Cramer-Von Mises Variance Estimators for Simulations

INFORMS Journal on Computing

et al. 2007

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F or a steady-state simulation output process, we formulate efficient algorithms to compute certain estimators of the process variance parameter (i.e., the sum of covariances at all lags), where the estimators are derived in principle from overlapping batches separately and then averaged over all such batches. The algorithms require order-of-sample-size work to evaluate overlapping versions of the area and Cramér-von Mises estimators arising in the method of standardized time series. Recently, Alexopoulos et al. showed that, compared with estimators based on nonoverlapping batches, the estimators based on overlapping batches achieve reduced variance while maintaining similar bias asymptotically as the batch size increases. We provide illustrative analytical and Monte Carlo results for M/M/1 queue waiting times and for a first-order autoregressive process. We also present evidence that the asymptotic distribution of each overlapping variance estimator can be closely approximated using an appropriately rescaled chi-squared random variable with matching mean and variance.

Section: Nonoverlapping Batched Estimatorsmentioning

confidence: 99%

Efficient Computation of Overlapping Variance Estimators for Simulation

Alexopoulos

Argon

INFORMS Journal on Computing

et al. 2007

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“…An interesting question to investigate is that of using other, non-χ 2 variance estimators in the Rinott procedure. For example, the low variance and reasonable convergence properties of the overlapping batch means estimator (Meketon and Schmeiser 1984) or the STS Cramér-von Mises estimator (Goldsman, Kang, and Seila 1999) might make them attractive candidates for inclusion in Rinott.…”

Section: Discussionmentioning

confidence: 99%

Selection procedures with standardized time series variance estimators

Proceedings of the 31st Conference on Winter Simulation Simulation---a Bridge to the Future - WSC '99

Marshall

1999

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This article studies a modification of Rinott's two-stage procedure for selecting the normal population with the largest (or smallest) mean. The modification, which is appropriate for use in the simulation environment, uses in the procedure's first stage different variance estimators than the usual batch means (BM) variance estimator. In particular, we will use variance estimators arising from the method of standardized time series (STS). On the plus side, certain STS estimators have more degrees of freedom than that of the BM estimator. On the other hand, STS variance estimators tend to require larger sample sizes than the BM estimator in order to converge to their assumed distributions. These considerations result in trade-offs involving the procedure's achieved probability of correct selection as well as the procedure's expected sample size.

“…The estimator C 1 (g 0 ; n) has the same asymptotic variance as, but significantly larger bias than, the original level-0 CvM estimator C 0 (g 0 ; n), whose bias is about 5γ 1 /n (Example 3). Using Lagrange multipliers as in Goldsman et al [1999], we can find the polynomial weight function g(t) that minimizes the limiting variance Var[C 1 (g)] of the level-1 folded CvM estimator for σ 2 while satisfying the first-order unbiasedness constraint (G +¯Ḡ = 1) and Assumptions G. For example, the asymptotically minimum-variance, first-order unbiased, level-1 quadratic weight is g 1,2 (t) ≡ −180t 2 + 168t − 24, resulting in a limiting variance of Var[C 1 (g 1,2 )] = 72σ 4 /35 . = 2.057σ 4 -a bit larger than that of the analogous level-0 quadratically weighted CvM estimator, which equals 1.729σ 4 (Example 3).…”

Section: The Folded Cvm Estimatormentioning

confidence: 99%

Performance of folded variance estimators for simulation

Alexopoulos

Antonini

ACM Trans. Model. Comput. Simul.

et al. 2010

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We extend and analyze a new class of estimators for the variance parameter of a steady-state simulation output process. These estimators are based on "folded" versions of the standardized time series (STS) of the process, and are analogous to the area and Cramér-von Mises estimators calculated from the original STS. In fact, one can apply the folding mechanism more than once to produce an entire class of estimators, all of which reuse the same underlying data stream. We show that these folded estimators share many of the same properties as their nonfolded counterparts, with the added bonus that they are often nearly independent of the nonfolded versions. In particular, we derive the asymptotic distributional properties of the various estimators as the run length increases, as well as their bias, variance, and mean squared error. We also study linear combinations of these estimators, and we show that such combinations yield estimators with lower variance than their constituents. Finally, we consider the consequences of batching, and we see that the batched versions of the new estimators compare favorably to benchmark estimators such as the nonoverlapping batch means estimator.