Folded Variance Estimators for Stationary Time Series

This thesis is concerned with simulation output analysis. In particular, we are inter-ested in estimating the variance parameter of a steady-state output process. The estimationof the variance parameter has immediate applications in problems involving (i) the precisionof the sample mean as a point estimator for the steady-state mean and #956;X, and (ii) confidenceintervals for and #956;X. The thesis focuses on new variance estimators arising from Schrubensmethod of standardized time series (STS). The main idea behind STS is to let such seriesconverge to Brownian bridge processes; then their properties are used to derive estimatorsfor the variance parameter. Following an idea from Shorack and Wellner, we study differentlevels of folded Brownian bridges. A folded Brownian bridge is obtained from the standardBrownian bridge process by folding it down the middle and then stretching it so thatit spans the interval [0,1]. We formulate the folded STS, and deduce a simplified expressionfor it. Similarly, we define the weighted area under the folded Brownian bridge, and weobtain its asymptotic properties and distribution. We study the square of the weighted areaunder the folded STS (known as the folded area estimator ) and the weighted area under thesquare of the folded STS (known as the folded Cram??von Mises, or CvM, estimator) asestimators of the variance parameter of a stationary time series. In order to obtain resultson the bias of the estimators, we provide a complete finite-sample analysis based on themean-square error of the given estimators. Weights yielding first-order unbiased estimatorsare found in the area and CvM cases. Finally, we perform Monte Carlo simulations to testthe efficacy of the new estimators on a test bed of stationary stochastic processes, includingthe first-order moving average and autoregressive processes and the waiting time process ina single-server Markovian queuing system.