Asymptotic Properties of a Recursive Procedure for Simultaneous Estimation

Abstract

In this paper we consider a problem that arises in estimating the heavy traffic limit of a sojourn time distribution in a queueing network during the course of a medium traffic simulation. We need to estimate α =E[f(γ,M)], where γ is an unknown constant andMa random variable. More specifically, we are given an iid sequence of random vectors {(Xi,Mi), 1 ≤ i ≤ n}, with γ =E[Xi] andMihaving the same distribution asM.For known γ, we have a standard estimation problem, which we describe here. The standard estimate is unbiased and asymptotically (asn→ 8 ) consistent. There is also a central limit theorem for this estimator. For unknown γ, we provide two estimation procedures, one that requires two passes through the data (as well as storage of {Mi, 1 ≤ i ≤ n}), and another one, which is recursive, requiring only one pass through and bounded storage. The estimators obtained from these two procedures are shown to be strongly consistent, and central limit theorems are also proven for them.