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

TL;DR

This paper develops and analyzes statistical inference methods, including maximum pseudo-likelihood and prediction-based estimators, for discrete observations of affine stochastic delay differential equations, focusing on their efficiency and asymptotic properties.

Contribution

It introduces a general class of prediction-based estimating functions and derives their asymptotic properties, comparing their efficiency to pseudo-likelihood estimators in stochastic delay equations.

Findings

Optimal prediction-based estimator has better efficiency than pseudo-likelihood estimator.

Asymptotic properties of estimators are derived and validated through simulations.

Two specific affine stochastic delay equations are analyzed in detail.

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. A more general class of prediction-based estimating functions is investigated as well. 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. Two examples of affine stochastic delay equation are considered in detail.