Monitoring the mean with locally weighted averages for skewed processes : this dissertation is submitted for the degree of Master of Philosophy in Statistics, School of Mathematics and Computational Sciences, Massey University, New Zealand. EMBARGOED until 1 January 2023.
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Date
2022
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Massey University
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Abstract
Averaging functions are used in many research areas such as decision making, image processing, pattern recognition and statistics. The basic averaging function, arithmetic mean, is most widely used in statistical quality control to monitor a particular quality characteristic. However, other averaging functions such as weighted averages can be used in control charting to improve the probability of detection in process level shifts when a process distribution deviates from the normality assumption. This study focused on applying locally weighted averages as the control statistic in quality control charts to detect the process mean of a right-skewed process. Six weights were defined: Max-weight – based on the maximum distance; PDF-weight – based on the probability density function of the process; CoPDF-weight – based on the complement of the probability density function of the process; CDF-weight – based on the cumulative probability density function of the process; CoCDF-weight – based on the complement of the cumulative density function of the process; and Haz-weight – based on the hazard function of the process. Weighted average control charts; 𝑋̃𝑚𝑎𝑥, 𝑋̃ 𝑝𝑑𝑓, 𝑋̃ 1−𝑝𝑑𝑓, 𝑋̃ 𝑐𝑑𝑓, 𝑋̃ 1−𝑐𝑑𝑓, and 𝑋̃ ℎ𝑎𝑧 were proposed to monitor the process mean using the weighted averages based on Max-weight, PDF-weight, CoPDF-weight, CDF-weight, CoCDF-weight, and Haz-weight, respectively as the control statistic. First, the behaviour of these control statistics was explored for symmetric distributions using the standard normal distribution. Second, the performance of these control charts was compared to Shewhart 𝑋̅ control chart for right-skewed distributions using the average run length (𝐴𝑅𝐿) and the standard deviation of the run length (𝑆𝐷𝑅𝐿). Exponential and three gamma distributions were considered to illustrate positively skewed distributions in this study. Monte-Carlo simulations were used in evaluating the 𝐴𝑅𝐿s and 𝑆𝐷𝑅𝐿s and control limits for Phase II applications. Then Phase I control limits were established for all the distributions considered using bootstrapping. When the process is symmetric, 𝑋̅ control chart was suitable for monitoring the process mean as expected. On the other hand, 𝑋̃ 𝑐𝑑𝑓 and 𝑋̃ 1−𝑐𝑑𝑓 control charts were able to detect the variance of symmetric distributions. The importance of these results is that the weighted average control charts and the 𝑋̅ control chart can be plotted in the same graph facilitating to simultaneously detect the mean and the variance, this is discussed as joint monitoring in the literature. Weighted average control charts cannot monitor the process mean when the underlying distribution of the quality characteristic is identified as exponential. However, when the quality characteristic follows a gamma distribution, weighted averages outperformed the Shewhart 𝑋̅ control chart in a variety of situations. Therefore, the locally weighted averages proposed in this study are useful in monitoring the process mean for gamma-distributed data and variance of symmetric distributions.
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Embargoed until 1 January 2023