Adaptive inference for small diffusion processes based on sampled data

TL;DR

This paper develops adaptive estimators and testing methods for small diffusion processes from discrete data, providing asymptotic analysis and applications to epidemic models.

Contribution

It introduces two novel adaptive estimators for drift and diffusion parameters with proven asymptotic properties under small dispersion and large sample limits.

Findings

Adaptive estimators are consistent and asymptotically normal.

Test statistics have known asymptotic distributions under null hypothesis.

Simulation studies validate the theoretical properties and compare estimator performances.

Abstract

We consider parametric estimation and tests for multi-dimensional diffusion processes with a small dispersion parameter $\varepsilon$ from discrete observations. For parametric estimation of diffusion processes, the main target is to estimate the drift parameter and the diffusion parameter. In this paper, we propose two types of adaptive estimators for both parameters and show their asymptotic properties under $\varepsilon\to0$, $n\to\infty$ and the balance condition that $(\varepsilon n^\rho)^{-1} =O(1)$ for some $\rho>0$. Using these adaptive estimators, we also introduce consistent adaptive testing methods and prove that test statistics for adaptive tests have asymptotic distributions under null hypothesis. In simulation studies, we examine and compare asymptotic behaviors of the two kinds of adaptive estimators and test statistics. Moreover, we treat the SIR model which describes a simple epidemic spread for a biological application.