Statistical inference for discrete-time samples from affine stochastic delay differential equations

Abstract

Statistical inference for discrete time observations of an affine stochastic delay differential equation is considered. The main focus is on maximum pseudo-likelihood estimators, which are easy to calculate in practice. Also a more general class of prediction-based estimating functions is investigated. In particular, the optimal prediction-based estimating function and the asymptotic properties of the estimators are derived. The maximum pseudo-likelihood estimator is a particular case, and an expression is found for the efficiency loss when using the maximum pseudo-likelihood estimator rather than the computationally more involved optimal prediction-based estimator. The distribution of the pseudo-likelihood estimator is investigated in a simulation study. For models where the delay measure is concentrated on a finite set, an estimator obtained by discretization of the continuous-time likelihood function is presented, and its asymptotic properties are investigated. The estimator is very easy to calculate, but it is shown to have a significant bias when the sampling frequency is low. Two examples of affine stochastic delay equation are considered in detail.