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

Abstract

Assume that we observe a stochastic process (X(t)) t∈ [−r,T ] , which satisfies the linear stochastic delay differential equationwhere 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 * ϑ < 0, local asymptotic quadraticity is shown if v * ϑ = 0, and, under some additional conditions, local asymptotic mixed normality or periodic local asymptotic mixed normality is valid if v * ϑ > 0, where v * ϑ is an appropriately defined quantity. As an application, the asymptotic behaviour of the maximum likelihood estimator ϑ T of ϑ based on (X(t)) t∈[−r,T ] can be derived as T → ∞.

Asymptotic inference for linear stochastic differential equations with time delay

“…Finally in Section 3.3 the general case is considered, where the delay measure a is an arbitrary nite signed measure on [−r, 0]. The results for the uniform case have been published in Benke and Pap [10], and for the general case in Benke and Pap [11].…”

“…In Section 3.3 the main results of the general case are presented. The results have been published in Benke and Pap [11]. In this general model it can not be possible to determine the exact values of the parameter ϑ, where the appropriate local asymptotic property is valid.…”