Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author. Monitoring the Mean with Locally Weighted Averages for Skewed Processes by W M P M Wickramasinghe This dissertation is submitted for the degree of Master of Philosophy in Statistics School of Mathematics and Computational Sciences Massey University New Zealand February 2022 1 ii Declaration I hereby declare that this dissertation is my own work carried out under the supervision of Dr Jonathan Godfrey and Dr K Govindaraju. I have not submitted this work in whole or in part for any other degree or qualification in this or any other university. To the best of my knowledge, this work contains nothing which is the outcome of work done in collaboration with others, except as specified in the text and Acknowledgements. Manori Wickramasinghe February 2022 iii Acknowledgement This study would have been impossible without the support of many outstanding individuals. First and foremost, I would like to express my profound gratitude to my primary supervisor, Dr Jonathan Godfrey, who gave me great support. I am very grateful for his guidance, encouragement, constant optimism and confidence in me through good and bad times. Apart from the support mentioned above, you were very friendly, making me more comfortable, and I am genuinely grateful for that. Next, I owe a big thank you to my co-supervisor, Dr K. Govindaraju, for accepting me as a student, initiating this study in the first place, and providing valuable guidance, support, and quick responses. I am immensely grateful to Prof Catherine Whitby for the endless support and encouragement throughout my journey, especially in difficult times. My heartiest thanks go to Dr Nihal, Dr Natisha, Dr Mathieu and Prof. Mark Bebbington. Thank you all for your guidance. I would like to mention Dr Liz Godfrey. Thank you very much for reading my thesis and helping me. Olivia, Helen and Gazella, thank you for your encouraging words throughout my time in Massey. Thanks to the present and past postgrad students and colleagues I had the pleasure of working with for nearly two years. I was glad to have highly supportive friends and families in my immediate vicinity. Thank you very much for your kindness and support; Dinushi, Tina, Sachi, Randi and Ishani, thank you and your families for assisting me in numerous ways to reach the finishing line on time. Bhagya, you were always there for me. Thank you very much. Finally, I would like to express my sincere gratitude to Niro, Sumudu, Dakshina and Nisansala for their generous support and hospitality to my family and myself throughout our arrival and early days in New Zealand. Thank you so much, Niro, for always encouraging, supporting, and being by my side along this journey. What a blessing it is to have friends like you. iv I would also like to thank my family for their unconditional support throughout this journey. Finally, thank you to all of my friends and colleagues who encouraged and supported me. Last but not least, I owe thanks to the very special person in my life, my husband, Mangala, for his continued and unfailing love, encouragement, support and understanding throughout this study. I will never reach this finishing line without you. You were always with me during the good and bad times of this journey and helped me to keep things in perspective. Gangamini, Damhiru and Hasareli, my little cuties, thank you for being with me in all the ups and downs in my life. Gangamini, thank you for looking after your younger siblings disturbing me while studying at home. Love you so much. Your mother (Amma) would have never achieved the finishing line without your unconditional love. Words would never say how grateful I am to you all. v 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; �̃�𝑚𝑎𝑥, �̃�𝑝𝑑𝑓 , �̃�1−𝑝𝑑𝑓 , �̃�𝑐𝑑𝑓, �̃�1−𝑐𝑑𝑓 , and �̃�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 �̅� 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. vi When the process is symmetric, �̅� control chart was suitable for monitoring the process mean as expected. On the other hand, �̃�𝑐𝑑𝑓 and �̃�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 �̅� 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 �̅� 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. vii Table of Content Declaration .................................................................................................................................... ii Acknowledgement ....................................................................................................................... iii Abstract .......................................................................................................................................... v List of Tables...................................................................................................................................x List of Figures .............................................................................................................................. xii Glossary ....................................................................................................................................... xv Chapter 1 - Introduction ................................................................................................................ 1 1.1 Background of the Study................................................................................................... 1 1.2 Research Objectives ........................................................................................................... 2 1.3 Literature Review .............................................................................................................. 2 1.3.1 Averaging Functions ................................................................................................. 2 1.3.1.1 Arithmetic Mean .............................................................................................. 2 1.3.1.2 Weighted Average ........................................................................................... 3 1.3.1.3 Geometric and Harmonic Means ................................................................... 3 1.3.1.4 Ordered Weighted Average (OWA) .............................................................. 3 1.3.1.5 Density Based Weighted Average .................................................................. 4 1.3.2 Control Charts for the Process Mean ....................................................................... 4 1.3.3 Application of Averaging Functions in Statistical Quality Control Charts .......... 8 1.4 Layout of the Chapters ...................................................................................................... 9 Chapter 2 - Methodology ............................................................................................................ 11 2.1 Introduction ..................................................................................................................... 11 2.2 Proposed Control Statistic ............................................................................................... 11 2.2.1 Maximum Distance Based Weighted (Max-weight) Average ............................. 11 2.2.2 Density Based Weighted (PDF-weight) Average .................................................. 12 2.2.3 Complement of Density Based Weighted (CoPDF-weight) Average.................. 12 2.2.4 Hazard Function Based Weighted (Haz-weight) Average .................................. 13 viii 2.2.5 Cumulative Function Based Weighted (CDF-weight) Average ............................. 13 2.2.6 Complement of Cumulative Function Based Weighted (CoCDF-weight) Average ………………………………………………………………………………………..13 2.3 Probability Distributions ................................................................................................. 14 2.3.1 Normal Distribution .................................................................................................. 14 2.3.2 Exponential Distribution ........................................................................................ 14 2.3.3 Gamma Distribution ............................................................................................... 14 2.4 Phase I and Phase II Control Chart ................................................................................ 15 2.5 Average Run Length (𝐴𝑅𝐿) ............................................................................................. 15 2.6 Monte Carlo Simulation .................................................................................................. 16 2.7 Bootstrapping ................................................................................................................... 16 2.8 Comparison of Weighted Average Control Charts with Shewhart 𝑿 Control Chart in Phase II ............................................................................................................................. 17 2.9 Implementation of Control Charts in Phase I ................................................................ 20 2.10 Summary .......................................................................................................................... 21 Chapter 3 - Results ....................................................................................................................... 22 3.1 Introduction ..................................................................................................................... 22 3.2 Symmetric Distributions ................................................................................................. 22 3.3 Positively Skewed Distributions..................................................................................... 29 3.3.1 Exponential Distribution ........................................................................................ 29 3.3.2 Gamma Distribution .............................................................................................. 30 3.4 Summary .......................................................................................................................... 62 Chapter 4 - Discussion ................................................................................................................. 64 4.1 Introduction ..................................................................................................................... 64 4.2 Weighted Averages ......................................................................................................... 64 4.2.1 Unweighted Average - 𝑿 .......................................................................................... 65 4.2.2 Maximum Distance Based Weighted Average - 𝑿𝒎𝒂𝒙 ........................................ 66 4.2.3 Probability Density Based Weighted Average - 𝑿𝒑𝒅𝒇 ........................................... 67 ix 4.2.4 Hazard Function Based Weighted Average - 𝑿𝒉𝒂𝒛 ............................................. 68 4.2.5 Cumulative Density Function Based Weighted Average - 𝑿𝒄𝒅𝒇 .......................... 68 4.2.6 Complement of Cumulative Density Function Based Weighted Average - 𝑿𝟏 − 𝒄𝒅𝒇 ………………………………………………………………………………………..69 4.3 Performance of Weighted Average Control Charts ...................................................... 70 4.3.1 Symmetric Distributions ............................................................................................ 71 4.3.2 Positively Skewed Distribution................................................................................. 71 4.4 Implementation of Control Charts for Phase I ................................................................. 74 4.4.1 Symmetric Distributions ............................................................................................ 74 4.3.1.1 Example ................................................................................................................ 74 4.4.2 Positively Skewed Distributions ............................................................................... 76 4.4.2.1 Exponential Distribution ..................................................................................... 76 4.3.2.2. Gamma Distribution ........................................................................................... 77 4.5. Detecting Shifts in Variance of the Normal Distribution ............................................... 85 4.5.1 Joint Monitoring of Mean and Variance ................................................................ 85 4.5.2 Monitoring an Increase in Variance When Mean In-control .................................. 86 4.6 Summary............................................................................................................................. 88 Chapter 5 - Conclusions and Future Study ................................................................................ 89 5.1 Introduction ........................................................................................................................ 89 5.2 General Conclusion ............................................................................................................ 89 5.2.1 Weighted Average Control Charts ........................................................................... 89 5.2.1.1 Symmetric Distributions ...................................................................................... 89 5.2.1.2 Positively skewed distribution............................................................................ 89 5.2.2 Summary of Research Objectives and Achievements ............................................. 91 5.3 Future Research Opportunities ......................................................................................... 92 References ..................................................................................................................................... 93 x List of Tables Table 3.1: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift, variance in-control - N(0,1) .................... 24 Table 3.2: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for variance increase, mean in-control - N(0,1)............................. 24 Table 3.3: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift, variance increases - N(0,1) ..................... 26 Table 3.4: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift, variance increases - N(0,1) ................ 27 Table 3.5: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - exp(1) ...................................................... 31 Table 3.6: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift - exp(1) ................................................. 31 Table 3.7: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - change in one parameter - Gam(0.5,1) . 33 Table 3.8: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift - change in one parameter - Gam(0.5,1) ……. ........................................................................................................................... 34 Table 3.9: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for mean shifts - change in both parameters - same direction - Gam(0.5,1) ................................................................................................................. 36 Table 3.10: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - change in both parameters - opposite direction - Gam(0.5,1) ............................................................................................... 38 Table 3.11: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for mean downward mean shifts - change in both parameters - opposite direction - Gam (0.5,1) .............................................................................. 40 Table 3.12: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for parameters shift, mean in-control - Gam(0.5,1) ...................... 42 Table 3.13: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - change in one parameter - Gam(1.5,2) . 44 Table 3.14: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift - change in one parameter - Gam(1.5,2)…………………………………….. .......................................................... 45 Table 3.15: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for mean shift - change in both parameters - same direction - Gam(1.5,2) ................................................................................................................. 47 Table 3.16: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - change in both parameters - opposite directions - Gam(1.5,2) ............................................................................................. 49 Table 3.17: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift - change in both parameters - opposite direction - Gam(1.5,2) ............................................................................................... 51 Table 3.18: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for parameters shift, mean in-control - Gam(1.5,2) ...................... 52 Table 3.19: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - change in one parameter - Gam(2,1) .... 55 Table 3.20: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift - change in one parameter - Gam(2,1)…………. .................................................................................................... 56 Table 3.21: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for mean shifts - change in both parameters - same direction - Gam(2,1) .................................................................................................................... 58 xi Table 3.22: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - change in both parameters - opposite directions – Gam(2,1) ............................................................................................... 59 Table 3.23: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift - change in both parameters - opposite direction - Gam(2,1).................................................................................................. 61 Table 3.24: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for parameters shift, mean in-control - Gam(2,1) ......................... 63 Table 4.1: Observed skewness of control statistic for sample size five ................................. 64 Table 4.2: Observed skewness of control statistic for sample size ten .................................. 64 Table 4.3: Control limits for Phase I - N(0,1) distributed data ............................................... 74 Table 4.4: Control limits for Phase I - exp(1) distributed data ............................................... 76 Table 4.5: Control limits for Phase I - Gam(0.5,1) distributed data ....................................... 77 Table 4.6: Control limits for Phase I - Gam(1.5,2) distributed data ....................................... 80 Table 4.7: Revised control limits for Phase I - Gam(1.5,2) distributed data .......................... 80 Table 4.8: Control limits for Phase I - Gam(2,1) distributed data .......................................... 83 Table 4.9: Revised control limits for Phase I - Gam(2,1) distributed data ............................. 83 xii List of Figures Figure 2.1: Gamma density plots ............................................................................................. 19 Figure 3.1: N(0,1) density curve with the means of control statistics ................................... 22 Figure 3.2: In-control and out-of-control pdfs for upward mean shift, variance in-control - N(0,1) ...................................................................................................................... 23 Figure 3.3: In-control and out-of-control pdfs for variance increases, mean in-control - N(0,1).. .................................................................................................................... 25 Figure 3.4: In-control and out-of-control pdfs for upward mean shift, variance increase - N(0,1) ...................................................................................................................... 28 Figure 3.5: In-control and out-of-control pdfs for downward mean shift, variance increase - N(0,1) .................................................................................................................... 28 Figure 3.6: exp(1) density curve with means of control statistics ......................................... 29 Figure 3.7: In-control and out-of-control pdfs for mean shift - exp(1) ................................. 30 Figure 3.8: Gam(0.5,1) density curve with the means of control statistics ........................... 32 Figure 3.9: In-control and out-of-control pdfs for upward mean shift - change in one parameter - Gam(0.5,1) .......................................................................................... 35 Figure 3.10: In- control and out-of-control pdfs for downward mean shift - change in one parameter – Gam(0.5,1) ......................................................................................... 35 Figure 3.11: In-control and out-of-control pdfs – change in both parameters - same direction - Gam(0.5,1) ............................................................................................................ 37 Figure 3.12: In-control and out-of-control pdfs for upward mean shift - change in both parameters - opposite direction - Gam(0.5,1) ...................................................... 39 Figure 3.13: In-control and out-of-control pdfs for downward mean shift - change in both parameters - opposite direction - Gam(0.5,1) ...................................................... 41 Figure 3.14: In-control and out-of-control pdfs for shifts in both parameters, mean in-control - Gam(0.5,1) ............................................................................................................ 41 Figure 3.15: Gam(1.5,2) density curve with means of control statistics ................................. 43 Figure 3.16: In-control and out-of-control pdfs for upward mean shift - change in one parameter - Gam (1.5,2) ......................................................................................... 46 Figure 3.17: In-control and out-of-control pdfs for downward mean shift - change in one parameter - Gam (1.5,2) ......................................................................................... 46 Figure 3.18: In-control and out-of-control pdfs - change in both parameters - same direction - Gam(1.5,2) ............................................................................................................ 48 xiii Figure 3.19: In-control and out-of-control pdfs for upward mean shift - change in both parameters - opposite direction - Gam(1.5,2) ...................................................... 48 Figure 3.20: In-control and out-of-control pdfs for downward mean shift - change in both parameters - opposite direction - Gam(1.5,2) ...................................................... 50 Figure 3.21: In-control and out-of-control pdfs for both parameters shift, mean in-control - Gam(1.5,2) .............................................................................................................. 53 Figure 3.22: Gam(2,1) density curve with means of control statistics .................................... 53 Figure 3.23: In-control and out-of-control pdfs for upward mean shift - change in one parameter - Gam(2,1) ............................................................................................. 54 Figure 3.24: In-control and out-of-control pdfs for downward mean shift - change in one parameter - Gam(2,1) ............................................................................................. 54 Figure 3.25: In-control and out-of-control pdfs - change in both parameters - same direction - Gam(2,1) ............................................................................................................... 57 Figure 3.26: In-control and out-of-control pdfs for upward mean shift - change in both parameters - opposite direction – Gam(2,1) ......................................................... 60 Figure 3.27: In-control and out-of-control pdfs for downward mean shifts - change in both parameters - opposite direction - Gam(2,1) ......................................................... 60 Figure 3.28: In-control and out-of-control pdfs for both parameters shfit, mean in-control - Gam(2,1) ................................................................................................................. 62 Figure 4.1: Histograms of empirical distributions of �̅� ......................................................... 65 Figure 4.2: Histograms of empirical distributions of �̃�𝑚𝑎𝑥.................................................... 66 Figure 4.3: Histograms of empirical distributions of �̃�𝑝𝑑𝑓 .................................................... 67 Figure 4.4: Histograms of empirical distributions of �̃�ℎ𝑎𝑧 ..................................................... 68 Figure 4.5: Histograms of empirical distributions of �̃�𝑐𝑑𝑓 ................................................... 69 Figure 4.6: Histograms of empirical distributions of �̃�(1−𝑐𝑑𝑓) .............................................. 70 Figure 4.7: Control charts for Phase I - N(0,1) distributed data ............................................ 75 Figure 4.8: �̅� control chart – example .................................................................................... 76 Figure 4.9: Control charts for Phase I - exp(1) distributed data ............................................ 78 Figure 4.10: Control charts for Phase I - Gam(0.5,1) distributed data .................................... 79 Figure 4.11: Control chart for Phase I - Gam(1.5,2) distributed data ...................................... 81 Figure 4.12: Revised control charts for Phase I - Gam(1.5,2) distributed data ....................... 82 Figure 4.13: Control charts for Phase I - Gam(2,1) distributed data ...................................... 84 Figure 4.14: Revised control charts for Phase I - Gam(2,1) distributed data .......................... 85 Figure 4.15: 𝐴𝑅𝐿 curve for variance increase, mean in-control - N(0,1) ................................. 87 xiv Figure 4.16: �̅� and �̃�𝑐𝑑𝑓 joint control chart - N(0,1) .................................................................. 87 Figure 4.17: �̅� and �̃�(1−𝑐𝑑𝑓) joint control chart - N(0,1) ............................................................ 88 xv Glossary 𝐴𝑅𝐿 Average Run Length 𝑆𝐷𝑅𝐿 Standard Deviation of Run Length SPC Statistical Process Control IN in-control OC out-of-control LCL Lower Control Limit CL Center Line UCL Upper Control Limit 𝑥𝑖𝑗 𝑗𝑡ℎ observation of the 𝑖𝑡ℎ sample 𝑛 Sample size 𝑚 Number of samples �̅� Sample average/ Unweighted average �̃�𝑚𝑎𝑥 Maximum distance based weighted average �̃�𝑝𝑑𝑓 Probability density function based weighted average �̃�1−𝑝𝑑𝑓 Complement of probability density function based weighted average �̃�ℎ𝑎𝑧 Hazard function based weighted average �̃�𝑐𝑑𝑓 Cumulative function based weighted average �̃�1−𝑐𝑑𝑓 Complement of cumulative function based weighted average 1 Chapter 1 - Introduction 1.1 Background of the Study Statistical quality control refers to applying statistical techniques to enhance product quality. In statistical quality control, a sample is drawn from an ongoing manufacturing process and examined for the interested quality characteristic because no manufacturing process produces all items of acceptable quality. Every production process has a specific stable pattern of variation. This natural variation is known as chance variation, and they are practically impossible to eliminate. The source of other variations can often be identified and therefore eliminated, and these are known as assignable variations. When the process runs without this assignable variation, it is said to be stable or in-control. The main purpose of statistical quality control is to identify whether the process is in-control or not. The determination of process stability is done via a control chart. Shewhart first introduced the �̅� control chart (Montgomery, 2020), and plenty of research has been conducted in constructing control charts for various situations. The distribution of the quality characteristic is assumed to be normal in Shewhart �̅� control chart. However, in a real-life scenario, the validity of this assumption is questionable since there are situations where the process cannot be modelled by a normal distribution. For example, the false alarm (signal indicating an out-of-control) rate will increase in a Shewhart �̅� chart as the skewness of the quality characteristic increases (Yourstone & Zimmer, 1992). The average (arithmetic mean) of the sample is used for examining the quality characteristic in the Shewhart �̅� control chart. The arithmetic mean gives equal weight to all observations in the sample. Therefore, it is inappropriate when the sample contains extreme values or individual observations with differing levels of importance. Thus, the arithmetic average is not a good representation of a skewed process. As a solution, each observation can be given a weight based on its relative importance, resulting in a weighted average. Hence a weighted average can be a better representation of a skewed process. Although the use of locally weighted averaging functions in constructing a control chart is not studied previously, this Chapter 1 - Introduction 2 research proposes their use for determining a weighted average as a control statistic in implementing a control chart for skewed processes. This implementation has to be done in two Phases. Phase I establishes the control limits, and Phase II uses these control limits in monitoring the process mean. 1.2 Research Objectives The use of a weighted average as a control statistic in implementing a control chart for skewed processes was studied by identifying the following objectives as the focus of the research. 1 Identify locally weighted averages that can be used to monitor a positively skewed process. 2 Explore the behaviour of the weighted averages in symmetric and positively skewed processes. 3 Compare and contrast the performance of weighted average control charts with the Shewhart �̅� chart. 4 Implementation of Phase I control chart for weighted averages. 5 Identify other possible uses of weighted average control charts. 1.3 Literature Review 1.3.1 Averaging Functions Averaging functions extract and aggregate information from a sample. The output of an averaging function is a single value that represents the information contained in several input values. The output value lies between the minimum and maximum inputs. Beliakov et al. (2016) discussed averaging functions and their properties. Some commonly used averaging functions are listed below. 1.3.1.1 Arithmetic Mean The most widely used averaging function is the arithmetic mean. For example, in statistical quality control charts, the arithmetic mean is used as the control statistic to Chapter 1 - Introduction 3 monitor the process mean. The arithmetic mean is defined as, ∑ 𝑥𝑖 𝑛 𝑖=1 𝑛⁄ , where 𝑥𝑖 ‘s are the observations, and n is the sample size. 1.3.1.2 Weighted Average When the inputs are given weights according to their relative contribution to the total value, it provides a weighted average of the input data. The weight has two properties. • The weight should be a value between zero and 1 (𝑤𝑖 ∈ [0,1]) • The summation of the weights should equal one (∑ 𝑤𝑖 = 1) A weighted average is defined as, ∑ 𝑤𝑖 𝑛 𝑖=1 𝑥𝑖, where 𝑥𝑖 ‘s are the observations, 𝑤𝑖 is the weight associated with 𝑥𝑖 and n is the sample size. 1.3.1.3 Geometric and Harmonic Means The geometric mean is defined as (∏ 𝑥𝑖 𝑛 𝑖=1 )1 𝑛⁄ . The harmonic mean is defined as 𝑛(∑ 1 𝑥𝑖 ⁄𝑛 𝑖=1 ) −1 . Here 𝑥𝑖 ‘s are the observations, and n is the sample size. They are widely used when discussing the rates. 1.3.1.4 Ordered Weighted Average (OWA) The ordered weighted average (OWA) is an averaging function introduced by Yager (1988). It differs from the weighted arithmetic mean in terms of weights, where OWA considers the size of the input. OWA is defined as, ∑ 𝑤𝑖 𝑛 𝑖=1 𝑥(𝑖), where 𝑤𝑖 is the weight associated with 𝑥𝑖, 𝑛 is the sample size, 𝑥(𝑖) is the 𝑖𝑡ℎ ordered observation in the sample, 𝑤𝑖 ≥ 0, and ∑ 𝑤𝑖 = 1. The weight can be chosen using the methods based on data, a measure of dispersion and weight generating functions etc. Xu and Da (2002) introduced the ordered weighted geometric (OWG) operator based on the geometric mean similar to the OWA operator. OWA has been used in various areas such as database systems, neural networks, decision making, mathematical programming and image processing. For more information, refer; Torra and Godo (2002), Chiclana et al. (2002), Bustince et al. (2011), and Farias et al. (2018). Chapter 1 - Introduction 4 1.3.1.5 Density Based Weighted Average Angelov and Yager (2013) proposed a weighted average based on the relative density of the data sample and is defined as ∑ 𝑤𝑖(𝑋)𝑛 𝑖=1 𝑥𝑖, where 𝑤𝑖(𝑋) is the density based weight. Sadiq and Tesfamariam (2007) used the probability density functions, which extended the method of creating OWA weights using the normal distribution function. The use of probability in density based weighting and OWA is discussed by Merigó (2010). A probabilistic-weighted-average (PWA) and probabilistic-ordered- weighted-averaging (POWA) is proposed by Merigo (2012a) and Merigo (2012b), respectively. Su et al. (2016) considered commonly used probability distributions for determining the weighting vector in weighted averages. Averaging is commonly utilized in a variety of applications. For example, instead of arithmetic means, averaging functions were used to compute the empirical risk by Shibzukhov (2017), Paternain et al. (2015) considered averaging functions to develop an image reduction algorithm, Lin and Jiang (2014) used weighted averages in decision making, etc. For more information, see Allen (1988), Lecomte (2014), Pogromsky and Matveev (2015), and Wilkin et al. (2014). In addition, identification of the stability of the mean of a process in statistical quality control is done using averages. However, the use of locally weighted averages in constructing control charts is minimal. Therefore, this study is focused on applying locally weighted averages to control charts. 1.3.2 Control Charts for the Process Mean Control charts are frequently used to identify the status of a process in statistical process control (SPC). These statuses are in-control (IC) and out-of-control (OC) of a process measured using two control limits: Upper control limit (UCL) and Lower control limit (LCL). In general, the Shewhart �̅� control chart is used in process monitoring by assuming the underlying process follows a normal distribution. However, process data violate the assumption of normality in many situations. Chapter 1 - Introduction 5 The effect of non-normality in constructing �̅� control chart has been studied by Burrows (1962), Schilling and Nelson (1976), Balakrishnan and Kocherlakota (1986), Yourstone and Zimmer (1992), Spedding and Rawlings (1994), Pyzde (1995), and Amhemad (2010). Tukey’s 𝜆-family of symmetric distributions was used to discuss the impact of a process departure from normality by Chan et al. (1988). These studies showed that there is an extensive influence on the performance of �̅� control chart when the process is skewed. Various techniques have been discussed to deal with the problem which arose from the non-normality of the underlying distribution. The first method is data transformation. The data is transformed to follow a normal distribution, and by using these transformed data, the control chart is then constructed. Chou et al. (1998) used a simulation study to show that the mean- squared error of the standard deviation estimate is increased when the process deviates from the normal assumption. As a solution, they discussed the suitability of the Johnson transformation of the data. The difficulty in this approach is determining a suitable technique and interpreting the results since the control chart is drawn in a different scale of measurements. A second approach is to increase the sample size. Then by applying the central limit theorem, the distribution of the sample mean is approximated to a normal distribution. However, increasing the sample size is not always easy and can be expensive. A third option is to use asymmetric control limits to monitor the process mean of a skewed process. Chen and Kuo (2007) compared the symmetric and asymmetric control limits for �̅� chart for skewed distributions. Gamma distribution was used to illustrate positively skewed distribution and Johnson unbounded distribution demonstrates the negatively skewed distribution. They found that when the process is positively skewed, the symmetric control limits show a lower average run length (𝐴𝑅𝐿) than the asymmetric control limits for large shifts in the mean. In contrast, symmetric control limits show a lower 𝐴𝑅𝐿 than the asymmetric control limits for Chapter 1 - Introduction 6 small shifts in the mean when the process is negatively skewed. Thus, choosing the control limits for monitoring a quality characteristic of a skewed process would be based on the direction of the skewness and the shift of the process. Another approach is to assume that the exact distribution of the data is known and to derive the control limits. Kantam et al. (2006) determined control charts and the corresponding control limits for the process mean and process range, where the underlying distribution is log-logistic. Control charts for the process mean, where the data comes from an exponential-gamma distribution, were discussed by Rao and Kumar (2015), while Adewara et al. (2020) examined the data from a Gompertz distribution. The limitation of this technique is that the identification of the distribution needs to be accurate. Otherwise, the decisions made by examining the control chart may be faulty. Another way of handling skewed data is to design control charts based on heuristic methods. The prevailing heuristic methods are the weighted variance (WV), the scaled weighted variance (SWV), the weighted standard deviation (WSD) and the skewness correction (SC) proposed by Bai and Choi (1995), Castagliola (2000), Chang and Bai (2001) and Chan and Cui (2003), respectively. Bai and Choi (1995) suggested the weighted variance control chart (WV- �̅�) by splitting the skewed distribution into two parts from the mean. The standard deviation is calculated for the two parts separately. The upper control limit produced by this approach is larger than the lower limit if the data is positively skewed, while it is shorter if the data is negatively skewed. For symmetric distributions, the limits are the same as the usual �̅� chart. Another method of obtaining the WV control limits was proposed by Choobineh and Ballard (1987) by splitting the skewed distribution into two parts from the mean of the process, and two symmetrical probability density functions are defined for each piece. The upper control limit (UCL) and lower control limit (LCL) are instituted using these probability density functions. Scaled weighted variance control chart (SWV-�̅�) proposed by Castagliola (2000) was an enhancement of the weighted variance control chart (WV- �̅�). Even though the data is split into two Chapter 1 - Introduction 7 functions in WV method, they are not probability density functions, and the replaced probability density functions are bell-shaped but not normally distributed. Therefore, a scaled factor is used to derive the control limits. The SWV−�̅� control chart is better in identifying downward shifts than the WV-�̅� chart and the �̅� chart. Chang and Bai (2001) introduced the weighted standard deviation (WSD) control chart. First, the standard deviation of the skewed process is decomposed into two parts at its mean to create two symmetric distributions. Then the standard deviations of these symmetric distributions have been used in constructing the control limits. Finally, the sum of these two standard deviations should be equal to the standard deviation of the original distribution. The skewness correction (SC) method proposed by Chan and Cui (2003) revised the Shewhart 3𝜎 control limits by adding a skewness adjustment factor to the control limits. The skewness correction method was extended by Mehmood et al. (2020) for unknown parameters and unknown skewed probability distributions. The skewness correction factor defined by Mehmood depends on the skewness, dispersion, sample size and the number of estimation samples. Pongpullponsak et al. (2007) compare the efficiency of the control charts for skewed distributions. They examined the WV control chart, SWV control chart, empirical quantiles method, and extreme-value theory for skewed data. The average run length (𝐴𝑅𝐿) was used to determine the performance of the control charts. A univariate control chart was introduced utilising the skewness correction method, denoted as SC−�̅�𝑆 by Zain et al. (2020). They demonstrated the SC−�̅�𝑆 control chart performs well for the non-normal data by using a simulation study and a health care example. Although SC control charts perform better than the other heuristic control charts in detecting out-of-control signals for skewed distributions, the WV−�̅� charts are easier to implement than the SC control charts. The synthetic control chart outperforms the Shewhart �̅� control chart when the underlying distribution is normal. Khoo et al. (2008) presented a new synthetic control chart for monitoring shifts in the process mean of skewed populations using Chapter 1 - Introduction 8 the WV technique. Positively skewed distributions; weibull, lognormal and gamma are used with the normal distribution to compare the performance of this new control chart with the existing control charts using the Monte Carlo simulations. Although the WSD−�̅� control chart shows less 𝐴𝑅𝐿 than the WV−�̅� control chart, the synthetic WSD−�̅� control chart gives high 𝐴𝑅𝐿 than the synthetic WV−�̅� control chart. It was extended to a synthetic scaled weighted variance (synthetic SWV) control chart by Castagliola and Khoo (2009) and demonstrated synthetic SWV−�̅� control chart was superior to the synthetic WV−�̅� control chart for detecting the downward shifts in the mean and the results were robust with the degree of the skewness. Lin and Chou (2007) discussed the performance of variable parameter (VP) �̅� charts when the data is not normally distributed. The gamma and t distributions were considered to illustrate that the VP−�̅� control chart was more effective in detecting small mean shifts regardless of the nature of the distribution. 1.3.3 Application of Averaging Functions in Statistical Quality Control Charts Using averaging functions in control charts have been found in the literature. For example, a distance-based weight was used in constructing a control chart when the data contains outliers by Kao (2016). He used the dispersion of data represented by the distance between the data and the mean, where the shorter distances were assigned a smaller weight. In contrast, outliers with a considerable distance were given more significant weight. A weighted average control chart and an inter-quartile control chart monitor the mean and the range, respectively. The average and range of a subgroup were replaced from a trimmed mean and the trimmed mean of the subgroup ranges, respectively, to implement a new control chart named the trimmed �̅� control chart by Langenberg and Iglewicz (1986). The control limits were found by using an averaging function instead of the subgroup mean used in �̅� control chart. Schoonhoven and Does (2010) studied non-normality in �̅� control chart and established new control limits based on a pooled sample Chapter 1 - Introduction 9 standard deviation and Gini’s mean sample differences. Further, Karagöz (2018) proposed a modified Shewhart (MS), a modified weighted variance (MWV), and a modified skewness correction (MSC) method to construct control limits by replacing the overall mean by a trimmed mean. The extensive literature review found that locally weighted averages were largely underused and were not assessed properly in the context of constructing control charts. Therefore, this study focused on identifying a locally weighted average as the control statistic to implement a control chart for positively skewed underlying distribution. 1.4 Layout of the Chapters Chapter 1: Introduction Chapter 1 includes a background of the study followed by the objectives. A comprehensive review of the use of control charts for process mean and averaging functions is presented, and then use of averaging functions in control charts for the skewed process is discussed. Chapter 2: Methodology Chapter 2 defines the locally weighted averages that can be used as control statistic and describes the Phase I and Phase II applications of statistical control charts and the tools used to implement the control charts: Monte-Carlo simulation and Bootstrapping. In addition, the methodologies for finding the control limits for Phase I application and obtaining the average run lengths (𝐴𝑅𝐿) and standard deviation of run lengths (𝑆𝐷𝑅𝐿) are detailed. Chapter 3: Results The results obtained in Phase II monitoring (simulation results of identifying the 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 of control charts constructed) with appropriate descriptive analysis are included in Chapter 3. Chapter 1 - Introduction 10 Chapter 4: Discussion A detailed discussion on the results in Chapter 3 and the construction of Phase I control charts are discussed in Chapter 4. Moreover, the distribution and the behaviour of the weighted averages are analysed. The detection of a shift in variance of a normal distribution is also included. Chapter 5: Conclusions and Future Study The major findings of the study will be summarised in Chapter 5. In addition, a summary of research objectives, achievements, and limitations are included with the future research directions. 11 Chapter 2 - Methodology 2.1 Introduction This chapter includes the definitions of proposed weighted averages and the probability distributions used in this study, including the methodologies for comparing each weighted average control chart against the state of the art, Shewhart �̅� control chart and for constructing control limits. Further, the Monte-Carlo simulation and the bootstrapping methodologies are discussed. 2.2 Proposed Control Statistic Six weighted averages are proposed as new control statistic to detect the shifts in mean and variance from the in-control distribution where the underlying distribution is symmetric or positively skewed. Let 𝑋𝑖𝑗 denote the 𝑗𝑡ℎ observation of the 𝑖𝑡ℎ sample of a particular quality characteristic where 𝑖 = 1,2, … , 𝑚 , 𝑗 = 1,2, … , 𝑛 , 𝑚 is the number of samples, and 𝑛 is the sample size. The unweighted average is defined as, �̅� = ∑ ∑ 𝑋𝑖𝑗 𝑛 𝑗 𝑚 𝑖 𝑛 × 𝑚 Unweighted average is used as the default control statistic in Shewhart �̅� control chart. The weights proposed in this study explored the possible scenario of assigning a value to the data according to its relative importance. Six possibilities were considered as follows. 2.2.1 Maximum Distance Based Weighted (Max-weight) Average The weight based on the maximum distance is referred to as Max-weight and denoted as 𝑊𝑚𝑎𝑥 . The Max-weight was defined using the distance from the maximum value in the sample to the observation as, 𝑊𝑗(𝑚𝑎𝑥) = |𝑋𝑖(𝑛) − 𝑋𝑖𝑗| ∑ |𝑋𝑖(𝑛) − 𝑋𝑖𝑗|∀𝑗 , Chapter 2 - Methodology 12 where 𝑋𝑖(𝑛) is the maximum observation in the 𝑖𝑡ℎ sample. A weighted average �̃�𝑚𝑎𝑥 is defined as �̃�𝑚𝑎𝑥 = ∑ 𝑊𝑗(𝑚𝑎𝑥) × 𝑋𝑖𝑗 𝑛 𝑗=1 . �̃�𝑚𝑎𝑥 is proposed as a control statistic in this study, and the control chart designed using the statistic �̃�𝑚𝑎𝑥 is denoted as the �̃�𝑚𝑎𝑥 control chart. 2.2.2 Density Based Weighted (PDF-weight) Average The probability density function of the underlying distribution was considered to define a novel weight referred to as PDF-weight and denoted as 𝑊𝑝𝑑𝑓 . The probability density at the observation 𝑋𝑖𝑗 denoted by 𝑓(𝑋𝑖𝑗) is observed and 𝑊𝑝𝑑𝑓 is defined as 𝑊𝑗(𝑝𝑑𝑓) = 𝑓(𝑋𝑖𝑗) ∑ 𝑓(𝑋𝑖𝑗)∀𝑗 . A weighted average �̃�𝑝𝑑𝑓 is defined as �̃�𝑝𝑑𝑓 = ∑ 𝑊𝑗(𝑝𝑑𝑓) × 𝑋𝑖𝑗 𝑛 𝑗=1 . �̃�𝑝𝑑𝑓 is proposed as a control statistic in this study, and the control chart designed using the statistic �̃�𝑝𝑑𝑓 is denoted as the �̃�𝑝𝑑𝑓 control chart. 2.2.3 Complement of Density Based Weighted (CoPDF-weight) Average The complement of the probability density function of the underlying distribution was considered to define a weight referred to as CoPDF-weight and denoted as 𝑊1−𝑝𝑑𝑓 . The complement of the probability density at the observation 𝑋𝑖𝑗 is observed and 𝑊1−𝑝𝑑𝑓 and is defined as, 𝑊𝑗(1−𝑝𝑑𝑓) = 1 − 𝑓(𝑋𝑖𝑗) ∑ (1 − 𝑓(𝑋𝑖𝑗))∀𝑗 . A weighted average �̃�1−𝑝𝑑𝑓 is defined as �̃�1−𝑝𝑑𝑓 = ∑ 𝑊𝑗(1−𝑝𝑑𝑓) × 𝑋𝑖𝑗 𝑛 𝑗=1 . �̃�1−𝑝𝑑𝑓 is proposed as a control statistic in this study, and the control chart designed using the statistic �̃�1−𝑝𝑑𝑓 is denoted as the �̃�1−𝑝𝑑𝑓 control chart. Chapter 2 - Methodology 13 2.2.4 Hazard Function Based Weighted (Haz-weight) Average The weight based on the hazard function (ℎ𝑖𝑗) is referred to as Haz-weight and denoted as 𝑊ℎ𝑎𝑧 . The hazard function is defined as ℎ𝑖𝑗 = 𝑓(𝑋𝑖𝑗) 1 − 𝑃(𝑋 ≤ 𝑋𝑖𝑗) , and the Haz-weight is defined as 𝑊𝑗(ℎ𝑎𝑧) = ℎ𝑖𝑗 ∑ ℎ𝑖𝑗∀𝑗 . A weighted average �̃�ℎ𝑎𝑧 is defined as �̃�ℎ𝑎𝑧 = ∑ 𝑊𝑗(ℎ𝑎𝑧) × 𝑋𝑖𝑗 𝑛 𝑗=1 . �̃�ℎ𝑎𝑧 is proposed as a control statistic in this study, and the control chart designed using the statistic �̃�ℎ𝑎𝑧 is denoted as the �̃�ℎ𝑎𝑧 control chart. 2.2.5 Cumulative Function Based Weighted (CDF-weight) Average The CDF-weight is defined using the cumulative probability of the underlying distribution and denoted as 𝑊𝑐𝑑𝑓 defined as 𝑊𝑗(𝑐𝑑𝑓) = 𝑃(𝑋 ≤ 𝑋𝑖𝑗) ∑ 𝑃(𝑋 ≤ 𝑋𝑖𝑗)∀𝑗 . A weighted average �̃�𝑐𝑑𝑓 is defined as �̃�𝑐𝑑𝑓 = ∑ 𝑊𝑗(𝑐𝑑𝑓) × 𝑋𝑖𝑗 𝑛 𝑗=1 . �̃�𝑐𝑑𝑓 is proposed as a control statistic in this study, and the control chart designed using the statistic �̃�𝑐𝑑𝑓 is denoted as the �̃�𝑐𝑑𝑓 control chart. 2.2.6 Complement of Cumulative Function Based Weighted (CoCDF- weight) Average The weight based on the complement of the cumulative probability function is referred to as CoCDF-weight and denoted as 𝑊(1−𝑐𝑑𝑓) and is defined as 𝑊𝑗(1−𝑐𝑑𝑓) = 1 − 𝑃(𝑋 ≤ 𝑋𝑖𝑗) ∑ (1 − 𝑃(𝑋 ≤ 𝑋𝑖𝑗))∀𝑗 Chapter 2 - Methodology 14 A weighted average �̃�(1−𝑐𝑑𝑓) is defined as �̃�(1−𝑐𝑑𝑓) = ∑ 𝑊𝑗(1−𝑐𝑑𝑓) × 𝑋𝑖𝑗 𝑛 𝑗=1 . �̃�(1−𝑐𝑑𝑓) is proposed as a control statistic in this study, and the control chart designed using the statistic �̃�(1−𝑐𝑑𝑓) is denoted as �̃�(1−𝑐𝑑𝑓) control chart. 2.3 Probability Distributions 2.3.1 Normal Distribution The normal distribution with mean 𝜇 and standard deviation 𝜎 denoted 𝑁(𝜇, 𝜎2) is defined as 𝑓(𝑥) = 1 𝜎√2𝜋 𝑒− 1 2 ( 𝑥−𝜇 𝜎 ) 2 ; for − ∞ < 𝑥 < ∞ , −∞ < 𝜇 < ∞ and 𝜎 > 0. When 𝜇 = 0 and 𝜎2 = 1, the normal distribution is referred to as a standard normal distribution. The sample average �̅� of a normal random variable follows a normal distribution with the mean 𝜇 and standard deviation 𝜎 √𝑛⁄ , where 𝑛 is the sample size. 2.3.2 Exponential Distribution An exponential random variable 𝑋 with rate 𝜆 is denoted as 𝑋~exp (𝜆). The probability density function is given by, 𝑓(𝑥) = 𝜆𝑒−𝜆𝑥 ; 𝑥 ≥ 0. The mean and the variance of the exponential distribution is 1 𝜆⁄ and 1 𝜆2⁄ , respectively. For random samples derived from an exponential distribution, the sample average �̅� follows a gamma distribution with shape parameter 𝑛 and the scale parameter 1 𝑛𝜆⁄ where 𝑛 is the sample size (Larsen & Marx, 2005). 2.3.3 Gamma Distribution The gamma distribution with shape parameter 𝛼 and scale parameter 𝛽 is defined as, 𝑓(𝑥) = 1 𝛽𝛼Γ(𝛼) 𝑥𝛼−1𝑒 − 𝑥 𝛽 ; 𝑥 > 0 and 𝛼, 𝛽 > 0 Chapter 2 - Methodology 15 and denoted as 𝐺𝑎𝑚(𝛼, 𝛽). The mean and the variance of the gamma distribution is 𝛼𝛽 and 𝛼𝛽2, respectively. When 𝛼 = 1, the gamma distribution becomes an exponential distribution with mean 1 𝛽⁄ . When the sampling distribution of 𝑋 follows a gamma distribution, then �̅� also follows a gamma distribution with shape and scale parameters 𝑛𝛼 and 𝛽 𝑛⁄ , respectively (Larsen & Marx, 2005). 2.4 Phase I and Phase II Control Chart In practice, the structure of the statistical process control consists of two phases, Phase I is known as the retrospective phase, and Phase II is known as the prospective or monitoring phase. The aim of Phase I is to understand the process and determine the stability of the process. In Phase I, the data is evaluated to verify no assignable causes of variations and determine the process control limits. The control limits specified in Phase I are trial control limits, and these control limits are modified until there are no assignable causes of variations present. After these iterative steps, the unknown parameters are estimated. The Phase II control limits can be determined using the estimations in Phase I and used in monitoring the process. When the observed control statistic lies within the control limits, the process is said to be in control, and if the control statistic goes beyond the control limits, the process is said to be out of control. Phase I is crucial because the process monitoring (Phase II) efficacy depends on Phase I estimates. (Montgomery, 2020) 2.5 Average Run Length (𝑨𝑹𝑳) Average run length (𝐴𝑅𝐿) is the average number of samples before an out-of-control signal is exhibited. The 𝐴𝑅𝐿 is defined as 1 𝑃⁄ , where 𝑃 is the probability of a point exceeding the control limits. The in control 𝐴𝑅𝐿, denoted by 𝐴𝑅𝐿0 is longer and the out-of-control 𝐴𝑅𝐿 (𝐴𝑅𝐿1) is shorter. When a process follows a normal distribution, it gives 𝑃 = 0.0027 for an in control �̅� chart. Therefore, the 𝐴𝑅𝐿0 for a �̅� chart with three-sigma limits is 370. 𝐴𝑅𝐿 can be used to evaluate the performance of control charts (Aroian & Levene, 1950). Control charts have the same false alarm rate when their in-control 𝐴𝑅𝐿 are equal. 𝐴𝑅𝐿1 of two control charts with the same 𝐴𝑅𝐿0 can be Chapter 2 - Methodology 16 directly compared and the control chart with less 𝐴𝑅𝐿1 value outperforms the other control chart. 2.6 Monte Carlo Simulation Monte Carlo simulation is a mathematical technique that is used in modelling by generating random samples. A random sample is generated from the probability distribution to obtain the parameters of interest. Then the simulation is conducted using those parameters. Kalos and Whitlock (2009) summarized the steps in Monte Carlo simulation as, 1. Define possible inputs and identify the statistical probability distribution. 2. Generate possible inputs through random sampling from the probability distribution over the domain. 3. Perform simulation with these input parameters. 4. Aggregate and analyze the output results statistically. Monte-Carlo simulation assumes that the distribution is known, and therefore, it was used to find the run length distribution of the control statistic in Phase II in this study. 2.7 Bootstrapping Efron and Lepage (1992) discussed the bootstrap method. The main concept of bootstrapping is generating samples from the original data with replacement. It is a powerful method of making inferences about a statistic when the distribution of the statistic is unknown. The steps in bootstrapping are, 1. Draw a random sample of size 𝑛. 2. Calculate the statistic of interest. 3. Generate 𝐵 random samples of size 𝑛 from the original data, known as the bootstrap sample. 4. Construct the sampling distribution of the statistic of interest from the bootstrap sample. Chapter 2 - Methodology 17 In practice, the distribution of the data is unknown. Therefore, the distribution of the observations is identified, and parameters are estimated. However, the distributions of the weighted averages are also unknown. Hence, bootstrapping is used in constructing the control limits for Phase I in this study. 2.8 Comparison of Weighted Average Control Charts with Shewhart �̅� Control Chart in Phase II The performance of the Shewhart �̅� chart and the weighed average control charts (�̃� charts) were compared in terms of their 𝐴𝑅𝐿s. A control chart is superior when it has the lowest 𝐴𝑅𝐿1 because it detects process changes quickly. 𝐴𝑅𝐿s of the weighted average control charts and the �̅� control chart were estimated using the Monte Carlo simulation approach. The simulation consists of two stages. In the first simulation stage, the in-control control limits were found for known parameters, and the 𝐴𝑅𝐿 for the control charts were obtained by simulating the run length distribution in the second stage. The steps of each stage are described below. Stage I: Determined the control limits Step 1: Generated a random sample of size 𝑛 (5 and 10) from the in-control process. Step2: Calculated the unweighted average �̅� and weighted averages, �̃�𝑚𝑎𝑥, �̃�𝑝𝑑𝑓, �̃�1−𝑝𝑑𝑓 , �̃�𝑐𝑑𝑓 , �̃�(1−𝑐𝑑𝑓) and �̃�ℎ𝑎𝑧. Step 3: For each sample size, Steps 1 and 2 were repeated for 𝑁 = 1, 000, 000 times to obtain the empirical distribution of the control statistic (�̅� and �̃�s). Step 4: Found the 𝛼𝑡ℎ and (1 − 𝛼)𝑡ℎ percentiles of the empirical distributions of �̅� and �̃�s to obtain the control limits of the control statistic (𝛼 = 0.01). Stage 2: Obtained the 𝑨𝑹𝑳 and 𝑺𝑫𝑹𝑳 Step 1: Generated a random sample of size 𝑛 (5 and 10) from the particular distribution and obtained the control statistics; �̅� and �̃�s. Chapter 2 - Methodology 18 Step 2: Repeated Step 1 until a false alarm occurred and counted the number of occurrences for the control statistic, which refers to the run length of the control statistic. Step 3: Repeated Step 2 for 𝑁 (20000) times to obtain the run length distributions. Step 4: Calculated the mean and the standard deviation of the run-length distributions, which refers to 𝐴𝑅𝐿0 and 𝑆𝐷𝑅𝐿0 of the control chart. Step 5: Changed the parameter values and repeated Steps 1 to 3 and calculated the mean and standard deviation of the run-length distributions, which refers to 𝐴𝑅𝐿1 and 𝑆𝐷𝑅𝐿1 for several out-of-control states. The standard normal distribution was considered to examine the behaviour of the weighted averages when the underlying distribution is symmetric. The Average run length (𝐴𝑅𝐿) and the standard deviation of the run length (𝑆𝐷𝑅𝐿) were found for all weighted averages and the unweighted average for several shifts. Shifts occurred by an increase in the mean while keeping variance in-control, an increase in the variance while keeping mean in-control, then increasing the mean and the variance simultaneously and finally increasing the variance while the mean decreased was considered. Exponential and gamma distributions were used to discuss the performance of the weighted averages when the underlying distribution is positively skewed. The standard exponential distribution, 𝑒𝑥𝑝(1), was considered in this study. The weighted averages, �̃�𝑚𝑎𝑥 , �̃�𝑝𝑑𝑓, �̃�(1−𝑝𝑑𝑓), �̃�𝑐𝑑𝑓 and �̃�(1−𝑐𝑑𝑓) were discussed as the control statistic and �̃�ℎ𝑎𝑧 was not considered because the hazard function is a constant (𝜆). So �̃�ℎ𝑎𝑧 is the same as �̅�. The 𝐴𝑅𝐿 and the 𝑆𝐷𝑅𝐿 were determined for shifts occurring in the mean in order to discuss the performance of the weighted average control charts. The shape of the gamma distribution is based on the selection of the parameters. This study considered three gamma distributions: a shape parameter less than one, a shape parameter greater than one and less than the scale parameter and a shape parameter Chapter 2 - Methodology 19 greater than one and the scale parameter. Figure 2.1 shows the shapes of the densities used in this study. Figure 2.1: Gamma density plots The mean of the gamma distribution depends on both parameters. A change in the mean can occur in three ways: a change in any one parameter or both parameters, which may cause an out-of-control state in the process. Four ways considered in this study were: 1. The mean was shifted by changing only the shape parameter. 2. The mean was shifted by changing only the scale parameter. 3. The mean was shifted by changing both parameters (both parameters either were increased or decreased, one parameter was increased while the other parameter was decreased) 4. The mean was in-control, but the parameters were shifted (which shifts the variance of the process from in-control to out-of-control). A shift in the mean affects the 𝐴𝑅𝐿 of the control chart. The performance of the weighted average control charts will be discussed in terms of the 𝐴𝑅𝐿 of the particular control chart. The in-control 𝐴𝑅𝐿 was used as 100 (𝛼 = 0.01) in the simulations of this study. The control limits for the different control statistic were found using the steps Chapter 2 - Methodology 20 highlighted in Stage 1 to have the 𝐴𝑅𝐿0 approximately 100. Then, the 𝐴𝑅𝐿 of the control charts were found using the steps stated in Stage 2 for several shifts in the parameters, making the in-control mean into out-of-control. The 𝐴𝑅𝐿1 from each weighted average control chart was compared with the 𝐴𝑅𝐿1 of Shewhart �̅� control chart to determine the performance of the weighted average control charts. 2.9 Implementation of Control Charts in Phase I Control limits established for Phase I is repetitive. First, data was randomly generated. Then the distribution of the observations was identified, and the parameters were estimated. However, the distribution of the weighted averages is often unknown. Therefore, the bootstrap technique was used in this study to find the control limits in Phase I. The steps for constructing control limits were as follows: Step 1: Considered 𝑚 (25) subgroups of size 𝑛 (5 or 10) generated from the known distribution. Step 2: Obtained a random sample of size 𝑛 with replacement from the data generated in Step 1. Step 3: Calculated the unweighted and weighted averages from the bootstrap sample drawn in Step 2. Step 4: Repeated Steps 2 and 3 for 𝐵 (100,000) number of bootstrap samples to obtain the empirical distribution of the averages. Step 5: Found the 𝛼𝑡ℎ, 0.5𝑡ℎ and (1 − 𝛼)𝑡ℎ percentiles of the empirical distributions as the control limits (𝛼 = 0.0027). Using the bootstrap control limits, the �̅� control chart and weighted average control charts were drawn to identify whether the process is in control or not. If the process was identified as out of control, the control limits need to be revised. Otherwise, the control limits were used in Phase II for monitoring the process. Chapter 2 - Methodology 21 2.10 Summary The weighted averages proposed in section 2.2 were considered as the control statistic to construct the control charts in this study. The performance of the weighted average control charts was compared to the Shewhart �̅� control chart using 𝐴𝑅𝐿. The 𝐴𝑅𝐿s obtained, and relative descriptive analysis are presented in Chapter 3. 22 Chapter 3 - Results 3.1 Introduction This chapter presents the tables of 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for the control charts discussed in this study for different situations with their relative descriptive analyses. Also, the performance of the proposed weighted average control charts is compared to the existing Shewhart �̅� control chart. Standard normal, exponential and gamma distributions were used as the underlying distributions, and the parameters are assumed to be known. Since the 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 are approximately in the same order, the conclusions made on the performance of the control charts based on 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 are also the same. 3.2 Symmetric Distributions The standard normal distribution was considered for illustrating the performance of weighted average control charts when the underlying distribution is symmetric. Figure 3.1 shows the 𝑁(0,1) density curve with the mean of the control statistic calculated from a randomly generated empirical distribution for subgroup size ten. The mean of the control statistic based on CDF-weight and Haz-weight were positioned to the right, while Max-weight and CoCDF-weight were positioned to the left of the unweighted average. The weighted average based on the PDF-weight overlapped with the unweighted average for symmetric distributions. Figure 3.1: N(0,1) density curve with the means of control statistics Chapter 3 – Results 23 The Shewhart 𝑋 ̅ control chart gave less 𝐴𝑅𝐿1 compared to the weighted average control charts discussed in this study in detecting the mean shifts. The corresponding 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 are given in Table 3.1. The upward and downward mean shifts reacted similarly because of the symmetry. Figure 3.2 shows the in-control and out-of-control probability density functions for upward shifts of the mean. The density functions shifted to the right as the mean increased from its in-control value and shifted to the left for the mean decreased from its in-control value. Figure 3.2: In-control and out-of-control pdfs for upward mean shift, variance in-control - N(0,1) The 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for variance increases, while the mean is in-control, are given in Table 3.2. The weighted average control charts �̃�𝑚𝑎𝑥, �̃�ℎ𝑎𝑧, �̃�𝑐𝑑𝑓 and �̃�(1−𝑐𝑑𝑓) showed less 𝐴𝑅𝐿1 compared to the 𝑋 ̅ control chart in detecting an increase in the variance when the mean is in its in-control state. The smallest 𝐴𝑅𝐿1 was given by �̃�(1−𝑐𝑑𝑓) and �̃�𝑐𝑑𝑓 control charts. Figure 3.3 demonstrates the in-control and out-of-control pdf for variance increase, and it shows that the pdfs were flattened as the variance increases. Chapter 3 – Results 24 Table 3.1: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift, variance in-control - N(0,1) Table 3.2: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for variance increase, mean in-control - N(0,1) Sample size Mean Standard Deviation �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�ℎ𝑎𝑧 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 n=5 0.00 1 100.67 100.43 99.64 98.57 100.45 100.02 100.62 100.16 100.18 99.86 100.02 98.9 0.20 1 56.60 56.48 58.20 57.61 66.33 65.85 88.12 87.57 78.19 77.83 60.98 60.29 0.50 1 13.88 13.39 16.60 16.06 20.62 20.1 35.36 34.86 26.20 25.63 19.67 19.24 1.00 1 2.73 2.18 3.49 2.95 4.60 4.08 7.03 6.48 4.86 4.33 4.61 4.06 1.50 1 1.28 0.60 1.53 0.9 1.96 1.37 2.23 1.67 1.72 1.12 2.00 1.42 n=10 0.00 1 100.68 100.40 98.08 97.48 99.7 99.08 100.17 99.66 100.27 99.28 100.05 99.21 0.20 1 37.17 36.70 41.32 40.82 47.90 47.56 70.18 69.43 57.87 57.22 48.38 47.71 0.50 1 6.26 5.73 7.90 7.42 10.17 9.67 18.03 17.5 12.39 11.90 11.27 10.76 1.00 1 1.39 0.73 1.68 1.07 2.13 1.55 2.85 2.31 2.08 1.49 2.56 2.00 1.50 1 1.02 0.13 1.06 0.26 1.20 0.49 1.2 0.5 1.09 0.31 1.38 0.72 Sample size Mean Standard Deviation Variance �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�ℎ𝑎𝑧 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 n=5 0.00 1.0 1.00 100.67 100.43 99.64 98.57 100.45 100.02 100.62 100.16 100.18 99.86 100.02 98.9 0.00 1.2 1.44 31.46 31.04 29.04 28.45 50.01 49.6 26.21 25.77 26.08 25.67 26.31 25.67 0.00 1.5 2.25 11.71 11.16 8.72 8.21 24.42 24.03 7.37 6.85 7.22 6.70 7.27 6.76 0.00 2.0 4.00 5.06 4.55 3.32 2.78 11.46 10.90 2.77 2.21 2.71 2.16 2.73 2.17 n=10 0.00 1.0 1.00 100.68 100.40 98.08 97.48 99.7 99.08 100.17 99.66 100.27 99.28 100.05 99.21 0.00 1.2 1.44 31.23 30.80 32.02 31.53 51.71 51.5 20.15 19.67 21.00 20.55 20.81 20.32 0.00 1.5 2.25 11.68 11.18 10.57 9.96 27.33 26.79 4.48 3.97 4.69 4.19 4.70 4.17 0.00 2.0 4.00 5.04 4.52 4.12 3.57 14.00 13.49 1.76 1.16 1.79 1.19 1.79 1.19 Chapter 3 - Results 25 Figure 3.3: In-control and out-of-control pdfs for variance increases, mean in-control - N(0,1) Table 3.3 provides the 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for simultaneous mean and variance increase. The �̃�𝑐𝑑𝑓 chart showed less 𝐴𝑅𝐿1 than the proposed weighted average control charts in this study and the �̅� control chart for detecting the shifts that increase the mean and the variance. When the increase in the variance is small compared to the increase in the mean, the �̅� control chart provided a lower 𝐴𝑅𝐿 than the �̃�𝑐𝑑𝑓 control chart. However, when the sample size increases, both �̃�𝑐𝑑𝑓 and �̅� charts led to approximately equal 𝐴𝑅𝐿1. Figure 3.4 shows the in-control and the out-of-control pdf discussed in Table 3.3. As the mean and the variance increased, the density curve shifted to the right and flattened. 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for the mean shift downward while the variance increases are given in Table 3.4. The �̃�(1−𝑐𝑑𝑓) control chart showed the lowest 𝐴𝑅𝐿1 among the discussed weighted average control charts and the �̅� control chart in detecting the mean shifts downward with increased variance. When the mean shift is large and the increase in the variance is comparatively low, the 𝐴𝑅𝐿1 of �̅� control chart was lower than the 𝐴𝑅𝐿1 of the �̃�(1−𝑐𝑑𝑓) control chart. However, with increasing sample size, the �̃�(1−𝑐𝑑𝑓) control chart showed less 𝐴𝑅𝐿1. Further, �̃�𝑚𝑎𝑥 and �̃�ℎ𝑎𝑧 control charts gave lower 𝐴𝑅𝐿1 than the �̅� control chart for small downward and upward mean shifts, respectively. Figure 3.5 shows the in-control and the out-of-control pdf for the mean decreases and variance increases, and it demonstrates that the density curve moves to the left and is flattened. Chapter 3 - Results 26 Table 3.3: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift, variance increases - N(0,1) Sample size Mean Standard Deviation Variance �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�ℎ𝑎𝑧 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 n=5 0.0 1.0 1.00 100.67 100.43 99.64 98.57 100.45 100.02 100.62 100.16 100.18 99.86 100.02 98.9 0.2 1.2 1.44 22.97 22.45 29.93 29.49 40.39 39.99 19.38 18.86 18.36 17.91 28.33 27.87 1.5 2.25 10.03 9.50 12.26 11.69 22.48 22.09 5.82 5.3 5.66 5.14 8.81 8.30 2.0 4.00 4.82 4.28 4.78 4.24 11.16 10.64 2.46 1.88 2.41 1.84 3.08 2.54 0.5 1.2 1.44 8.89 8.38 14.29 13.72 19.01 18.49 10.31 9.82 9.03 8.53 17.80 17.28 1.5 2.25 5.83 5.27 10.31 9.81 15.42 15.02 4.00 3.45 3.80 3.27 9.57 9.09 2.0 4.00 3.77 3.24 5.31 4.81 9.83 9.28 2.06 1.48 2.01 1.43 3.58 3.03 1.0 1.2 1.44 2.58 2.02 4.01 3.49 5.78 5.25 3.75 3.2 3.14 2.59 5.89 5.39 1.5 2.25 2.44 1.87 4.53 3.99 6.97 6.44 2.28 1.7 2.12 1.54 6.45 5.95 2.0 4.00 2.27 1.70 4.30 3.77 6.84 6.35 1.57 0.95 1.53 0.90 3.98 3.44 1.5 1.2 1.44 1.35 0.69 1.81 1.2 2.59 2.03 1.81 1.21 1.57 0.95 2.65 2.09 1.5 2.25 1.44 0.80 2.23 1.67 3.59 3.06 1.49 0.86 1.42 0.77 3.57 3.03 2.0 4.00 1.53 0.90 2.77 2.22 4.51 4.00 1.28 0.6 1.26 0.57 3.57 3.03 n=10 0.00 1.0 1.00 100.68 100.40 98.08 97.48 99.7 99.08 100.17 99.66 100.27 99.28 100.05 99.21 0.2 1.2 1.44 17.58 17.06 27.42 26.87 34.78 34.3 11.89 11.36 11.27 10.79 28.80 28.35 1.5 2.25 8.78 8.27 11.8 11.2 23.01 22.45 3.33 2.78 3.36 2.82 6.54 6.04 2.0 4.00 4.54 4.01 4.25 3.7 13.26 12.71 1.56 0.93 1.58 0.95 2.06 1.48 0.5 1.2 1.44 4.89 4.36 8.83 8.23 11.6 11.07 5.22 4.69 4.55 4.01 16.95 16.58 1.5 2.25 3.88 3.34 8.69 8.14 12.44 11.95 2.2 1.63 2.15 1.57 9.24 8.74 2.0 4.00 3.03 2.47 5.17 4.63 10.65 10.13 1.35 0.68 1.35 0.68 2.57 2.00 1.0 1.2 1.44 1.45 0.81 2.10 1.52 3.02 2.47 1.85 1.25 1.62 1.01 4.33 3.80 1.5 2.25 1.53 0.90 2.77 2.22 4.5 3.98 1.36 0.7 1.31 0.64 6.88 6.35 2.0 4.00 1.62 1.00 3.54 2.98 6.24 5.7 1.13 -0.38 1.12 0.37 3.47 2.92 1.5 1.2 1.44 1.04 0.20 1.17 0.44 1.53 0.91 1.14 0.4 1.08 0.29 1.97 1.38 1.5 2.25 1.08 0.29 1.41 0.77 2.3 1.72 1.08 0.29 1.06 0.25 3.37 2.82 2.0 4.00 1.16 0.43 1.94 1.34 3.78 3.25 1.04 0.20 1.04 0.19 3.67 3.12 Chapter 3 – Results 27 Table 3.4: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift, variance increases - N(0,1) Sample size Mean Standard Deviation Variance �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�ℎ𝑎𝑧 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 n=5 0.0 1.0 1.00 100.67 100.43 99.64 98.57 100.45 100.02 100.62 100.16 100.18 99.86 100.02 98.9 -0.2 1.2 1.44 22.78 22.30 22.39 21.89 40.06 36.91 27.41 26.97 28.27 27.88 18.35 17.82 1.5 2.25 10.07 9.61 7.92 7.41 22.29 21.79 8.75 8.28 8.75 8.17 5.67 5.13 2.0 4.00 4.80 4.27 3.45 2.91 11.13 10.63 3.13 2.57 3.07 2.53 2.43 1.86 -0.5 1.2 1.44 8.84 8.32 10.27 9.74 18.88 18.37 17.23 16.65 17.91 17.40 9.06 8.55 1.5 2.25 5.80 5.27 4.82 4.27 15.37 14.88 9.56 9.09 9.56 9.01 3.83 3.29 2.0 4.00 3.78 3.25 2.65 2.08 9.83 9.31 3.67 3.14 3.57 3.04 2.01 1.43 -1.0 1.2 1.44 2.58 2.02 3.42 2.87 5.77 5.2 5.74 5.17 5.93 5.37 3.15 2.61 1.5 2.25 2.44 1.88 2.38 1.82 6.97 6.46 6.43 5.89 6.48 5.95 2.12 1.54 2.0 4.00 2.29 1.71 1.78 1.18 6.83 6.28 4.14 3.64 3.98 3.46 1.54 0.91 -1.5 1.2 1.44 1.35 0.69 1.67 1.06 2.6 2.04 2.59 2.03 2.66 2.11 1.57 0.94 1.5 2.25 1.44 0.79 1.49 0.86 3.57 3.03 3.52 2.98 3.56 3.03 1.41 0.76 2.0 4.00 1.54 0.91 1.36 0.7 4.49 3.96 3.67 3.14 3.57 3.02 1.26 0.57 n=10 0.0 1.0 1.00 100.68 100.40 98.08 97.48 99.7 99.08 100.17 99.66 100.27 99.28 100.05 99.21 -0.2 1.2 1.44 17.83 17.32 14.96 14.46 35.23 34.62 25.68 25.09 29.04 28.71 11.25 10.74 1.5 2.25 8.83 8.34 5.42 4.9 23.16 22.74 6.04 5.52 6.55 6.04 3.35 2.80 2.0 4.00 4.55 4.01 2.59 2.02 13.35 12.87 2.01 1.43 2.06 1.48 1.57 0.95 -0.5 1.2 1.44 4.94 4.40 5.12 4.56 11.81 11.27 15.38 14.92 16.95 16.42 4.52 3.97 1.5 2.25 3.91 3.35 2.84 2.28 12.6 12.12 8.13 7.58 9.25 8.72 2.15 1.57 2.0 4.00 3.05 2.49 1.85 1.25 10.75 10.21 2.44 1.92- 2.56 2.01 1.34 0.68 -1.0 1.2 1.44 1.45 0.81 1.65 1.04 3.05 2.5 4.05 3.53 4.33 3.80 1.62 1.00 1.5 2.25 1.54 0.91 1.42 0.77 4.57 4.05 6.24 5.72 6.94 6.43 1.31 0.64 2.0 4.00 1.62 1.01 1.27 0.59 6.32 5.77 3.28 2.72 3.47 2.93 1.12 0.37 -1.5 1.2 1.44 1.04 0.20 1.08 0.3 1.55 0.92 1.87 1.27 1.97 1.37 1.08 0.29 1.5 2.25 1.08 0.30 1.08 0.29 2.31 1.74 3.15 2.61 3.37 2.82 1.06 0.25 2.0 4.00 1.16 0.43 1.07 0.28 3.83 3.29 3.51 2.98 3.67 3.15 1.03 0.19 Chapter 3 – Results 28 Figure 3.4: In-control and out-of-control pdfs for upward mean shift, variance increase - N(0,1) Figure 3.5: In-control and out-of-control pdfs for downward mean shift, variance increase - N(0,1) Chapter 3 – Results 29 3.3 Positively Skewed Distributions The exponential and gamma distributions were used to examine the behaviour of the weighted average control charts when the underlying distribution is positively skewed. 3.3.1 Exponential Distribution The rate of the exponential distribution was considered as one in this study. Exponential distribution occurs when the shape parameter equals one in gamma distribution. The probability density function was drawn with the means of the control statistics, calculated from an empirical distribution generated randomly for subgroup size ten is shown in Figure 3.6. The weighted averages based on the CoPDF- weight and CDF-weight are the same and positioned to the right of the unweighted average. In contrast, the weighted averages based on PDF-weight and the CoCDF- weight were similar and positioned to the left of the simple average. Also, the weighted average based on Max-weight was left to the unweighted average. Since the hazard function of the exponential distribution is a constant (𝜆), the weighted average X̃ℎ𝑎𝑧 was the same as the unweighted average 𝑋 ̅. Hence, the control statistic defined based on Haz-weight was not considered for the exponential distribution. Figure 3.6: exp(1) density curve with means of control statistics Chapter 3 – Results 30 Tables 3.5 and 3.6 give the 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for mean shifts upward and downward from the in-control value, respectively. The 𝑋 ̅ control chart gave the lowest 𝐴𝑅𝐿1 compared to all the weighted average control charts in detecting mean shifts. Figure 3.7 shows the in-control and out-of-control pdf for the increasing and decreasing mean changes. When the mean increases, the density curve was stretched to the right, while the density curve was shrunk to the left when the mean decreases from the in- control value. Figure 3.7: In-control and out-of-control pdfs for mean shift - exp(1) 3.3.2 Gamma Distribution Gam(0.5,1), Gam(1.5,2) and Gam(2,1) were considered as the in-control distributions to illustrate the shape parameter less than one, the shape parameter greater than one and less than the scale parameter and shape parameter greater than one and scale parameter, respectively. In addition, mean shifts that change the shape and scale parameters in various ways will be discussed for two sample sizes, 5 and 10 in the following sections. 3.3.2.1 Shape Parameter Less than One – Gam(0.5,1) Figure 3.8 shows the probability density function of Gam(0.5,1) with the means of the control statistics, calculated from a randomly generated empirical distribution of subgroup size ten. As illustrated in Figure 3.8, the mean of control statistic based on the CDF-weight was shifted to the right of the unweighted average. In contrast, all the other weighted averages were to the left of the unweighted average. Chapter 3 – Results 31 Table 3.5: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - exp(1) Table 3.6: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift - exp(1) Sample size rate Standard Deviation Variance �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�1−𝑝𝑑𝑓 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 n=5 1 1.00 1.00 98.98 98.70 99.45 98.83 99.02 98.57 98.59 98.26 98.59 98.26 99.02 98.57 0.90 1.11 1.23 65.09 64.84 72.28 71.90 78.47 78.14 74.32 74.12 74.32 74.12 78.47 78.14 0.75 1.33 1.78 23.15 22.57 30.73 30.07 43.28 42.58 32.15 31.56 32.15 31.56 43.28 42.58 0.50 2.00 4.00 4.04 3.52 5.90 5.38 12.82 12.27 5.65 5.15 5.65 5.15 12.82 12.27 0.25 4.00 16.00 1.26 0.57 1.54 0.90 3.76 3.22 1.42 0.77 1.42 0.77 3.76 3.22 n=10 1 1.00 1.00 101.1 100.42 100.33 60.79 100.48 99.49 98.35 97.68 98.35 97.68 100.48 99.49 0.90 1.11 1.23 57.19 56.45 61.31 16.92 77.13 76.68 64.68 63.85 64.68 63.85 77.13 76.68 0.75 1.33 1.78 14.32 13.77 17.48 2.13 36.93 36.29 19.5 18.92 19.5 18.92 36.93 36.29 0.50 2.00 4.00 2.19 1.62 2.69 0.27 9.66 9.18 2.82 2.27 2.82 2.27 9.66 9.18 0.25 4.00 16.00 1.03 0.18 1.07 60.79 3.2 2.64 1.06 0.26 1.06 0.26 3.2 2.64 Sample size rate Standard Deviation Variance �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�1−𝑝𝑑𝑓 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 n=5 1 1.00 1.00 98.98 98.70 99.45 98.83 99.02 98.27 98.59 98.26 98.59 98.26 99.02 98.27 1.20 0.83 0.69 87.49 86.83 92.60 92.50 98.69 98.24 84.17 83.65 84.17 83.65 98.69 98.24 1.50 0.67 0.44 39.90 39.62 48.55 48.07 56.00 55.33 41.14 40.86 41.14 40.86 56.00 55.33 2.00 0.50 0.25 14.56 13.98 19.53 19.04 22.11 21.53 15.47 14.90 15.47 14.9 22.11 21.53 2.50 0.40 0.16 7.26 6.73 10.38 9.86 11.06 10.52 7.86 7.31 7.86 7.31 11.06 10.52 n=10 1 1.00 1.00 101.1 100.42 100.33 100.16 100.48 99.49 98.35 97.68 98.35 97.68 100.48 99.49 1.20 0.83 0.69 60.66 60.1 64.85 64.16 80.74 79.7 61.34 60.6 61.34 60.6 80.74 79.7 1.50 0.67 0.44 17.39 16.86 20.47 19.94 32.03 31.71 19.2 18.73 19.2 18.73 32.03 31.71 2.00 0.50 0.25 4.65 4.14 5.77 5.25 9.25 8.69 5.46 4.93 5.46 4.93 9.25 8.69 2.50 0.40 0.16 2.23 1.65 2.75 2.2 4.08 3.55 2.65 2.1 2.65 2.1 4.08 3.55 Chapter 3 – Results 32 Figure 3.8: Gam(0.5,1) density curve with the means of control statistics Tables 3.7 and 3.8 summaries the 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for the mean increase and decreases occurred by a shift in either shape or scale parameter, respectively. The weighted average control charts �̃�𝑚𝑎𝑥, �̃�𝑝𝑑𝑓, �̃�ℎ𝑎𝑧 and �̃�(1−𝑐𝑑𝑓) showed lower 𝐴𝑅𝐿1 compared to �̅� chart in detecting a shift in the mean, which occurs due to a change in the shape parameter. The �̃�ℎ𝑎𝑧 control chart indicated the lowest 𝐴𝑅𝐿1 compared to other control charts. In contrast, the �̅� chart was better for detecting mean shifts from a change in the scale parameter. Also, the 𝐴𝑅𝐿1 turned out to be shorter with an increase in the sample size. Further, �̃�𝑐𝑑𝑓 control chart was insensitive for a slight decrease in the shape parameter, and all the control charts were insensitive in detecting a slight decrease in the scale parameter when the sample size is five. Figures 3.9 and 3.10 summaries the in-control and out-of-control pdf for the mean shifts occurred by a single parameter. When the scale increases, the density curve was expanded to the right, whereas the scale decreases, the density curve was shrunk to the left. A decrease in the shape parameter could move the density curve to the left, and an increase in the shape parameter could flattened the density curve and moved to the right. Chapter 3 - Results 33 Table 3.7: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - change in one parameter - Gam(0.5,1) S am p le si ze S h ap e S ca le M ea n V ar ia n ce S k ew n es s �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart �̃�ℎ𝑎𝑧 Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 n=5 0.5 1 0.5 0.5 2.8 100.52 99.62 100.81 100.27 99.47 99.13 100.04 99.99 99.29 98.63 98.95 98.55 0.70 1.00 0.7 0.7 2.4 51.60 51.17 37.09 36.73 38.51 37.89 125.45 125.94 37.78 37.18 36.76 36.44 1.00 1.00 1.0 1.0 2.0 12.43 11.92 7.56 7.06 8.31 7.84 54.61 53.74 8.04 7.61 7.49 6.98 1.50 1.00 1.5 1.5 1.6 2.99 2.44 1.97 1.38 2.32 1.75 11.93 11.36 2.26 1.69 1.98 1.39 2.00 1.00 2.0 2.0 1.4 1.50 0.86 1.19 0.48 1.38 0.73 3.93 3.40 1.35 0.68 1.20 0.49 0.50 1.20 0.6 0.7 2.8 52.06 51.67 60.89 60.53 70.43 69.51 61.12 61.08 74.39 74.14 54.95 55.13 0.50 1.50 0.8 1.1 2.8 19.84 19.31 27.43 26.98 45.10 44.38 25.78 25.07 46.79 46.11 23.99 23.62 0.50 2.00 1.0 2.0 2.8 7.28 6.76 10.82 10.29 25.58 25.30 9.56 9.04 25.86 25.55 9.36 8.84 0.50 2.50 1.3 3.1 2.8 4.10 3.59 6.10 5.57 17.52 16.93 5.28 4.77 17.36 16.87 5.42 4.85 n=10 0.5 1 0.5 0.5 2.8 99.69 98.97 99.92 99.52 99.38 98.4 99.85 99.49 99.36 98.45 98.38 97.25 0.70 1.00 0.7 0.7 2.4 30.18 29.64 20.45 19.88 19.01 18.37 91.42 89.93 18.09 17.43 17.18 16.59 1.00 1.00 1.0 1.0 2.0 5.12 4.59 3.22 2.66 3.20 2.63 24.45 24.18 2.94 2.38 2.70 2.14 1.50 1.00 1.5 1.5 1.6 1.39 0.74 1.15 0.41 1.23 0.53 3.97 3.45 1.16 0.43 1.09 0.32 2.00 1.00 2.0 2.0 1.4 1.03 0.18 1.01 0.08 1.03 0.19 1.53 0.89 1.01 0.12 1.00 0.05 0.50 1.20 0.6 0.7 2.8 42.06 41.55 47.93 47.24 69.96 69.06 49.72 49.22 72.31 71.85 47.25 47.17 0.50 1.50 0.8 1.1 2.8 12.46 11.96 15.67 15.17 43.24 42.39 15.88 15.60 40.73 39.67 17.24 16.56 0.50 2.00 1.0 2.0 2.8 4.01 3.47 5.15 4.63 24.50 24.12 5.16 4.66 20.77 20.28 6.05 5.49 0.50 2.50 1.3 3.1 2.8 2.30 1.73 2.86 2.29 16.80 16.20 2.76 2.21 13.64 13.11 3.44 2.90 Chapter 3 - Results 34 Table 3.8: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shift - change in one parameter - Gam(0.5,1) S am p le si ze S h ap e S ca le M ea n V ar ia n ce S k ew n es s �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart �̃�ℎ𝑎𝑧 Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 n = 5 0.5 1.00 0.50 0.50 2.83 100.52 99.62 100.81 100.27 99.47 99.13 100.04 99.99 99.29 98.63 98.95 98.55 0.40 1.00 0.40 0.40 3.16 48.75 48.22 42.82 42.47 41.22 40.57 49.54 48.81 48.78 47.40 37.96 37.51 0.30 1.00 0.30 0.30 3.65 15.99 15.41 11.78 11.26 10.22 9.70 19.34 18.85 14.75 14.29 8.87 8.42 0.25 1.00 0.25 0.25 4.00 9.17 8.64 6.36 5.83 5.21 4.64 11.59 10.96 8.25 7.69 4.57 4.01 0.50 0.80 0.40 0.32 2.83 109.10 108.57 112.97 112.90 129.79 130.31 104.21 103.33 104.49 104.13 127.87 127.96 0.50 0.50 0.25 0.13 2.83 41.23 40.96 52.51 51.89 124.75 122.39 41.91 41.20 48.16 48.00 81.32 80.14 0.50 0.25 0.13 0.03 2.83 9.64 9.10 14.73 14.22 72.36 72.51 9.85 9.32 11.04 10.53 34.55 33.88 n = 10 0.5 1.00 0.50 0.50 2.83 99.69 98.97 99.92 99.52 99.38 98.4 99.85 99.49 99.36 98.45 98.38 97.25 0.40 1.00 0.40 0.40 3.16 39.27 38.82 32.81 32.29 36.72 35.78 45.40 45.02 31.79 31.03 33.16 32.60 0.30 1.00 0.30 0.30 3.65 10.58 10.07 7.47 6.95 7.79 7.33 16.58 16.02 6.95 6.41 6.71 6.22 0.25 1.00 0.25 0.25 4.00 5.80 5.29 3.98 3.45 3.73 3.23 10.10 9.53 3.61 3.08 3.26 2.71 0.50 0.80 0.40 0.32 2.83 79.51 78.18 81.93 81.03 130.28 128.52 76.59 76.40 88.79 87.57 129.80 129.84 0.50 0.50 0.25 0.13 2.83 14.85 14.32 17.34 16.83 127.51 127.59 15.49 15.10 23.82 23.30 78.66 77.89 0.50 0.25 0.13 0.03 2.83 2.33 1.74 2.87 2.33 73.91 74.42 2.55 1.99 3.64 3.11 30.93 30.56 Chapter 3 - Results 35 Figure 3.9: In-control and out-of-control pdfs for upward mean shift - change in one parameter - Gam(0.5,1) Figure 3.10: In- control and out-of-control pdfs for downward mean shift - change in one parameter – Gam(0.5,1) Table 3.9 presented the 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for mean shifts when both parameters were shifted simultaneously to the same direction. The �̃�𝑚𝑎𝑥 and �̃�ℎ𝑎𝑧 control charts provided lower 𝐴𝑅𝐿1 than the �̅� control chart for both upward and downward mean shifts. All the other control charts had longer 𝐴𝑅𝐿1 in comparison to the �̅� control chart. Figure 3.11 shows the in-control and out-of-control pdfs for the mean shifts from their in-control states when both parameters were shifted to the same direction. The density curve was flattened and expanded to the right when both parameters were increased. When both parameters decrease, the density curve was shrunk to the left. Chapter 3 - Results 36 Table 3.9: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for mean shifts - change in both parameters - same direction - Gam(0.5,1) S am p le si ze S h ap e S ca le M ea n V ar ia n ce S k ew n es s �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart �̃�ℎ𝑎𝑧 Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 Mean increase n=5 0.5 1.00 0.50 0.50 2.83 100.52 99.62 100.81 100.27 99.47 99.13 100.04 99.99 99.29 98.63 98.95 98.55 0.70 1.20 0.84 1.01 2.39 19.33 18.75 16.96 16.54 22.83 22.20 45.23 44.39 22.21 21.53 15.57 15.07 1.00 1.20 1.20 1.44 2.00 5.70 5.19 4.28 3.76 5.78 5.23 18.39 17.60 5.60 5.04 4.02 3.49 1.00 1.50 1.50 2.25 2.00 2.91 2.35 2.59 2.02 4.06 3.48 6.69 6.08 3.92 3.36 2.34 1.79 1.50 1.20 1.80 2.16 1.63 1.89 1.29 1.50 0.87 1.93 1.34 4.96 4.40 1.87 1.28 1.45 0.81 n=10 0.5 1.00 0.50 0.50 2.83 99.69 98.97 99.92 99.52 99.38 98.4 99.85 99.49 99.36 98.45 98.38 97.25 0.70 1.20 0.84 1.01 0.70 9.78 9.30 8.07 7.56 12.02 11.58 25.40 24.85 10.63 10.24 7.25 6.75 1.00 1.20 1.20 1.44 1.00 2.51 1.95 1.95 1.35 2.52 1.96 7.53 7.01 2.24 1.67 1.73 1.13 1.00 1.50 1.50 2.25 1.00 1.50 0.86 1.35 0.68 2.08 1.46 2.87 2.30 1.81 1.21 1.27 0.59 1.50 1.20 1.80 2.16 1.50 1.12 0.37 1.05 0.22 1.17 0.45 1.95 1.36 1.01 0.33 1.03 0.16 Mean decrease n=5 0.5 1.00 0.50 0.50 2.83 100.52 99.62 100.81 100.27 99.47 99.13 100.04 99.99 99.29 98.63 98.95 98.55 0.40 0.80 0.32 0.26 3.16 35.63 35.11 32.26 31.83 38.52 38.13 39.83 39.65 35.88 35.09 31.61 31.29 0.40 0.50 0.20 0.10 3.16 15.53 15.03 16.22 15.74 29.98 29.38 17.95 17.37 16.61 16.08 19.90 19.10 0.25 0.80 0.20 0.16 4.00 7.16 6.61 5.26 4.73 4.85 4.35 9.65 9.04 6.57 6.12 4.08 3.57 0.25 0.50 0.13 0.06 4.00 4.30 3.75 3.55 3.02 4.13 3.62 5.75 5.25 4.08 3.57 3.21 2.67 0.25 0.25 0.06 0.02 4.00 2.23 1.65 2.13 1.56 3.17 2.60 2.84 2.26 2.18 1.59 2.33 1.76 n=10 0.5 1.00 0.50 0.50 2.83 99.69 98.97 99.92 99.52 99.38 98.4 99.85 99.49 99.36 98.45 98.38 97.25 0.40 0.80 0.32 0.26 3.16 20.90 20.44 18.14 17.65 34.02 33.01 27.49 27.16 19.62 19.06 26.90 26.55 0.40 0.50 0.20 0.10 3.16 5.82 5.29 5.69 5.15 26.32 25.46 7.60 7.05 6.86 6.34 16.35 15.72 0.25 0.80 0.20 0.16 4.00 3.95 3.41 2.92 2.37 3.48 2.90 7.00 6.49 2.84 2.26 2.89 2.36 0.25 0.50 0.13 0.06 4.00 2.02 1.44 1.71 1.10 2.96 2.43 3.23 2.71 1.79 1.20 2.25 1.67 0.25 0.25 0.06 0.02 4.00 1.14 0.40 1.11 0.35 2.32 1.77 1.48 0.84 1.15 0.41 1.65 1.04 Chapter 3 - Results 37 Figure 3.11: In-control and out-of-control pdfs - change in both parameters - same direction - Gam(0.5,1) The mean of the process was varied by changing the parameters in the opposite direction. 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shifts are given in Tables 3.10. The weighted average control charts �̃�𝑚𝑎𝑥, �̃�𝑝𝑑𝑓 , �̃�(1−𝑐𝑑𝑓) and �̃�ℎ𝑎𝑧 gave lower 𝐴𝑅𝐿1 than the �̅� control chart while �̃�𝑐𝑑𝑓 gave higher 𝐴𝑅𝐿1 than the �̅� control chart for the shifts occurred by an increase in the shape parameter and decrease in the scale parameter. Although the �̃�𝑝𝑑𝑓 and �̃�(1−𝑐𝑑𝑓) charts showed lower 𝐴𝑅𝐿1 than 𝐴𝑅𝐿0 for both sample sizes, the �̅� and �̃�𝑐𝑑𝑓 control charts were insensitive to small changes in the mean. Further, �̃�𝑚𝑎𝑥 and �̃�ℎ𝑎𝑧 control charts were insensitive to small shifts when the sample size is five and gave smaller 𝐴𝑅𝐿1 compared to 𝐴𝑅𝐿0 for subgroup size 10. In contrast, �̃�𝑐𝑑𝑓 control chart gave lower 𝐴𝑅𝐿1 compared to all the other control charts for shifts occurred by a decrease in the shape parameter and increase in the scale parameter. Moreover, �̃�ℎ𝑎𝑧 control chart also resulted less 𝐴𝑅𝐿1 than the �̅� control chart for a significant increase in the mean. Figure 3.12 shows the in-control and out-of-control pdf considered for the mean increased by shifting the parameters opposite. The density curve was flattened and moved to the right when the shape increases and scale decreases, while the density curve was condensed to the left when the shape decreases and scale increases. Chapter 3 - Results 38 Table 3.10: 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for upward mean shift - change in both parameters - opposite direction - Gam(0.5,1) S am p le si ze S h ap e S ca le M ea n V ar ia n ce S k ew n es s �̅� Chart �̃�𝑚𝑎𝑥 Chart �̃�𝑝𝑑𝑓 Chart �̃�𝑐𝑑𝑓 Chart �̃�(1−𝑐𝑑𝑓) Chart �̃�ℎ𝑎𝑧 Chart 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 𝐴𝑅𝐿 𝑆𝐷𝑅𝐿 n = 5 0.5 1.00 0.50 0.50 2.83 100.52 99.62 100.81 100.27 99.47 99.13 100.04 99.99 99.29 98.63 98.95 98.55 0.70 0.80 0.56 0.45 2.39 223.79 223.37 116.90 116.40 82.44 82.55 421.90 423.10 81.97 81.47 136.67 136.71 1.00 0.80 0.80 0.64 2.00 46.48 45.82 18.68 18.16 15.03 14.40 290.86 292.89 14.84 14.40 21.21 20.69 2.00 0.50 1.00 0.50 1.41 33.51 32.87 5.13 4.61 3.24 2.71 1530.52 1533.92 3.22 2.67 8.13 7.53 1.50 0.80 1.20 0.96 1.63 7.22 6.68 3.30 2.76 3.19 2.66 54.62 54.38 3.11 2.58 3.61 3.04 2.00 0.80 1.60 1.28 1.41 2.50 1.93 1.50 0.87 1.61 0.99 12.94 12.45 1.57 0.95 1.58 0.96 0.40 1.50 0.60 0.90 3.16 30.17 29.54 39.76 39.45 39.06 38.46 27.77 27.26 56.22 55.83 32.20 32.14 0.25 2.50 0.63 1.56 4.00 10.85 10.35 11.80 11.30 6.45 5.90 8.21 7.69 15.60 15.08 6.93 6.39 0.40 2.00 0.80 1.60 3.16 12.01 11.57 20.71 19.89 31.95 31.40 11.95 11.75 43.30 42.67 17.27 16.64 0.40 2.50 1.00 2.50 3.16 6.37 5.87 11.56 11.05 25.92 25.50 6.82 6.36 32.70 32.43 10.07 9.45 n = 10 0.5 1.00 0.50 0.50 2.83 99.69 98.97 99.92 99.52 99.38 98.4 99.85 99.49 99.36 98.45 98.38 97.25 0.70 0.80 0.56 0.45 2.39 189.13 188.36 94.03 93.41 39.35 38.41 424.71 423.86 41.55 41.03 72.64 72.51 1.00 0.80 0.80 0.64 2.00 20.11 19.69 8.42 7.87 4.84 4.31 181.85 178.36 4.68 4.14 6.51 5.99 2.00 0.50 1.00 0.50 1.41 7.83 7.30 2.01 1.42 1.19 0.47 526.59 532.16 1.18 0.45 1.73 1.13 1.50 0.80 1.20 0.96 1.63 2.54 1.98 1.52 0.90 1.36 0.70 17.89 17.36 1.29 0.61 1.36 0.70 2.00 0.80 1.60 1.28 1.41 1.20 0.49 1.04 0.20 1.05 0.22 3.61 3.10 1.02 0.16 1.02 0.15 0.40 1.50 0.60 0.90 3.16 25.73 25.12 36.22 35.78 38.41 37.61 21.19 20.76 54.13 53.80 31.48 31.03 0.25 2.50 0.63 1.56 4.00 9.94 9.40 13.18 12.69 4.71 4.13 5.77 5.25 8.64 8.09 5.44 4.97 0.40 2.00 0.80 1.60 3.16 7.72 7.17 12.60 12.12 34.61 33.72 7.24 6.69 49.16 48.61 15.63 15.06 0.40 2.50 1.00 2.50 3.16 3.82 3.30 5.93 5.43 29.87 29.40 3.80 3.22 36.92 36.51 8.53 7.96 Chapter 3 - Results 39 Figure 3.12: In-control and out-of-control pdfs for upward mean shift - change in both parameters - opposite direction - Gam(0.5,1) The 𝐴𝑅𝐿 and 𝑆𝐷𝑅𝐿 for downward mean shifts are given in Table 3.11. The �̃�𝑚𝑎𝑥, �̃�𝑝𝑑𝑓 and �̃�ℎ𝑎𝑧 control charts gave small 𝐴𝑅𝐿1 compared to the �̅� control chart for the downward mean shift occurred by shape parameter decreases and scale parameter increases. In addition, �̃�𝑐𝑑𝑓 control chart gave the lower 𝐴𝑅𝐿1 for small mean shifts while �̃�(1−𝑐𝑑𝑓) showed lower 𝐴𝑅𝐿1 for significant mean shifts when compared to �̅� control chart. The lowest 𝐴𝑅𝐿1 was given by �̃�ℎ𝑎𝑧 control chart. �̃�ℎ𝑎𝑧 and �̃�𝑝𝑑𝑓 control charts were insensitive in detecting downward mean shifts occurred by increasing the shape parameter and decreasing the scale parameter. However, 𝐴𝑅𝐿1 was smaller than 𝐴𝑅𝐿0 for �̃�𝑐𝑑𝑓 control chart when the subgroup size is ten and �̅�, �̃�𝑚𝑎𝑥 and �̃�(1−𝑐𝑑𝑓) control charts for significant mean shifts. However, the �̃�𝑐𝑑𝑓 control chart showed smaller 𝐴𝑅𝐿1 than �̅� control chart. Figure 3.13 shows the in-control and out-of-control pdf for the downward mean shifts when the parameters were shifted in the opposite direction. The density curve was moved to the right when the mean was decreased by shape increases and scale decreases, and vice versa. Chapter 3 - Results 40 Table 3.11: 𝐴