Optimality of estimators for misspecified semi-Markov models

TL;DR

This paper investigates the asymptotic properties and efficiency of maximum likelihood estimators in semi-Markov models, demonstrating their robustness and optimality even under model misspecification.

Contribution

It provides a heuristic derivation of the asymptotic distributions and efficiency of MLEs in semi-Markov models, including under misspecification.

Findings

Estimators are asymptotically efficient for correctly specified models.

MLEs remain asymptotically efficient under model misspecification.

Heuristic derivation of asymptotic distributions of estimators.

Abstract

Suppose we observe a geometrically ergodic semi-Markov process and have a parametric model for the transition distribution of the embedded Markov chain, for the conditional distribution of the inter-arrival times, or for both. The first two models for the process are semiparametric, and the parameters can be estimated by conditional maximum likelihood estimators. The third model for the process is parametric, and the parameter can be estimated by an unconditional maximum likelihood estimator. We determine heuristically the asymptotic distributions of these estimators and show that they are asymptotically efficient. If the parametric models are not correct, the (conditional) maximum likelihood estimators estimate the parameter that maximizes the Kullback--Leibler information. We show that they remain asymptotically efficient in a nonparametric sense.