On Goodness-of-fit Testing for Ergodic Diffusion Process with Shift Parameter

TL;DR

This paper develops and analyzes goodness-of-fit tests for ergodic diffusion processes with an unknown shift parameter, demonstrating their asymptotic parameter-free distribution and evaluating their consistency through simulations.

Contribution

Introduces two new Cramer-Von Mises type tests for ergodic diffusion processes with shift parameters, showing their asymptotic distribution independence and assessing their performance.

Findings

Test statistics have asymptotically parameter-free distributions.

Both tests are consistent for the null hypothesis.

Simulation studies support theoretical findings.

Abstract

A problem of goodness-of-fit test for ergodic diffusion processes is presented. In the null hypothesis the drift of the diffusion is supposed to be in a parametric form with unknown shift parameter. Two Cramer-Von Mises type test statistics are studied. The first one is based on local time estimator of the invariant density, the second one is based on the empirical distribution function. The unknown parameter is estimated via the maximum likelihood estimator. It is shown that both the limit distributions of the two test statistics do not depend on the unknown parameter, so the distributions of the tests are asymptotically parameter free. Some considerations on the consistency of the proposed tests and some simulation studies are also given.