Nonparametric Statistical Process Control

SUBHABRATA CHAKRABORTI, PHD is Professor of Statistics and Morrow Faculty Excellence Fellow at the University of Alabama, Tuscaloosa, AL , USA. He is a Fellow of the American Statistical Association and an elected member of the International Statistical Institute. Professor Chakraborti has contributed in a number of research areas, including censored data analysis and income inference. His current research interests include development of statistical methods in general and nonparametric methods in particular for statistical process control. He has been a Fulbright Senior Scholar to South Africa and a visiting professor in several countries, including India, Holland and Brazil. Cited for his mentoring and collaborative work with students and scholars from around the world, Professor Chakraborti has presented seminars, delivered keynote/plenary addresses and conducted research workshops at various conferences. MARIEN ALET GRAHAM, PHD is a senior lecturer at the Department of Science, Mathematics and Technology Education at the University of Pretoria, Pretoria, South Africa. She holds an Y1 rating from the South African National Research Foundation (NRF). Her current research interests are in Statistical Process Control, Nonparametric Statistics and Statistical Education. She has published several articles in international peer review journals and presented her work at various conferences.

• About the Authors xiiiPreface xvAbout the companion website xix1 Background/Review of Statistical Concepts 1Chapter Overview 11.1 Basic Probability 11.2 Random Variables and Their Distributions 31.3 Random Sample 121.4 Statistical Inference 161.5 Role of the Computer 222 Basics of Statistical Process Control 23Chapter Overview 232.1 Basic Concepts 232.1.1 Types of Variability 232.1.2 The Control Chart 252.1.3 Construction of Control Charts 292.1.4 Variables and Attributes Control Charts 302.1.5 Sample Size or Subgroup Size 312.1.6 Rational Subgrouping 312.1.7 Nonparametric or Distribution-free 342.1.8 Monitoring Process Location and/or Process Scale 362.1.9 Case K and Case U 372.1.10 Control Charts and Hypothesis Testing 372.1.11 General Steps in Designing a Control Chart 392.1.12 Measures of Control Chart Performance 392.1.12.1 False Alarm Probability (FAP) 412.1.12.2 False Alarm Rate (FAR) 432.1.12.3 The Average Run-length (ARL) 432.1.12.4 Standard Deviation of Run-length (SDRL) 442.1.12.5 Percentiles of Run-length 442.1.12.6 Average Number of Samples to Signal (ANSS) 482.1.12.7 Average Number of Observations to Signal (ANOS) 482.1.12.8 Average Time to Signal (ATS) 482.1.12.9 Number of Individual Items Inspected (I) 492.1.13 Operating Characteristic Curves (OC-curves) 502.1.14 Design of Control Charts 512.1.14.1 Sample Size, Sampling Frequency, and Variable Sample Sizes 512.1.14.2 Variable Control Limits 542.1.14.3 Standardized Control Limits 562.1.15 Size of a Shift 572.1.16 Choice of Control Limits 592.1.16.1 k-sigma Limits 592.1.16.2 Probability Limits 603 Parametric Univariate Variables Control Charts 63Chapter Overview 633.1 Introduction 643.2 Parametric Variables Control Charts in Case K 643.2.1 Shewhart Control Charts 653.2.2 CUSUM Control Charts 673.2.3 EWMA Control Charts 723.3 Types of Parametric Variables Charts in Case K: Illustrative Examples 773.3.1 Shewhart Control Charts 773.3.1.1 Shewhart Control Charts for Monitoring Process Mean 773.3.1.2 Shewhart Control Charts for Monitoring Process Variation 793.3.2 CUSUM Control Charts 843.3.3 EWMA Control Charts 873.4 Shewhart, EWMA, and CUSUM Charts: Which to Use When 903.5 Control Chart Enhancements 913.5.1 Sensitivity Rules 913.5.2 Runs-type Signaling Rules 953.5.2.1 Signaling Indicators 973.6 Run-length Distribution in the Specified Parameter Case (Case K) 1103.6.1 Methods of Calculating the Run-length Distribution 1103.6.1.1 The Exact Approach (for Shewhart and some Shewhart-type Charts) 1103.6.1.2 The Markov Chain Approach 1113.6.1.3 The Integral Equation Approach 1283.6.1.4 The Computer Simulations (the Monte Carlo) Approach 1283.7 Parameter Estimation Problem and Its Effects on the Control Chart Performance 1313.8 Parametric Variables Control Charts in Case U 1333.8.1 Shewhart Control Charts in Case U 1333.8.1.1 Shewhart Control Charts for the Mean in Case U 1333.8.1.2 Shewhart Control Charts for the Standard Deviation in Case U 1343.8.2 CUSUM Chart for the Mean in Case U 1373.8.3 EWMA Chart for the Mean in Case U 1373.9 Types of Parametric Control Charts in Case U: Illustrative Examples 1383.9.1 Charts for the Mean 1383.9.2 Charts for the Standard Deviation 1413.9.2.1 Using the Estimator Sp 1443.10 Run-length Distribution in the unknown Parameter Case (Case U) 1533.10.1 Methods of Calculating the Run-length Distribution and Its Properties: The Conditioning/Unconditioning Method 1533.10.1.1 The Shewhart Chart for the Mean in Case U 1533.10.1.2 The Shewhart Chart for the Variance in Case U 1693.10.1.3 The CUSUM Chart for the Mean in Case U 1703.10.1.4 The EWMA Chart for the Mean in Case U 1713.11 Control Chart Enhancements 1723.11.1 Run-length Calculation for Runs-type Signaling Rules in Case U 1723.12 Phase I Control Charts 1743.12.1 Phase I X-chart 1743.13 Size of Phase I Data 1763.14 Robustness of Parametric Control Charts 177Appendix 3.1 Some Derivations for the EWMA Control Chart 178Appendix 3.2 Markov Chains 180Appendix 3.3 Some Derivations for the Shewhart Dispersion Charts 1844 Nonparametric (Distribution-free) Univariate Variables Control Charts 187Chapter Overview 1874.1 Introduction 1874.2 Distribution-free Variables Control Charts in Case K 1894.2.1 Shewhart Control Charts 1894.2.1.1 Shewhart Control Charts Based on Signs 1894.2.1.2 Shewhart Control Charts Based on Signed-ranks 1964.2.2 CUSUM Control Charts 2024.2.2.1 CUSUM Control Charts Based on Signs 2024.2.2.2 A CUSUM Sign Control Chart with Runs-type Signaling Rules 2034.2.2.3 Methods of Calculating the Run-length Distribution 2034.2.2.4 CUSUM Control Charts Based on Signed-ranks 2054.2.3 EWMA Control Charts 2084.2.3.1 EWMA Control Charts Based on Signs 2084.2.3.2 EWMA Control Charts Based on Signs with Runs-type Signaling Rules 2104.2.3.3 Methods of Calculating the Run-length Distribution 2104.2.3.4 EWMA Control Charts Based on Signed-ranks 2144.2.3.5 An EWMA-SR control chart with runs-type signaling rules 2164.2.3.6 Methods of Calculating the Run-length Distribution 2164.3 Distribution-free Control Charts in Case K: Illustrative Examples 2194.3.1 Shewhart Control Charts 2194.3.2 CUSUM Control Charts 2294.3.3 EWMA Control Charts 2434.4 Distribution-free Variables Control Charts in Case U 2534.4.1 Shewhart Control Charts 2544.4.1.1 Shewhart Control Charts Based on the Precedence Statistic 2544.4.1.2 Shewhart Control Charts Based on the Mann–Whitney Test Statistic 2754.4.2 CUSUM Control Charts 2814.4.2.1 CUSUM Control Charts Based on the Exceedance Statistic 2814.4.2.2 CUSUM Control Charts Based on the Wilcoxon Rank-sum Statistic 2854.4.3 EWMA Control Charts 2874.4.3.1 EWMA Control Charts Based on the Exceedance Statistic 2874.4.3.2 EWMA Control Charts Based on the Wilcoxon Rank-sum Statistic 2904.5 Distribution-free Control Charts in Case U: Illustrative Examples 2934.5.1 Shewhart Control Charts 2934.5.2 CUSUM Control Charts 2954.5.3 EWMA Control Charts 3024.6 Effects of Parameter Estimation 3074.7 Size of Phase I Data 3074.8 Control Chart Enhancements 3084.8.1 Sensitivity and Runs-type Signaling Rules 308Appendix 4.1 Shewhart Control Charts 311Appendix 4.1.1 The Shewhart-Prec Control Chart 311Appendix 4.2 CUSUM Control Charts 312Appendix 4.2.1 The CUSUM-EX Control Chart 312Appendix 4.2.2 The CUSUM-rank Control Chart 314Appendix 4.3 EWMA Control Charts 317Appendix 4.3.1 The EWMA-SN Control Chart 317Appendix 4.3.2 The EWMA-SR Control Chart 318Appendix 4.3.3 The EWMA-EX Control Chart 319Appendix 4.3.4 The EWMA-rank Control Chart 3235 Miscellaneous Univariate Distribution-free (Nonparametric) Variables Control Charts 325Chapter Overview 3255.1 Introduction 3255.2 Other Univariate Distribution-free (Nonparametric) Variables Control Charts 3265.2.1 Phase I Control Charts 3265.2.1.1 Introduction 3265.2.1.2 Phase I Control Charts for Location 3315.2.2 Special Cases of Precedence Charts 3435.2.2.1 The Min Chart 3435.2.2.2 The CUMIN Chart 3465.2.3 Control Charts Based on Bootstrapping 3485.2.3.1 Methodology 3515.2.4 Change-point Models 3535.2.5 Some Adaptive Charts 3575.2.5.1 Introduction 3575.2.5.2 Variable Sampling Interval (VSI) and Variable Sample Size (VSS) Charts 3585.2.5.3 Other Adaptive Schemes 3595.2.5.4 Properties and Performance Measures of Adaptive Charts 3605.2.5.5 Adaptive Nonparametric Control Charts 362Appendix A Tables 369Appendix B Programmes 381References 413Index 425