[Paper Review] Nondegeneracy of Random Field and Estimation of Diffusion

Masayuki Uchida, Nakahiro Yoshida|arXiv (Cornell University)|Dec 22, 2012

Statistical Methods and Inference3 citations

TL;DR

This paper establishes a quasi likelihood analysis for diffusion processes observed at high frequency by proving a polynomial-type large deviation inequality for the quasi likelihood random field. The key contribution is a novel nondegeneracy criterion for the random index $\chi_0$, achieved via a tensor bundle section argument, enabling asymptotic normality and convergence of moments for estimators under high-frequency sampling.

ABSTRACT

We construct a quasi likelihood analysis for diffusions under the high-frequency sampling over a finite time interval. For this, we prove a polynomial type large deviation inequality for the quasi likelihood random field. Then it becomes crucial to prove nondegeneracy of a key index chi_0. By nature of the sampling setting, chi_0 is random. This makes it difficult to apply a naive sufficient condition, and requires a new machinery. In order to establish a quasi likelihood analysis, we need quantitative estimate of the nondegeneracy of chi_0. The existence of a nondegenerate local section of a certain tensor bundle associated with the statistical random field solves this problem.

Motivation & Objective

• Address the challenge of nondegeneracy in quasi likelihood analysis for diffusions under high-frequency discrete sampling.
• Overcome the difficulty of a random, non-degenerate index $\chi_0$ that cannot be handled by classical deterministic conditions.
• Develop a quantitative nondegeneracy criterion for the statistical random field to enable moment convergence and higher-order inference.
• Establish a framework for quasi maximum likelihood and Bayesian estimation with convergence of moments under high-frequency asymptotics.
• Provide a general analytical machinery applicable to locally asymptotically mixed normal (LAMN) experiments in stochastic processes.

Proposed method

• Formulate the stochastic regression model as a stochastic integral equation with unknown drift and diffusion coefficient depending on a parameter $\theta$.
• Define the quasi likelihood random field $\mathbb{Z}_n(u)$ based on high-frequency discrete observations over a finite time interval.
• Prove a polynomial-type large deviation inequality for the quasi likelihood field to control tail probabilities and enable moment convergence.
• Introduce a novel nondegeneracy condition via the existence of a nondegenerate local section of a tensor bundle associated with the statistical field.
• Use Malliavin calculus and regularity estimates to control derivatives of the log-likelihood field uniformly over parameter space.
• Apply the Ibragimov-Has’minskii-Kutoyants scheme in a stochastic setting, leveraging Yoshida's (2005) framework for LAQ models.

Experimental results

Research questions

• RQ1 How can nondegeneracy of the random index $\chi_0$ be established when it is inherently random due to sampling design?
• RQ2 What conditions ensure the existence of a nondegenerate local section of the tensor bundle that controls the quasi likelihood field?
• RQ3 Can a polynomial-type large deviation inequality be derived for high-frequency diffusion models with unobserved drift?
• RQ4 Under what conditions does the quasi maximum likelihood estimator achieve convergence of moments in the LAMN setting?
• RQ5 How can the analytical machinery be extended to handle singular or irregular points in the state space of the diffusion process?

Key findings

• A polynomial-type large deviation inequality is established for the quasi likelihood random field under high-frequency sampling, enabling higher-order asymptotic analysis.
• The nondegeneracy of $\chi_0$ is proven via the existence of a nondegenerate local section of a tensor bundle, a novel condition that avoids reliance on continuous supporting functions.
• Convergence of moments for the quasi maximum likelihood and Bayesian estimators is established under the new nondegeneracy criterion.
• Theoretical results are extended to cases where the diffusion process may hit singular points, provided it escapes quickly into regular regions.
• The framework supports model selection and information criteria via moment convergence, as demonstrated in related work by Uchida (2010).
• The method applies to multi-dimensional diffusions and time-inhomogeneous models, including stochastic volatility models, under high-frequency observation.