Local Asymptotic Mixed Normality via Transition Density Approximation and an Application to Ergodic Jump-Diffusion Processes

TL;DR

This paper establishes conditions under which local asymptotic mixed normality holds for jump-diffusion models, enabling the development of efficient estimators and tests for discretely observed processes.

Contribution

It weakens existing conditions for local asymptotic mixed normality and applies transition density approximation to jump-diffusion models, including parameter estimation and hypothesis testing.

Findings

Local asymptotic mixed normality is shown for jump-diffusion models.

Efficient quasi-maximum-likelihood and Bayes estimators are constructed.

Asymptotically most powerful tests are developed for model parameters.

Abstract

We study sufficient conditions for local asymptotic mixed normality. We weaken the sufficient conditions in Theorem 1 of Jeganathan (Sankhya Ser. A 1982) so that they can be applied to a wider class of statistical models including a jump-diffusion model. Moreover, we show that local asymptotic mixed normality of a statistical model generated by approximated transition density functions is implied for the original model. Together with density approximation by means of thresholding techniques, we show local asymptotic normality for a statistical model of discretely observed jump-diffusion processes where the drift coefficient, diffusion coefficient, and jump structure are parametrized. As a consequence, the quasi-maximum-likelihood and Bayes-type estimators proposed in Shimizu and Yoshida (Stat. Inference Stoch. Process. 2006) and Ogihara and Yoshida (Stat. Inference Stoch. Process. 2011) are shown to be asymptotically efficient in this model. Moreover, we can construct asymptotically uniformly most powerful tests for the parameters.