Properties of Standardized Time Series Weighted Area Variance Estimators

Abstract

We wish to estimate the variance of the sample mean from a continuous-time stationary stochastic process. This article expands on the results of a technical note (Goldsman and Schruben 1990) by using the theory of standardized time series to investigate weighted generalizations of Schruben's area variance estimator. We find a simple expression for the bias of the weighted area variance estimator, and we give weights which yield variance estimators with lower asymptotic bias than certain other popular estimators. We use the weighted area variance estimators to derive asymptotically valid confidence interval estimators (CIEs) for the mean of a stationary stochastic process. Although the weighted area CIEs have the same asymptotic expected value and variance of the length as Schruben's area CIE, we show that the new CIEs sometimes yield coverages which are closer to the nominal value.simulation output analysis, confidence intervals, standardized time series

Boundary Crossing Probabilities for the Cumulative Sample Mean

“…Many researchers have used standardized time series methods to estimate the mean and variance of data that have some degree of serial dependency (Batur, Goldsman, and Kim [4], Goldsman, Meketon, and Schruben [9]). We take a different approach and exploit the properties of Brownian bridges to derive the probability the cumulative mean stays within some distance from the long-term mean and the true mean.…”

SPSTS: A sequential procedure for estimating the steady-state mean using standardized time series

“….} are uniformly integrable (Durrett, 2005), then we have lim m→∞ Var[A ( f ; b, m)] = 2σ 4 /b (Foley and Goldsman, 1999;Goldsman et al 1990). If one chooses weights having F = F = 0, then the resulting area estimator is first-order unbiased since its bias is o(1/m).…”

“…Damerdji and Goldsman (1995) gave conditions such that the estimator A ( f ; b, m) is strongly consistent as both b, m → ∞ in an appropriate fashion. (Schruben, 1983), (Goldsman et al 1990), and f cos, j (t) = √ 8π jcos(2π jt), j = 1, 2, . .…”

An improved standardized time series Durbin–Watson variance estimator for steady-state simulation

“…There are a number of different techniques in the literature devoted to the estimation of σ 2 , e.g., the methods of nonoverlapping batch means (NBM) [16], overlapping batch means (OBM) [15], and standardized time series (STS) [17]. Among the estimators based on the STS methodology are the so-called area [14] and Cramér-von Mises (CvM) [12] estimators. Goldsman et al [11] combine these two types of estimators to obtain new STS estimators-the Durbin-Watson (DW) and jackknifed DW (JDW)-with competitive bias and lower asymptotic variance than many of their competitors.…”

“…We shall also assume uniform integrability [6] of the squares of the estimators, thereby establishing the limiting variance of each estimator. As in [14], the STS area estimator for σ 2 and its limiting functional are given by…”