On the Goodness-of-Fit Tests for Some Continuous Time Processes

Abstract

We present a review of several results concerning the construction of the Cramér-von Mises and Kolmogorov-Smirnov type goodnessof-fit tests for continuous time processes. As the models we take a stochastic differential equation with small noise, ergodic diffusion process, Poisson process and self-exciting point processes. For every model we propose the tests which provide the asymptotic size α and discuss the behaviour of the power function under local alternatives. The results of numerical simulations of the tests are presented.

Goodness of fit test for ergodic diffusions by tick time sample scheme

“…Note that the Kolmogorov-Smirnov statistic for ergodic diffusion process was studied in Fournie (1992), see also Fournie and Kutoyants (1993) for more details, while the weak convergence of the empirical process was proved in Negri (1998) (see van der Vaart and van Zanten (2005) for further developments). Dachian and Kutoyants (2008) and Negri and Nishiyama (2009) proposed some asymptotically distribution free tests. Recently Kutoyants (2009) proposed some Crámer-von Mises type tests based on the empirical distribution function and the local time estimator of the invariant density; the proposed test is asymptotically distribution free after a suitable transformation of the test statistics.…”

On goodness-of-fit tests for parametric hypotheses in perturbed dynamical systems using a minimum distance estimator

“…There are several possibilities to solve this problem in the case of the limit (1). One of them is to find a linear transformation L [·] of the random function u (·), such that L [u] (t) = W (t), where W (t) , 0 ≤ t ≤ 1 is some Wiener process.…”

“…Many authors wrote that similar transformation can be obtained in the case of other estimators with limit (2), but as we know this work (construction of the linear transformation with other estimators) was not done. The problem of GoF testing for the model of continuous time observations of diffusion process, with a simple null hypothesis Θ = {ϑ 0 }, was studied in [1] and [10]. Suppose that the observed diffusion process under hypothesis is…”

Review on Goodness of Fit Tests for Ergodic Diffusion Processes by Different Sampling Schemes

“…Note that the Kolmogorov–Smirnov statistics for ergodic diffusion processes were studied in Fournie (1992), see also Fournie and Kutoyants (1993) for more details, while the weak convergence of the empirical process was proved in Negri (1998) (see van der Vaart and van Zanten, 2005 for further developments). More recently Dachian and Kutoyants (2008) propose modifications both of the Kolmogorov–Smirnov and of the Cramer–von Mises test and they prove that such tests are asymptotically distribution free. They discuss also the behaviour of the power function under local alternatives.…”