在單一長回合 (one long run) 的穩態系統模擬中，由於觀察值之間幾乎不為獨立同分配，若使用樣本變異數作為母體變異數τ的估計量會產生偏誤，於是許多學者提出不同的方法估計τ，常見的方法為依序將觀察值不重複(無重疊分批)或重複(重疊分批)分入相鄰的批次中，利用批次的統計量估計 τ，而大部分使用重疊分批求得的估計量，其表現皆比使用無重疊分批得到的估計量好，但由於重疊分批使批次之間擁有相關性，因此與變異數估計量相關的抽樣分配不服從卡方分配及 Student’s t 分配，所以在進行區間估計時使用卡方分配或 Student’s t 分配會產生偏誤，除非能找出與該估計量相關的抽樣分配。
本研究利用泛函中央極限定理(functional central limit theorem; FCLT)及連續映射定理(continuous mapping theorem; CMT)推導重疊標準化時間序列變異數估計量(overlapping standardized time series variance estimator; OSTS estimator)之漸近分配：OSTS-χ2分配及OSTS-t分配，並搭配數值法 (Numerical Method) 產生建立信區間時所需的OSTS-χ2分配表及OSTS-t分配表；產生分配表的基本方法為蒙地卡羅法 (Monte Carlo Method) ，而本研究使用的數值法的概念為將漸近分配利用線性組合表示並搭配拉普拉斯變換 (Laplace Transforms) ，隨後即可求得特定分位數的值，相較於蒙地卡羅法是較為省時且兼具精準的方法。
經由自迴歸模型(autoregressive model)的實例驗證，發現使用OSTS變異數估計量搭配本研究的 OSTS-χ 2 分配表及 OSTS-t 分配表，在批次內觀察值數目為足夠大的條件下，建立母體參數的信賴區間並不會產生偏誤，除了再次驗證 OSTS 變異數估計量的良好表現，也說明即使重疊分批的變異數估計方法不遵守統計的邏輯，但若能求得該估計量的分配，它會是個表現很好的估計量。

Batching is a classic approach for estimating the variance of the sample mean obtained from a stationary data process. Nonoverlapping batch means (NBM) estimator has long been the most-common foundation for statistical inference. Overlapping batch means (OBM) estimator, despite their inherent statistical efficiency have not been a popular basis for statistical inference, primarily because their dependence makes the classical χ 2 and Student’s t coefficients inappropriate. Overlapping standardized time series (OSTS) estimator, which is more efficient than OBM estimator, similarly creates dependence that negate the use of the classical sampling distributions; without a statistical-inference foundation, use of the OSTS estimator of the standard error languishes. This study is intended to provide a foundation for OSTS statistical inference by defining the OSTS-χ 2 and OSTS-t distributions, analogous to the classical χ2 and Student’s t distributions. We develop an eigenvalue-based numerical algorithm for computing the distributions’ cumulative probabilities, which we use to present a table of quantile values for each OSTS distribution. The experimental results show that when the value of the batch size is chosen large enough, we closely approximated the distributions’ quantile values with a numerical method that uses eigenvalues to reflect the covariances of the overlapping batch areas. Access to these distribution quantiles simplifies and improves statistical inference based on the OSTS estimator of the sample-mean’s standard error.

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