Folded variance estimators for stationary time series

This thesis is concerned with simulation output analysis. In particular, we are interested in estimating the variance parameter of a steady-state output process. The estimation of the variance parameter has immediate applications in problems involving (i) the precision of the sample mean as a point estimator for the steady-state mean muX, and (ii) confidence intervals for muX. The thesis focuses on new variance estimators arising from Schruben's method of standardized time series (STS). The main idea behind STS is to let such series converge to Brownian bridge processes; then their properties are used to derive estimators for the variance parameter. Following an idea from Shorack and Wellner, we study different levels of folded Brownian bridges. A folded Brownian bridge is obtained from the standard Brownian bridge process by "folding" it down the middle and then "stretching" it so that it spans the interval [0,1]. We formulate the folded STS, and deduce a simplified expression for it. Similarly, we define the weighted area under the folded Brownian bridge, and we obtain its asymptotic properties and distribution. We study the square of the weighted area under the folded STS (known as the folded area estimator) and the weighted area under the square of the folded STS (known as the folded Cramer - von Mises, or CvM, estimator ) as estimators of the variance parameter of a stationary time series. In order to obtain results on the bias of the estimators, we provide a complete finite-sample analysis based on the mean-square error of the given estimators. Weights yielding first-order unbiased estimators are found in the area and CvM cases. Finally, we perform Monte Carlo simulations to test the efficacy of the new estimators on a test bed of stationary stochastic processes, including the first-order moving average and autoregressive processes and the waiting time process in a single-server Markovian queuing system.