One-parameter statistical model for linear stochastic differential equation with time delay

TL;DR

This paper analyzes the asymptotic properties of the likelihood function and maximum likelihood estimator for a linear stochastic delay differential equation with a single parameter, revealing different limit behaviors depending on the sign of a key quantity.

Contribution

It introduces a detailed asymptotic analysis of the likelihood function and estimator for a linear stochastic delay differential equation with a single parameter, including cases with different asymptotic normality behaviors.

Findings

Proves local asymptotic normality when v_* < 0

Shows local asymptotic quadraticity when v_* = 0

Establishes local asymptotic mixed normality or periodic LAMN when v_* > 0

Abstract

Assume that we observe a stochastic process $(X(t))_{t\in[-r,T]}$, which satisfies the linear stochastic delay differential equation \[ \mathrm{d} X(t) = \vartheta \int_{[-r,0]} X(t + u) \, a(\mathrm{d} u) \, \mathrm{d} t + \mathrm{d} W(t) , \qquad t \geq 0 , \] where $a$ is a finite signed measure on $[-r, 0]$. The local asymptotic properties of the likelihood function are studied. Local asymptotic normality is proved in case of $v_\vartheta^* < 0$, local asymptotic quadraticity is shown if $v_\vartheta^* = 0$, and, under some additional conditions, local asymptotic mixed normality or periodic local asymptotic mixed normality is valid if $v_\vartheta^* > 0$, where $v_\vartheta^*$ is an appropriately defined quantity. As an application, the asymptotic behaviour of the maximum likelihood estimator $\widehat{\vartheta}_T$ of $\vartheta$ based on $(X(t))_{t\in[-r,T]}$ can be derived as $T \to \infty$.