Goodness-of-Fit Tests for Ornstein-Uhlenbeck Process

TL;DR

This paper develops goodness-of-fit tests for the Ornstein-Uhlenbeck process, a type of linear stochastic differential equation, demonstrating that the test statistics' distributions are independent of unknown parameters.

Contribution

It introduces two Cramer-von Mises type tests based on empirical distribution and local time estimators, with limit distributions unaffected by unknown parameters.

Findings

Test statistics' limit distributions are parameter-independent.

Proposed tests effectively assess model fit for Ornstein-Uhlenbeck processes.

The methods are applicable even with composite hypotheses involving unknown parameters.

Abstract

We consider the goodness of fit testing problem for linear stochastic differential equation (Ornstein-Uhlenbeck process). The basic hypothesis is supposed to be composite with two-dimensional unknown parameter. We study two goodness of fit tests of Cramer-von Mises type based on empirical distribution function and on local time estimator of the invariant density. It is shown that the limit distributions of the underlying statistics under hypothesis do not depend on the unknown parameter.