Confidence Interval Estimation Using Standardized Time Series

Abstract

Observations of a stationary stochastic process can be transformed into a standardized time series. This paper presents a lemma giving the asymptotic properties of this standardized series under quite general conditions. In particular, the conditions are satisfied by stationary discrete-event simulations. Confidence intervals can be constructed using this lemma. For illustration, we develop two easily computed interval estimators for the process mean. When independent replications of the series are available, such as in computer simulation experiments, these interval estimators may be combined with the classical confidence interval estimator. These interval estimators also tend to compensate for simulation initialization bias if the sign of the bias is known. In an empirical study using three elementary simulated processes, the interval estimators presented here compare favorably with the classical interval estimator. In a recent paper, Goldsman and Schruben (Goldsman, D., L. Schruben. 1982. Asymptotic properties of some confidence interval estimators. Tech. Rep. 544, School of O.R.I.E., Cornell University, Ithaca, NY.) show that the asymptotic properties of the confidence intervals presented in this paper strictly dominate those of classical confidence intervals.

Cumulative mean bounds for quality control analysis

“…To calculate the value of CMB, we rely on properties of standardized time series developed in Schruben (1983). Standardized time series based on m samples of observations rescale the data to form a new time series over the range of time t ∈ [0, 1] according to the following function:…”

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

“…In order to give a measure of the precision ofȲ n or to build a confidence interval for µ, we can also estimate the variance parameter, σ 2 ≡ lim n→∞ nVar(Ȳ n ). 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.…”

“…Under the Standing Assumptions, [10] and [17] show that T n ⇒ B, a Brownian bridge on [0, 1], i.e., a Gaussian process with E[B(t)] = 0 and Cov(B(s), B(t)) = min(s, t) − st. The cited references use this fact to show that all of the variance estimators considered herein converge to limiting random variables having expectation σ 2 .…”

Monitoring autocorrelated processes using a distribution-free tabular CUSUM chart with automated variance estimation

“…Fortunately, the simulation literature provides a number of variance-estimation techniques based on the following methods for analysis of steady-state simulation outputs: autoregressive representation (Fishman, 1971); nonoverlapping batch means (Fishman and Yarberry, 1997); overlapping batch means (Alexopoulos et al, 2007a); spectral analysis (Lada and Wilson, 2006); and standardized time series (STS) (Schruben, 1983). Although accurate and efficient estimation of the variance parameter is an important research problem by itself, in this article we are more interested in developing automated variance-estimation procedures that can be effectively incorporated into distribution-free SPC charts.…”