Designing and Analyzing Sample Paths

Abstract

As in Ch. 5, we take as our objective the estimation of

$$\mu = \mu \left( g \right) = \int_X {g\left( {\text{x}} \right){\text{d}}F\left( {\text{x}} \right)} $$

(1)

, where F denotes an m-dimensional d.f. on $X \subseteq {\mathbb{R}^m}$. Consider a Monte Carlo Markov sampling experiment composed of n independent replications, each of which begins in a state drawn from an initializing nonequilibrium distribution π₀. After a warm-up interval of k — 1 steps on each replication, sampling continues for t additional steps and one uses the n independent truncated sample paths or realizations, each of length t, to estimate µ. Whereas Ch. 5 concentrates on sample path generating algorithms and a conceptual understanding of convergence to an equilibrium state, this chapter focuses on sampling plan design and statistical inference. With regard to design, the chapter shows how the choices of k, n, π₀, and t affect computational and statistical efficiency. With regard to statistical inference, it describes methods for estimating the warm-up interval k that significantly mitigate the influence of the initial states drawn from the nonequilibrium distribution π₀. Also, it describes methods for computing asymptotically valid confidence intervals for µ in expression (1) as n → ∞ for fixed t, as t → ∞ for fixed n and as both n → ∞ and t → ∞. Because confidence intervals inevitably depend on variance estimates, we need to impose a moderately stronger restriction on g. Whereas the assumption $\int_X {{g^2}\left( {\text{x}} \right){\text{d}}F\left( {\text{x}} \right)} < \infty$ in Ch. 5 guarantees a finite variance for a single observation on a sample path, the assumption $\int_X {{g^4}\left( {\text{x}} \right){\text{d}}F\left( {\text{x}} \right)} < \infty$ is necessary for us to obtain consistent estimators of that variance and of other variances that play essential roles in the derivation of asymptotically valid confidence intervals for µ.