The Effects of Autocorrelation in the Estimation of Process Capability Indices." (1998)

Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1998 The ffecE ts of Autocorrelation in the Estimation of Process Capability Indices. Lawrence Lee Magee Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Magee, Lawrence Lee, "The Effects of Autocorrelation in the Estimation of Process Capability Indices." (1998). LSU Historical Dissertations and Theses. 6847. https://digitalcommons.lsu.edu/gradschool_disstheses/6847 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. 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THE EFFECTS OF AUTOCORRELATION IN THE ESTIMATION OF PROCESS CAPABILITY INDICES A Dissertation Submitted to the Graduate Faculty o f the Louisiana State University and Agricultural and Mechanical College in partial fulfillment o f the requirements for the degree o f Doctor o f Philosophy in The Interdepartmental Program in Business Administration in Inform ation Systems and Decision Sciences by Lawrence Lee Magee B.S., Louisiana State University, 1971 M.S., Louisiana State University, 1977 M.S., University o f Wisconsin, 1985 December 1998 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9922096 UMI Microform 9922096 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS In a very real sense, there are many individuals to thank for this work. Elementary, secondary, and university school teachers recede in time, but not in memory. Recently my major professor, Dr. Helmut Schneider, has provided the necessary guidance. More importantly, the quantity o f patience he has demonstrated can only be described as o f biblical proportion. I must thank the members o f my dissertation committee, Drs. Kwei Tang, Peter Kelle, Edward Watson, and Michael Salassi. Their helpful suggestions have produced a better work. Errors, o f course, must be attributed to me. My son, John and my sisters, Sharon, Deborah, Barbara, and Mary Ellen, deserve recognition for their moral and practical support during this period. Finally, my father. LeRoy and mother, Gloria are due incalculable portions o f gratitude and love. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ACKNOWLEDGMENTS......................................................................................................ii LIST OF TABLES................................................................................................................. v LIST OF FIGURES.............................................................................................................. vi ABSTRACT .........................................................................................................................vii CHAPTER 1. INTRODUCTION ..................................................................................................... 1 1.1. Process Capability Analysis ............................................................................ I 1.2. The Stability o f a Process ...............................................................................3 1.3. The Indices Cp, C p l, Cpu, and C p k ............................................................... 5 1.4. The P roportion tZq o f Product Outside Specification ...................................7 1.5. The (Cp/, Cpu) Indices as Reparameterization o f (p, cr) .............................. 8 1.6. The (Cpl, Cp, Cpu ) D iagram ........................................................................11 1.7. The (Cpl, Cpu) Indices as Measure of Actual versus Ideal (p, a ) ................ 14 1.8. A Compendium o f Bijections ........................................................................ 16 1.9. Criticisms o f the Indices Cp, Cpl, Cpu, and C p k .........................................17 1.10. Contribution o f the Research ....................................................................... 22 1.11. Organization o f the Research ....................................................................... 23 2. REVIEW OF THE LITERATURE....................................................................... 24 2.1. The Cp In d e x ................................................................................................ 25 2.2. The Cpk Index ............................................................................................... 27 2.3. The Cpm Index and Its Variants .................................................................. 32 2.4. The Effect o f Autocorrelation on ProcessCapability Indices ......................36 2.5. Non-Normality and Robustness ................................................................... 37 2.6. Multivariate Process Capability Indices........................................................43 2.7. Recent Approaches and Criticism ................................................................ 45 3. CAPABILITY INDICES UNDER NORMAL INDEPENDENCE.....................49 3.1. Why Some Real World Data Exhibit Norm ality ......................................... 49 3.2. Estimating Cp when p is Known and cr is Unknown ..................................52 3.3. Estimating Cp when B oth p and a are U n kn o w n ....................................... 64 3.4. Estimating (Cpl. Cp, Cpu) when p is Unknown and cr is Known .............. 68 3.5. Estimating (Cpl, Cp, Cpu) when p is Known and cr is Unknown ...............74 3.6. Estimating (Cpl, Cp, Cpu) when Both p and cr are Unknown .................... 82 3.7. Estimating Cpk when B oth p and a are U n k n o w n .................................. 100 4. CAPABILITY INDICES UNDER NORMAL CORRELATION..................... 105 4.1. Why Some Real World Data Exhibit Autocorrelation .............................. 106 4.2. The Indices (Cpl, Cp, Cpu) and the ARMA(p, q) M o d e l ..........................113 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3. The Effect o f Autocorrelation on the Sample Variance ........................... 114 4.4. A Low er B o u n d ......................................................................................... 116 4.5. Estimating Cp when p. is Known and ct is Unknown ............................... 119 4.6. Estimating Cp when Both p and a are Unknown ...................................... 136 4.7. Estimating (Cpl, Cp, Cpu ) when p is Unknown and a is Known 143 4.8. Estimating (Cpl, Cp, Cpu) when p. is Known and a is Unknown 147 4.9. Estimating (Cpl, Cp, Cpu) when Both p and <J are Unknown ................... 149 5. SUMMARY AND CONCLUSIONS.................................................................... 157 BIBLIOGRAPHY............................................................................................................... 160 APPEN D IX A DERIVATION OF AN IMPORTANT MEAN .................................................... 167 B. MOMENTS OF THE MULTIVARIATE NORMAL.......................................... 170 C. THE AR( 1) CORRELATION M ATRIX............................................................ 175 D DERIVATION OF AN IMPORTANT VARIANCE .......................................... 178 VITA ..................................................................................................................................

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