Empirical $L^2$-distance test statistics for ergodic diffusions

TL;DR

This paper introduces a new empirical $L^2$-distance test statistic for one-dimensional ergodic diffusions, demonstrating its asymptotic distribution free property and superior small-sample performance through Monte Carlo simulations.

Contribution

It proposes a novel $L^2$-distance based test statistic for ergodic diffusions using a quasi-likelihood approach, with proven asymptotic properties and improved small-sample performance.

Findings

Test statistic converges to a chi-squared distribution asymptotically.

The test performs better than existing methods in small samples.

Monte Carlo simulations validate the theoretical advantages.

Abstract

The aim of this paper is to introduce a new type of test statistic for simple null hypothesis on one-dimensional ergodic diffusion processes sampled at discrete times. We deal with a quasi-likelihood approach for stochastic differential equations (i.e. local gaussian approximation of the transition functions) and define a test statistic by means of the empirical $L^2$-distance between quasi-likelihoods. We prove that the introduced test statistic is asymptotically distribution free; namely it weakly converges to a $\chi^2$ random variable. Furthermore, we study the power under local alternatives of the parametric test. We show by the Monte Carlo analysis that, in the small sample case, the introduced test seems to perform better than other tests proposed in literature.