsequently to products of finite numbers of terms, thereby enabling efficient numerical computation.