On Multi-step Estimation of Delay for SDE

TL;DR

This paper introduces one-step and two-step maximum likelihood estimators for delay estimation in SDEs, overcoming computational difficulties of traditional MLEs by simplifying the estimation process while maintaining asymptotic properties.

Contribution

The paper proposes novel one-step and two-step MLE methods for delay estimation in SDEs that are easier to compute and asymptotically equivalent to traditional MLEs.

Findings

Proposed estimators are consistent and asymptotically normal.

The new estimators are computationally simpler than classical MLEs.

Application to stochastic Pantograph equation demonstrates practical relevance.

Abstract

We consider the problem of delay estimation by the observations of the solutions of several SDEs. It is known that the MLE for these models are consistent and asymptotically normal, but the likelihood ratio functions are not differentiable w.r.t. parameter and therefore numerical calculation of the MLEs has certain difficulties. We propose One-step and Two-step MLE, whose calculation has no such problems and provide estimator asymptotically equivalent to the MLE. These constructions are realized in two or three steps. First we construct preliminary estimators which are consistent and asymptotically normal, but not asymptotically efficient. Then we use these estimators and modified Fisher-score device to obtain One-step and Two-step MLEs. We suppose that its numerical realization is much more simple. Stochastic Pantograph equation is introduced and related statistical problems are discussed.