=2), the difference of the ARL between the nonnormal and estimation case with the normal and known variance case decreases as m increases. (6) For the simultaneous effect of autocorrelation, nonnormality, and estimation: we conclude that the three-dimensional effect is nonmonotic on the mean and standard deviation of run length. For Subproblem 2, we propose an algorithm to compute values of the sample size n and control-limit factor k that minimize the out-of-control ARL while keeping the in-control ARL at a specified value. Simulation experiments are run to compare the proposed Xbar chart design with the EWMAST (Exponentially Weighted Moving Average STationary), SCC (Special Cause Control), and ARMAST (Auto-Regressive Moving-Average STationary) charts. When the data process is ARMA(1, 1) (AutoRegressive of order 1 and Moving Average of order 1) process, the ARMAST chart outperforms the proposed Xbar chart, EWMA, and SCC because the ARMAST chart is designed based on the ARMA(1, 1) process. When the data process is autocorrelated with nonnormal distribution, the proposed Xbar chart often performs better than the ARMAST chart. For Subproblem 3, we propose two distribution-free methods (denoted as Methods 1 and 2) to compute the sample size n and control-limit factor k that minimize the out-of-control ARL while keeping the in-control ARL at a specified value. Methods 1 and 2 are based on the assumptions that sample means are independently and normally distributed and that sample means follow an AR(1) (AutoRegressive of order 1) process, respectively. We further modify Methods 1 and 2 by setting the sample size to be at least 30 to meet the central limit theorem. The modified Method 2 outperforms the modified Method 1. We compare the modified Method 2 with R&W (Runger and Willemain) and DFTC (Distribution-Free Tabular CuSum) charts. The R&W and DFTC charts perform better when the autocorrelation is small to moderate and the shift is small or large. Our modified Method 2 performs better when the autocorrelation is high and the shift is moderate to large. For Subproblem 4, we propose a model-free Xbar chart design for unknown autocorrelated data processes, when M observations of in-control quality characteristic measurements are given. The standard deviation of Xbar is estimated by the NBM (nonoverlapping batch means) method. The Xbar chart design parameters n and k and the NBM batch size omega are determined to minimize the out-of-control ARL while keeping the in-control ARL at a specified value. During the search of n, k, and omega, the ARL values are computed assuming that the sample means and batch means are independently and normally distributed. The computed optimal values of n and omega are further modified to be at least 30 for meeting the central limit theorem. Simulation experiments are run to evaluate the performance of the proposed model-free Xbar chart using AR(1) processes and ARTA(1) (AutoRegressive To Anything of order 1) processes with t and exponential marginal distributions as testing examples. The experimental results show that our model-free method performs well in general; it performs better when the autocorrelation is small or when the autocorrelation is large but the shift is small or moderate. The proposed model-free Xbar chart performs worse when the data process is ARTA(1) with an exponential marginal distribution.' > Airiti Library
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  1. [1] Adams, B. M., Woodal , W. H., and C. R. Superville (1994). Discussion on Run-Length Distributions of Special-Cause Control Charts for Correlated Processes. Technometrics 36, 19-22.
  2. [2] Adams, B.M. and L-T. Tseng (1998). Robustness of forecast-based monitoring scheme. Journal of Quality Technology 30, 328-339.
  3. [3] Alwan, L.C. and H.V. Roberts (1988). Time-series modeling for statistical process control. Journal of Business and Economic Statistics6, 87-95.
  4. [5] Borror, C.M., Montgomery, D.C., Runger, G.C. (1999). Robustness of the EWMA Control Chart to Non-normality. Journal of Quality Technology 31, 309-316.
  5. [6] Box, G.E.P., G.M. Jenkins, and G.C. Reinsel (1994). Time Series Analysis: Forecasting and Control, 3rd ed. New Jersey: Prentice-Hall, Inc.
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