Abstract
We discuss goodness-of-fit tests for stationary ergodic processes based on limit theorems for marked empirical processes viewed as elements of an $L^{2}$ space. Our limit theorems cover a weighted process that enables an Anderson–Darling-type test statistic for the goodness-of-fit tests to be proposed. The procedures presented for these goodness-of-fit tests are novel.
Highlights
With regard to the goodness-of-fit tests for stationary ergodic processes, we propose a Cramer–von Mises type statistic based on discrete time observations to test a simple hypothesis for a diffusion process and an Anderson–Darlingtype statistic for a nonlinear time series
We provide two limit theorems, Theorems 1.1 and 1.2 presented in Sections 1.1 and 1.2, which assert weak convergence of marked empirical processes in L2 space
Goodness-of-fit tests have been extensively studied in the literature because they are useful in deciding whether a mathematical model is acceptable to describe sampled data
Summary
With regard to the goodness-of-fit tests for stationary ergodic processes, we propose a Cramer–von Mises type statistic based on discrete time observations to test a simple hypothesis for a diffusion process and an Anderson–Darlingtype statistic for a nonlinear time series. Using a smoothed empirical process, Masuda et al [14] proposed Kolmogorov–Smirnovtype goodness-of-fit tests based on discrete time observations. They established not the weak convergence of Zn(x), which shall be defined in (1.1), but the weak convergence of its smoothed version. As for goodness-of-fit test for time series models, Koul and Stute [10] considered a simple hypothesis and one that is a parametric composite based on the notion of the martingale transformation (Khmaladze [8]). As for ergodic diffusion process models (continuous time observations), Kleptsyna and Kutoyants [9] and Kutoyants [12, 13] considered a parametric composite hypothesis, marked empirical processes were not treated. The limit theorems are presented in the following two subsections
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