Strong Approximations for Multivariate Integrated Empirical Processes: Two-Sample Testing, Change-Point Detection, and Inference with Estimated Parameters

Abstract

We investigate a broad class of multivariate $p$-fold integrated empirical processes obtained by iterated lower-orthant Stieltjes integration of the empirical distribution function on $[0,1]^d$. After deriving exact structural representations for the corresponding integrated distribution and empirical distribution functionals, we show that, in the scalarizable regime (including the univariate and effectively totally ordered/comonotone settings), the $p$-fold integrated empirical process admits a precise reduction to the ordinary multivariate empirical process, with leading term $(p!)^{-1} F(\mathbf{u})^p \alpha_n(\mathbf{u})$. In the genuinely multivariate partial-order setting, the first-order structure is governed instead by the chain functional and its influence function. This algebraic decomposition permits a systematic transfer of strong approximation results from the classical empirical process to its integrated counterparts. More specifically, on suitable probability spaces, we construct almost sure couplings of the integrated empirical process with appropriately weighted $F$-Brownian bridges and $F$-Kiefer processes, and we obtain explicit uniform approximation rates over $[0,1]^d$. These couplings yield asymptotic approximations for integrated Kolmogorov--Smirnov and Cram\'er--von Mises type functionals, as well as a functional law of the iterated logarithm. We then exploit the resulting Gaussian approximations in two inferential directions: first, for multivariate two-sample procedures based on modified $p$-fold integrated empirical contrasts; second, for change-point statistics constructed from sequential integrated empirical discrepancies, for which asymptotically valid Gaussian calibrations are derived. We further analyze plug-in versions of the integrated empirical process in parametric settings and establish asymptotic Gaussian approximations in the presence of estimated nuisance parameters. A simulation study in dimension $d=2$ complements the theory and illustrates the finite-sample behavior of the proposed procedures under dependence, marginal, and joint perturbation regimes.