The empirical identity process: Asymptotics and applications

When sampling independent observations drawn from the uniform distribution on the unit interval, as the sample size gets large the asymptotic behaviour of both the empirical distribution function and empirical quantile function is well known. In this article we study analogous asymptotic results for the function that is obtained by composing the empirical quantile function with the empirical distribution function. Since the former is the generalized inverse of the latter, the result will approximate the identity function. We define a scaled and centered version of this function—the empirical identity process—and prove it converges to a highly irregular limit process whose trajectories are not right-continuous and impossible to study using standard probability in metric spaces. However, when this process is integrated over time, and appropriately rescaled and centered, it becomes possible to define a functional limit theorem for it, which then converges to a randomly pinned Brownian motion. By applying these theoretical results, a new goodness-of-fit test is derived. We demonstrate that this test is very efficient when it is applied to data which come from a multimodal or mixture distribution, like the classic Old Faithful dataset. The Canadian Journal of Statistics 46: 656–672; 2018 © 2018 Société statistique du Canada.