Divergences Test Statistics for Discretely Observed Diffusion Processes

TL;DR

This paper introduces $$-divergence-based test statistics for hypothesis testing in discretely observed diffusion processes, deriving their asymptotic distributions and demonstrating their effectiveness through numerical analysis.

Contribution

It develops a new class of divergence-based tests for diffusion processes and derives their asymptotic behavior under specific sampling schemes.

Findings

Asymptotic distributions depend on the regularity of $$.

The tests perform well in small samples in terms of level and power.

Different divergence measures can be used within the proposed framework.

Abstract

In this paper we propose the use of $\phi$-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process $\de X_t = b(X_t, \theta)\de t + \sigma(X_t, \theta)\de W_t$, from discrete observations $\{X_{t_i}, i=0, ..., n\}$ with $t_i = i\Delta_n$, $i=0, 1, >..., n$, under the asymptotic scheme $\Delta_n\to0$, $n\Delta_n\to\infty$ and $n\Delta_n^2\to 0$. The class of $\phi$-divergences is wide and includes several special members like Kullback-Leibler, R\'enyi, power and $\alpha$-divergences. We derive the asymptotic distribution of the test statistics based on $\phi$-divergences. The limiting law takes different forms depending on the regularity of $\phi$. These convergence differ from the classical results for independent and identically distributed random variables. Numerical analysis is used to show the small sample properties of the test statistics in terms of estimated level and power of the test.