Statistica Sinica 18(2008), 1641-1651

AN EXTENSION OF PESKUN AND TIERNEY ORDERINGS

TO CONTINUOUS TIME MARKOV CHAINS

Fabrizio Leisen and Antonietta Mira

University of Insubria

Abstract: Peskun ordering is a partial ordering defined on the space of transition matrices of discrete time Markov chains. If the Markov chains are reversible with respect to a common stationary distribution $\pi$, Peskun ordering implies an ordering on the asymptotic variances of the resulting Markov chain Monte Carlo estimators of integrals with respect to $\pi$. Peskun ordering is also relevant in the framework of time-invariant estimating equations in that it provides a necessary condition for ordering the asymptotic variances of the resulting estimators. Tierney ordering extends Peskun ordering from finite to general state spaces. In this paper Peskun and Tierney orderings are extended from discrete time to continuous time Markov chains.

Key words and phrases: Asymptotic variance, covariance ordering, efficiency ordering, MCMC, time-invariance estimating equations.