Adaptive test for ergodic diffusions plus noise

TL;DR

This paper develops parametric tests for ergodic diffusion models with noise, utilizing quasi-likelihood functions, and demonstrates their effectiveness through simulations and real wind data analysis.

Contribution

It introduces new likelihood-ratio, Wald, and Rao-type tests for diffusion-plus-noise models based on quasi-likelihood functions, with validation through simulations and real data.

Findings

Test statistics converge in law under null hypotheses.

Tests are consistent under alternative hypotheses.

Application to wind data illustrates practical utility.

Abstract

We propose some parametric tests for ergodic diffusion-plus-noise model, which is a version of state-space modelling in statistics for stochastic diffusion equations. The test statistics are classified into three types: likelihood-ratio-type test statistic; Wald-type one; and Rao-type one. All the test statistics are constructed with quasi-likelihood-functions for local mean sequence of noised observation. We also simulate the behaviour of them for several practical hypothesis tests and check the convergence in law of test statistics under null hypotheses and consistency of the test under alternative ones. We apply the method for real data analysis of wind data, and examine some sets of the hypotheses mainly with respect to the structure of diffusion coefficient.