Output Assessment for Monte Carlo Simulations via the Score Statistic

Abstract

This article presents several applications of the score statistic in the context of output assessment for Monte Carlo simulations. We begin by observing that the expected value of the score statistic U is zero, and that when the inverse of the information matrix I = E(UU^T) exists, the asymptotic distribution of U^TI⁻¹U is χ². Thus, we may monitor the sample mean of this statistic throughout a simulation as a means to determine whether or not the simulation has been run for a sufficiently long time.

We also demonstrate a second convergence assessment method based upon the idea of path sampling, but first show how the score statistic can be used to accurately estimate the stationary density using only a small number of simulated values. These methods provide a powerful suite of tools which can be generically applied when alternatives such as the Rao-Blackwell density estimator are not available. Our second convergence assessment method is based upon these density estimates. By running several replications of the chain, the corresponding estimated densities may be compared to assess how “close” the chains are to one another and to the true stationary distribution. We explain how this may be done using both L¹ and L² distance measures.

We first illustrate these new methods via the analysis of MCMC output arising from some simulated examples, emphasizing the advantages of our methods over existing diagnostics. We further illustrate the utility of our methods with three examples: analyzing a set of real time series data, a collection of censored survival data, and bivariate normal data using a model with a nonidentified parameter.

Key Words:

• Convergence diagnostics
• Markov chain Monte Carlo
• Density estimation
• Path sampling