Goodness-of-Fit Tests for Perturbed Dynamical Systems

TL;DR

This paper develops goodness-of-fit tests for stochastic differential equations with small diffusion, analyzing their power and potential for distribution-free testing under known trend coefficients.

Contribution

It introduces several goodness-of-fit tests, including Cramer-von Mises, Kolmogorov-Smirnov, and Chi-Square, for perturbed dynamical systems with small noise.

Findings

Analyzed power functions for specific close alternatives.

Discussed construction of tests based on local time.

Explored asymptotically distribution-free tests for composite hypotheses.

Abstract

We consider the goodness of fit testing problem for stochastic differential equation with small diffiusion coefficient. The basic hypothesis is always simple and it is described by the known trend coefficient. We propose several tests of the type of Cramer-von Mises, Kolmogorov-Smirnov and Chi-Square. The power functions of these tests we study for a special classes of close alternatives. We discuss the construction of the goodness of fit test based on the local time and the possibility of the construction of asymptotically distribution free tests in the case of composite basic hypothesis.