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...and the quality of the sample mean is assessed by its variance var([=.Y]). The overall goal of this paper concerns the estimation of var([=.Y]). Figure 1 shows the relationships among [theta], [=.Y], var([=.Y]) and [^.V]([=.Y]), where [^.V]([=.Y]) is an estimator of var([=.Y]).
Various estimators of the variance of the sample mean have been proposed. For example, regenerative (Crane and Iglehart, 1975; Crane and Lemoine, 1977; Glynn and Iglehart, 1986), spectral (Heidelberger and Welch, 1981; Priestley, 1981) Non-overlapping Batch Means (NBM) (Schmeiser, 1982), Overlapping Batch Means (OBM) (Meketon and Schmeiser, 1984), Partial-overlapping Batch Means (PBM) (Welch, 1987), standardized time series (Schruben, 1983; Glynn and Iglehart, 1990), and its variations (Foley and Goldsman, 1999; Goldsman et al., 1999). All of these estimators are developed assuming that the sample size n is known in advance and all require O(n) storage space, at least in their original forms.
To our knowledge, the Dynamic Non-overlapping Batch Means (DNBM) estimator proposed by Yeh and Schmeiser (2000) is the only existing variance estimator requiring O(1) storage space for any sample size n. We now propose another type of estimator, which we call the Dynamic Partial-overlapping Batch Means (DPBM) estimator, that also requires O(1) storage space and has better statistical performance in terms of Mean Squared Error (MSE) than that for DNBM estimators.
This paper is organized as follows. In Section 2.1 we review Batch Means (BM) estimators, including asymptotic and finite-sample results. In Section 2.2 we review DNBM estimators. In Section 3, we propose and evaluate the proposed DPBM estimators. Section 4 is the summary.
2. Literature review
2.1. BM estimators
The definition of the BM estimator for var([=.Y]) with batch size m and batch shift s, 1 [less than or equal to] s [less than or equal to] n - m, is
[^.V](m, s) = [[[summation].sub.h=1.sup.b] ([bar.Y.sub.s(h-1)+1,m] - [=.Y.sub.n])[.sup.2]]/[d.sub.b]. (1)
where the hth batch mean is
[bar.Y.sub.s(h-1)+1,m] = [m.summation over (j=1)] [Y.sub.s(h-1)+j]/m, (2)
the number of batches b = [??](n - m + s)/s[??] (where [??]k[??] is the greatest integer smaller than or equal to k), the overall mean [=.Y.sub.n] = [[summation].sub.i=1.sup.n] [Y.sub.i]/n, and the constant [d.sub.b] = b((n/m) - 1). That is, BM estimators first divide observations [Y.sub.1],..., [Y.sub.n] into b batches (not necessarily non-overlapping) each with size m, then compute the sum of the squared distances of each batch mean from the overall mean, finally multiplied by an appropriate constant to ensure unbiasedness for the case of independent identically distributed (i.i.d.) observations.
[FIGURE 1 OMITTED]
The NBM estimator with batch size m is the special case obtained when s = m, and is denoted by [^.V.sup.N](m) (see Fig. 2(a)). The OBM estimator with batch size m is the special case obtained when s = 1, and is denoted by [^.V.sup.O](m) (see Fig. 2(b)). The PBM estimator with batch size m is the special case obtained when 1 m, and is denoted by [^.V.sup.S](m) (see Fig. 2(d)). Throughout this paper, we focus on PBM estimators and use NBM and OBM as base-line estimators. (Spaced BM estimators are mentioned here only for completeness.)
First, we review asymptotic properties of BM estimators. A comparison among asymptotic bias and variance results for BM estimators with different values of s is given in Table 1. The second column of Table 1 is the shift s. The third column gives the corresponding types of BM estimators. The fourth column gives asymptotic relative variance results, which...
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