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

Based on $m$-fold integrated empirical measures, we study three new classes of goodness-of-fits tests, generalizing Anderson-Darling, Cram\'er-von Mises, and Watson statistics, respectively, and examine the corresponding limiting stochastic processes. The limiting null distributions of the statistics all lead to explicitly solvable cases with closed-form expressions for the corresponding Karhunen-Lo\`{e}ve expansions and covariance kernels. In particular, the eigenvalues are shown to be $\frac1{k(k+1)\cdots (k+2m-1)}$ for the generalized Anderson-Darling, $\frac1{(\pi k)^{2m}}$ for the generalized Cram\'er-von Mises, and $\frac1{2\pi\lceil k/2\rceil^{2m}}$ for the generalized Watson statistics, respectively. The infinite products of the resulting moment generating functions are further simplified to finite ones so as to facilitate efficient numerical calculations. These statistics are capable of detecting different features of the distributions and thus provide a useful toolbox for goodness-of-fit testing.