[Paper Review] Estimating discontinuous periodic signals in a non-time homogeneous diffusion process

Reinhard Hoepfner, Yury A. Kutoyants|arXiv (Cornell University)|Mar 29, 2009

Stochastic processes and statistical mechanics27 references3 citations

TL;DR

This paper studies parameter estimation for a time-inhomogeneous diffusion process with a discontinuous periodic signal introduced via an unknown phase shift $\vartheta$. By establishing limit theorems for martingales and functionals under positive Harris recurrence, it proves local asymptotic normality in a non-standard framework, showing convergence of likelihood ratios to a double-sided Brownian motion model. The key result is that the Bayes estimator achieves lower quadratic risk than the maximum likelihood estimator in this limit experiment, confirming its superiority in this non-regular setting.

ABSTRACT

We consider a diffusion $(ξ_t)_{t\ge 0}$ with some $T$-periodic time dependent input term contained in the drift: under an unknown parameter $\vth\inΘ$, some discontinuity - an additional periodic signal - occurs at times $kT{+}\vth$, $k\in\bbn$. Assuming positive Harris recurrence of $(ξ_{kT})_{k\in\bbn_0}$ and exploiting the periodicity structure, we prove limit theorems for certain martingales and functionals of the process $(ξ_t)_{t\ge 0}$. They allow to consider the statistical model parametrized by $\vth\inΘ$ locally in small neighbourhoods of some fixed $\vth$, with radius $1/n$ as $ to$. We prove convergence of local models to a limit experiment studied by Ibragimov and Khasminskii [IH 81] and discuss the behaviour of estimators under contiguous alternatives.

Motivation & Objective

• To analyze statistical inference for a diffusion process with a time-inhomogeneous drift containing a periodic discontinuity parameterized by $\vartheta$.
• To establish limit theorems for martingales and functionals in a non-stationary, periodic diffusion setting.
• To investigate convergence of local models to a limit experiment with likelihood ratios of the form $\exp(W_u - \frac{1}{2}|u|)$, as studied by Ibragimov and Khasminskii.
• To compare the asymptotic performance of maximum likelihood and Bayes estimators in this non-regular, non-$L^2$-differentiable model.
• To derive a local asymptotic minimax bound and show that a Bayes sequence attains this bound on shrinking neighborhoods of radius $1/n$.

Proposed method

• Prove positive Harris recurrence of the $T$-segmented process $((\xi_{kT+s})_{0\leq s\leq T})_{k\in\mathbb{N}_0}$ using the periodic structure of the semigroup.
• Establish strong laws of large numbers and limit theorems for functionals and martingales derived from the continuous-time diffusion $\xi_t$.
• Use Le Cam’s Third Lemma and asymptotic equivariance to analyze estimators under contiguous alternatives.
• Construct local models at $\vartheta$ with local scale $1/n$ as $n \to \infty$, showing convergence to a limit experiment with likelihood ratios $\widetilde{L}^{u/0} = \exp(\widetilde{W}(uJ_\vartheta) - \tfrac{1}{2}|uJ_\vartheta|)$.
• Analyze the limit experiment using double-sided Brownian motion $\widetilde{W}$, where parametrization is $\tfrac{1}{2}$-Hölder continuous in Hellinger distance.
• Derive the local asymptotic minimax bound and show that a Bayes estimator with uniform prior attains this bound, proving its optimality in quadratic risk.

Experimental results

Research questions

• RQ1 How does the statistical model for a time-inhomogeneous diffusion with a discontinuous periodic signal behave asymptotically under local parametrization at $\vartheta$?
• RQ2 What is the limiting experiment to which the local models of the diffusion process converge as $n \to \infty$?
• RQ3 How do the maximum likelihood and Bayes estimators compare in terms of asymptotic quadratic risk in this non-regular model?
• RQ4 Can a local asymptotic minimax bound be derived in this setting, and is it attainable?
• RQ5 Is the Bayes estimator strictly better than the maximum likelihood estimator in this limit experiment?

Key findings

• The local models at $\vartheta$ converge to a limit experiment with likelihood ratios $\widetilde{L}^{u/0} = \exp(\widetilde{W}(uJ_\vartheta) - \frac{1}{2}|uJ_\vartheta|)$, where $\widetilde{W}$ is a double-sided Brownian motion.
• The limit experiment is not locally asymptotically normal or $L^2$-differentiable, and lacks a sufficient statistic or central limit structure.
• The Bayes estimator with a uniform prior on $\mathbb{R}$ achieves the local asymptotic minimax bound for quadratic risk on shrinking neighborhoods of radius $1/n$.
• The Bayes estimator has strictly smaller variance than the maximum likelihood estimator in the limit experiment, confirming its superiority.
• The proof relies on asymptotic equivariance, Le Cam’s Third Lemma, and convergence of the posterior mean to the Bayes risk under the limit measure.
• The convergence of likelihood ratios is not uniform in $\vartheta$, and the authors avoid this assumption, instead using contiguous alternatives and detailed tail analysis of posterior densities.