Bias Properties of Budget Constrained Simulations

This paper addresses the bias characteristics of estimators produced from Monte Carlo simulations. If the computer time allocated to the simulation is t , then N ( t ), the number of replications completed by time t , is a renewal process. The simulation implications of a known exact expression for the expected value of a sample mean based on N ( t ) replications are explored and a similar exact expression for a sample mean based on N ( t ) + 1 replications is derived. Bias expansions for a sample mean based on N ( t ) or N ( t ) + 1 replications are obtained. The bias in the sample mean based on N ( t ) replications is at most of order o(1/ t ). Under suitable moment conditions, the bias decreases at a much faster rate than o(1/ t ); on the other hand, the estimator based on N ( t ) + 1 replications has bias of order 1/ t . The exact expressions also lead to simple and totally unbiased estimators. Using Taylor series, the bias expansions of a general function of means based on N ( t ) or N ( t ) + 1 replications are determined. The leading terms in these expansions are of order 1/ t , although the coefficients are different. Based on these expansions, a Tin-style adjusted estimator is proposed to reduce the bias. These expansions are specialized to the case of ratio estimation in regenerative simulation. Due to a cancellation effect, the ratio estimator based on N ( t ) + 1 cycles is biased only to order o(1/ t ) providing confirmation and reinterpretation of a result of M. Meketon and P. Heidelberger.