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## 陳 逸群 D937817

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**Batch Size EffectsandMethod of Minimal-MSE Linear**Combinationin the Analysis of Simulation Output 陳逸群 D937817**Reference Papers**• Bruce Schmeiser. 1981. Batch Size Effects in the Analysis of Simulation Output. Operations Research • Bruce Schmeiser.1988. Minimal-MSE Linear Combination of Variance Estimators of the Sample Mean. 1988 Winter Simulation Conference • Bruce Schmeiser. 2004. Simulation Output Analysis: A Tutorial Based on One Research Thread. 2004 Winter Simulation Conference**Source of**randomness Random number Output variates Point estimators Random variates U Y G X The value of the performance measure Background • All simulation experiments that we consider can be view as • In general, μ is used to measure the performance and is used to be the point estimator for μ • But, how to judge quality of the point estimator?**Performance measure**Quality measure Correlated data iid data Regerative ARMA STS.A. OBM NBM Batch Size Effect m* Minimal-MSE Linear Combination Background**X1**X2 Xn-1 Xn …….… X1 X(k-1)m+1 Xm+1 Xm+2 X2 X(k-1)m+2 X2m Xkm Xm … … …… … … Introduction: Batching • Batching: Transforming correlated observations into fewer uncorrelated and normally distributed batch means (Conway, 1963) when k ≦ k* batch size: m ; number of batches: k Non-overlapping Batching**Introduction : Batching**• For n observations, point estimator of μ is • For each batch means, the sample variance is • is an estimator of So, a (1-α)100% C.I. is ,where is the half width confidence interval**Introduction : Batch Size Effects**• The primary question is how to select the appropriate batch size m under constant n • Tradeoff issue • Less batches are used, half width is much longer, more information lost • More batches are used (k>k*), independency and nonnormality will not be hold • The most used criterion for evaluating m • Half width • Expected value of half width • Standard deviation of half width • Coefficient of variation of half width • Probability of coverage**Properties of the Half Width**• When k ≦ k*, the assumption of negligible dependence of nonnormality is satisfied, then • For sample variance Sk2, then**Properties of the Half Width**• Define a bias ratio , then we have • For the expected h.w., • For the variance of the h.w., • For the C.V. of the h.w.,**Result: Half Width**• When n remains constant, changing k does affect the properties of**Result: Half Width (cont’)**• For , ↘ as k↗ for all α. The rate of decrease is much larger for small values of k than for large values. is robust for k≧10. • For , ↘ as k↗ for all α. The rate of decrease is much larger for small values of k than for large values. • For C.V., k<10 is to be avoided because of high variability and for k>30 is worthwhile.**Result: Half Width (cont’)**• Less than 10 batches yield wide unstable interval; additional batches beyond 30 yield little additional improvement**Properties of Coverage**• Coverage Function ( smaller values, better performance ) which is more comprehensive than half width because considers the performance of and , and • Use δ to demonstrate the difference between μand μ1**Result: Coverage**• When α=0.01, 0.05, 0.1, comparison of β by number of batches k**Result: Coverage (cont’)**• The decrease in due to incrementing k by one. • Decreases as k increases • Decreases as α increases(additional batches have more value when α is small) • Is small when • With any α, then decrease is very small when k>30**Conclusion**• Using a small number of batches will allow the simulation run to be terminated earlier, and the research result can be used to determine the penalty for using a specific number of batches. • 10≦k≦30 is reasonable for most simulation situations due to • k＜10, 1. more loss in information; 2. high variability • k＞30, 1. worthwhile; 2. independency and nonnormality might be invalid • Both limits are subjective but will be useful for practitioners when selecting the number of batches**Minimal-MSE Problems**• Two cases of linear combination of estimators are studied • Constrained: The sum of weights is required to be 1 • Unconstrained: The sum of weights is not required to be 1**Minimal-MSE Problems (Constrained Weights)**• Proposition 1**Numerical Study**• The numerical study is to see how much the use of linear combination can improve • The component estimators are NBM, OBM, STS.A, NBM+ STS.A optimal batch size and their properties of individual estimators are presented first as follows**Numerical Study**• The individual component estimators that comprise the optimal linear combinations**Numerical Study**• Linear combinations composed of two different types estimators with constrained and unconstrained weights**Conclusion**• The formula for mse-optimal linear combinations are the functions of biases, variances, and covariances of the component estimators • The choice of component estimator type is relatively unimportant