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

Uwe Küchler , Michael Sørensen

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 25

MSC 2000

62M09 Non-Markovian processes: estimation

34K50 Stochastic delay 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 predictionbased 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.