透過您的圖書館登入
IP:3.238.57.9
  • 學位論文

變異數降低法用在估計Xbar管制圖之平均連串長度上

Variance Reduction Techniques for Estimating the Xbar Chart Average Run Length

指導教授 : 陳慧芬

摘要


本論文探討變異數降低法應用在Shewhart 管制圖平均串連長度(以ARL表示)之模擬估計上。當監控的資料過程{ }為具相關性的資料時,用數值分析方法來計算ARL可能會很困難;在此種情形之下,用模擬法來估計ARL是必須的,但是ARL的估計量變異數相當大。例如:若{ }為一獨立過程,管制內的串連長度(N)會服從期望值為 ,變異數為 的幾何分配,其中 為 超出管制界限外的機率,由於 通常為一極小機率值,所以 。故本研究的目的在於當ARL 無法使用解析法或數值分析方法來計算時,在ARL的估計量偏誤不明顯之下能有效的降低模擬估計量的變異數。 我們針對ARTA過程提出ARL 估計量之變異數降低法。ARTA是具有任意邊際分配且共變數矩陣的時間數列模式,透過自迴歸(Autoregressive,AR)模式轉換而得,因此可廣泛應用(Cario and Nelson 1998)。 常見變異數降低方法有:共同亂數法、對立變數法、重點抽樣法、控制變數法等(Law and Kelton 2000)。本研究考慮資料的三種情況:第一種情況是資料為獨立分配時;第二種情況是資料為ARTA間隔1且樣本數為1時;第三種情況是資料為ARTA間隔大於1且樣本數為大於2時。本研究使用重點抽樣法於資料為獨立分配時;使用馬可夫近似法以及random hazard法於資料為ARTA間隔1且樣本數為1時;使用控制變數法於料為ARTA間隔大於1且樣本數為大於2時。 根據模擬實驗結果發現:(1)在資料為獨立分配或ARTA間隔1且樣本數為1時使用random hazard的方法之下,對稱分配降低變異數的效果優於非對稱分配。(2)在資料為ARTA間隔1且樣本數為1時使用random hazard的方法之下,資料間在低相關性時的降低變異數效果優於高相關性。(3)在資料為ARTA間隔大於1且樣本數為大於2使用control variates的方法之下,變異數降低的效果會隨著樣本數增加而降低。

並列摘要


We propose variance reduction methods to reduce the variance for estimating the average run length in Shewart control chart. When the data { } are not independently, analytical computation of ARL may be difficult. In this case, the simulation approach is an obvious choice, but the variance of ARL is large. In the Shewhart chart, the in-control run length N for independent data is a geometric random variable with mean and variance , where is the probability that an point falls outside the control limits when the process is in control. Since is usually small, the resulting ARL (= ) is large and the standard deviation of the run length is almost as big as the ARL. This proposal considers variance reduction techniques to improve the simulation efficiency in estimating E(N) when analytical computation of E(N) is not possible. For our simulated experiments, we consider the correlated data process such as ARTA process to reduce the variance in estimating ARL. ARTA process is a correlated stationary process that takes a base AR (Autoregressive) process with a desired marginal distribution and can be applied widely (Cario and Nelson 1998). General variance reduction techniques (VRTs) include common random numbers, antithetic variates, control variates, important sampling, etc. (Law and Kelton 2000). Three cases are considered: (i) the data are independent, (ii) the ARTA lag p=1 and the sample size n=1, and (iii) . For case (i), we propose an importance sampling VRT. For case (ii), we propose the Markov’s approximation as a numerical approximation. We also propose the control variate VRTs for case (ii) if the simulation approach is used. For case (iii) we use the AR(1) ARL estimator as the control variate for variance reduction. According to our experiment results, we find that: (1) the effect of variance reduction in symmetric distribution is better than in non- symmetric distribution when the data process is independent and using the random-hazard VRT to correlation data. (2) The effect of variance reduction in low correlation is better than in high correlation using the random-hazard VRT to correlation data. (3) The effect of variance reduction is decreasing when the sample size is increasing using control-variates VRT to correlation data.

參考文獻


Cario, M.C. and B.L. Nelson (1998). Numerical Methods for Fitting and Simulating Autoregressive-To-Anything Processes. INFORMS Journal on Computing 10, 72–81.
Schemes Using Variance Reduction Techniques. Commun. Statist.—Simula. 22, 877–887.
Lucas, J.M. (1973). A Modified “V” Mask Control Scheme. Technometrics 15, 833–847.
Lucas, J.M. and M.S. Saccucci (1990). Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements. Technometrics 32, 1–12.
Page, E.S. (1954). Continuous Inspection Schemes. Biometrika 41, 100–114.

延伸閱讀