Description

# Nonparametric Statistical Process Control

## Authors: Chakraborti Subhabrata, Graham Marien

Language: Anglais## Subject for *Nonparametric Statistical Process Control *:

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## Description

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**A unique approach to understanding the foundations of statistical quality control with a focus on the latest developments in nonparametric control charting methodologies**

Statistical Process Control (SPC) methods have a long and successful history and have revolutionized many facets of industrial production around the world. This book addresses recent developments in statistical process control bringing the modern use of computers and simulations along with theory within the reach of both the researchers and practitioners. The emphasis is on the burgeoning field of nonparametric SPC (NSPC) and the many new methodologies developed by researchers worldwide that are revolutionizing SPC.

Over the last several years research in SPC, particularly on control charts, has seen phenomenal growth. Control charts are no longer confined to manufacturing and are now applied for process control and monitoring in a wide array of applications, from education, to environmental monitoring, to disease mapping, to crime prevention. This book addresses quality control methodology, especially control charts, from a statistician’s viewpoint, striking a careful balance between theory and practice. Although the focus is on the newer nonparametric control charts, the reader is first introduced to the main classes of the parametric control charts and the associated theory, so that the proper foundational background can be laid.

- Reviews basic SPC theory and terminology, the different types of control charts, control chart design, sample size, sampling frequency, control limits, and more
- Focuses on the distribution-free (nonparametric) charts for the cases in which the underlying process distribution is unknown
- Provides guidance on control chart selection, choosing control limits and other quality related matters, along with all relevant formulas and tables
- Uses computer simulations and graphics to illustrate concepts and explore the latest research in SPC

Offering a uniquely balanced presentation of both theory and practice, *Nonparametric Methods for Statistical Quality Control *is a vital resource for students, interested practitioners, researchers, and anyone with an appropriate background in statistics interested in learning about the foundations of SPC and latest developments in NSPC.

About the Authors

Preface

About the companion website

**1 Background/Review of Statistical Concepts**

Chapter Overview

1.1 Basic Probability

1.2 Random Variables and Their Distributions

1.3 Random Sample

1.4 Statistical Inference

1.5 Role of the Computer

**2 Basics of Statistical Process Control**

Chapter Overview

2.1 Basic Concepts

2.1.1 Types of Variability

2.1.2 The Control Chart

2.1.3 Construction of Control Charts

2.1.4 Variables and Attributes Control Charts

2.1.5 Sample Size or Subgroup Size

2.1.6 Rational Subgrouping

2.1.7 Nonparametric or Distribution-free

2.1.8 Monitoring Process Location and/or Process Scale

2.1.9 Case K and Case U

2.1.10 Control Charts and Hypothesis Testing

2.1.11 General Steps in Designing a Control Chart

2.1.12 Measures of Control Chart Performance

2.1.12.1 False Alarm Probability (*FAP*)

2.1.12.2 False Alarm Rate (*FAR*)

2.1.12.3 The Average Run-length (*ARL*)

2.1.12.4 Standard Deviation of Run-length (*SDRL*)

2.1.12.5 Percentiles of Run-length

2.1.12.6 Average Number of Samples to Signal (*ANSS*)

2.1.12.7 Average Number of Observations to Signal (*ANOS*)

2.1.12.8 Average Time to Signal (*ATS*)

2.1.12.9 Number of Individual Items Inspected (*I*)

2.1.13 Operating Characteristic Curves (*OC*-curves)

2.1.14 Design of Control Charts

2.1.14.1 Sample Size, Sampling Frequency, and Variable Sample Sizes

2.1.14.2 Variable Control Limits

2.1.14.3 Standardized Control Limits

2.1.15 Size of a Shift

2.1.16 Choice of Control Limits

2.1.16.1 *k*-sigma Limits

2.1.16.2 Probability Limits

**3 Parametric Univariate Variables Control Charts**

Chapter Overview

3.1 Introduction

3.2 Parametric Variables Control Charts in Case K

3.2.1 Shewhart Control Charts

3.2.2 CUSUM Control Charts

3.2.3 EWMA Control Charts

3.3 Types of Parametric Variables Charts in Case K: Illustrative Examples

3.3.1 Shewhart Control Charts

3.3.1.1 Shewhart Control Charts for Monitoring Process Mean

3.3.1.2 Shewhart Control Charts for Monitoring Process Variation

3.3.2 CUSUM Control Charts

3.3.3 EWMA Control Charts

3.4 Shewhart, EWMA, and CUSUM Charts: Which to Use When

3.5 Control Chart Enhancements

3.5.1 Sensitivity Rules

3.5.2 Runs-type Signaling Rules

3.5.2.1 Signaling Indicators

3.6 Run-length Distribution in the Specified Parameter Case (Case K)

3.6.1 Methods of Calculating the Run-length Distribution

3.6.1.1 The Exact Approach (for Shewhart and some Shewhart-type Charts)

3.6.1.2 The Markov Chain Approach

3.6.1.3 The Integral Equation Approach

3.6.1.4 The Computer Simulations (the Monte Carlo) Approach

3.7 Parameter Estimation Problem and Its Effects on the Control Chart Performance

3.8 Parametric Variables Control Charts in Case U

3.8.1 Shewhart Control Charts in Case U

3.8.1.1 Shewhart Control Charts for the Mean in Case U

3.8.1.2 Shewhart Control Charts for the Standard Deviation in Case U

3.8.2 CUSUM Chart for the Mean in Case U

3.8.3 EWMA Chart for the Mean in Case U

3.9 Types of Parametric Control Charts in Case U: Illustrative Examples

3.9.1 Charts for the Mean

3.9.2 Charts for the Standard Deviation

3.9.2.1 Using the Estimator *Sp *144

3.10 Run-length Distribution in the unknown Parameter Case (Case U)

3.10.1 Methods of Calculating the Run-length Distribution and Its Properties: The Conditioning/Unconditioning Method

3.10.1.1 The Shewhart Chart for the Mean in Case U

3.10.1.2 The Shewhart Chart for the Variance in Case U

3.10.1.3 The CUSUM Chart for the Mean in Case U

3.10.1.4 The EWMA Chart for the Mean in Case U

3.11 Control Chart Enhancements

3.11.1 Run-length Calculation for Runs-type Signaling Rules in Case U

3.12 Phase I Control Charts

3.12.1 Phase I *X*-chart

3.13 Size of Phase I Data

3.14 Robustness of Parametric Control Charts

Appendix 3.1 Some Derivations for the EWMA Control Chart

Appendix 3.2 Markov Chains

Appendix 3.3 Some Derivations for the Shewhart Dispersion Charts

**4 Nonparametric (Distribution-free) Univariate Variables Control Charts**

Chapter Overview

4.1 Introduction

4.2 Distribution-free Variables Control Charts in Case K

4.2.1 Shewhart Control Charts

4.2.1.1 Shewhart Control Charts Based on Signs

4.2.1.2 Shewhart Control Charts Based on Signed-ranks

4.2.2 CUSUM Control Charts

4.2.2.1 CUSUM Control Charts Based on Signs

4.2.2.2 A CUSUM Sign Control Chart with Runs-type Signaling Rules

4.2.2.3 Methods of Calculating the Run-length Distribution

4.2.2.4 CUSUM Control Charts Based on Signed-ranks

4.2.3 EWMA Control Charts

4.2.3.1 EWMA Control Charts Based on Signs

4.2.3.2 EWMA Control Charts Based on Signs with Runs-type Signaling Rules

4.2.3.3 Methods of Calculating the Run-length Distribution

4.2.3.4 EWMA Control Charts Based on Signed-ranks

4.2.3.5 An EWMA-SR control chart with runs-type signaling rules

4.2.3.6 Methods of Calculating the Run-length Distribution

4.3 Distribution-free Control Charts in Case K: Illustrative Examples

4.3.1 Shewhart Control Charts

4.3.2 CUSUM Control Charts

4.3.3 EWMA Control Charts

4.4 Distribution-free Variables Control Charts in Case U

4.4.1 Shewhart Control Charts

4.4.1.1 Shewhart Control Charts Based on the Precedence Statistic

4.4.1.2 Shewhart Control Charts Based on the Mann–Whitney Test Statistic

4.4.2 CUSUM Control Charts

4.4.2.1 CUSUM Control Charts Based on the Exceedance Statistic

4.4.2.2 CUSUM Control Charts Based on the Wilcoxon Rank-sum Statistic

4.4.3 EWMA Control Charts

4.4.3.1 EWMA Control Charts Based on the Exceedance Statistic

4.4.3.2 EWMA Control Charts Based on the Wilcoxon Rank-sum Statistic

4.5 Distribution-free Control Charts in Case U: Illustrative Examples

4.5.1 Shewhart Control Charts

4.5.2 CUSUM Control Charts

4.5.3 EWMA Control Charts

4.6 Effects of Parameter Estimation

4.7 Size of Phase I Data

4.8 Control Chart Enhancements

4.8.1 Sensitivity and Runs-type Signaling Rules

Appendix 4.1 Shewhart Control Charts

Appendix 4.1.1 The Shewhart-Prec Control Chart

Appendix 4.2 CUSUM Control Charts

Appendix 4.2.1 The CUSUM-EX Control Chart

Appendix 4.2.2 The CUSUM-rank Control Chart

Appendix 4.3 EWMA Control Charts

Appendix 4.3.1 The EWMA-SN Control Chart

Appendix 4.3.2 The EWMA-SR Control Chart

Appendix 4.3.3 The EWMA-EX Control Chart

Appendix 4.3.4 The EWMA-rank Control Chart

**5 Miscellaneous Univariate Distribution-free (Nonparametric) Variables Control Charts**

Chapter Overview

5.1 Introduction

5.2 Other Univariate Distribution-free (Nonparametric) Variables Control Charts

5.2.1 Phase I Control Charts

5.2.1.1 Introduction

5.2.1.2 Phase I Control Charts for Location

5.2.2 Special Cases of Precedence Charts

5.2.2.1 The Min Chart

5.2.2.2 The CUMIN Chart

5.2.3 Control Charts Based on Bootstrapping

5.2.3.1 Methodology

5.2.4 Change-point Models

5.2.5 Some Adaptive Charts

5.2.5.1 Introduction

5.2.5.2 Variable Sampling Interval (VSI) and Variable Sample Size (VSS) Charts

5.2.5.3 Other Adaptive Schemes

5.2.5.4 Properties and Performance Measures of Adaptive Charts

5.2.5.5 Adaptive Nonparametric Control Charts

Appendix A Tables

Appendix B Programmes

References

Index

**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.