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Weibull Models

Weibull Models

D. N. Prabhakar Murthy, Min Xie, Renyan Jiang

ISBN: 978-0-471-47326-8

Dec 2003

408 pages

Description

A comprehensive perspective on Weibull models

The literature on Weibull models is vast, disjointed, and scattered across many different journals. Weibull Models is a comprehensive guide that integrates all the different facets of Weibull models in a single volume.

This book will be of great help to practitioners in reliability and other disciplines in the context of modeling data sets using Weibull models. For researchers interested in these modeling techniques, exercises at the end of each chapter define potential topics for future research.

Organized into seven distinct parts, Weibull Models:

  • Covers model analysis, parameter estimation, model validation, and application
  • Serves as both a handbook and a research monograph. As a handbook, it classifies the different models and presents their properties. As a research monograph, it unifies the literature and presents the results in an integrated manner
  • Intertwines theory and application
  • Focuses on model identification prior to model parameter estimation
  • Discusses the usefulness of the Weibull Probability plot (WPP) in the model selection to model a given data set
  • Highlights the use of Weibull models in reliability theory

Filled with in-depth analysis, Weibull Models pulls together the most relevant information on this topic to give everyone from reliability engineers to applied statisticians involved with reliability and survival analysis a clear look at what Weibull models can offer.

Preface xiii

PART A OVERVIEW 1

Chapter 1 Overview 3

1.1 Introduction 3

1.2 Illustrative Problems 5

1.3 Empirical Modeling Methodology 7

1.4 Weibull Models 9

1.5 Weibull Model Selection 11

1.6 Applications of Weibull Models 12

1.7 Outline of the Book 15

1.8 Notes 16

Exercises 16

Chapter 2 Taxonomy for Weibull Models 18

2.1 Introduction 18

2.2 Taxonomy for Weibull Models 18

2.3 Type I Models: Transformation of Weibull Variable 21

2.4 Type II Models: Modification/Generalization of Weibull Distribution 23

2.5 Type III Models: Models Involving Two or More Distributions 28

2.6 Type IV Models: Weibull Models with Varying Parameters 30

2.7 Type V Models: Discrete Weibull Models 33

2.8 Type VI Models: Multivariate Weibull Models 34

2.9 Type VII Models: Stochastic Point Process Models 37

Exercises 39

PART B BASIC WEIBULL MODEL 43

Chapter 3 Model Analysis 45

3.1 Introduction 45

3.2 Basic Concepts 45

3.3 Standard Weibull Model 50

3.4 Three-Parameter Weibull Model 54

3.5 Notes 55

Exercises 56

Chapter 4 Parameter Estimation 58

4.1 Introduction 58

4.2 Data Types 58

4.3 Estimation: An Overview 60

4.4 Estimation Methods and Estimators 61

4.5 Two-Parameter Weibull Model: Graphical Methods 65

4.6 Standard Weibull Model: Statistical Methods 67

4.7 Three-Parameter Weibull Model 74

Exercises 82

Chapter 5 Model Selection and Validation 85

5.1 Introduction 85

5.2 Graphical Methods 86

5.3 Goodness-of-Fit Tests 89

5.4 Model Discrimination 93

5.5 Model Validation 94

5.6 Two-Parameter Weibull Model 95

5.7 Three-Parameter Weibull Model 99

Exercises 100

PART C TYPES I AND II MODELS 103

Chapter 6 Type I Weibull Models 105

6.1 Introduction 105

6.2 Model I(a)-3: Reflected Weibull Distribution 106

6.3 Model I(a)-4: Double Weibull Distribution 108

6.4 Model I(b)-1: Power Law Transformation 109

6.5 Model I(b)-2: Log Weibull Transformation 111

6.6 Model I(b)-3: Inverse Weibull Distribution 114

Exercises 119

Chapter 7 Type II Weibull Models 121

7.1 Introduction 121

7.2 Model II(a)-1: Pseudo-Weibull Distribution 122

7.3 Model II(a)-2: Stacy–Mihram Model 124

7.4 Model II(b)-1: Extended Weibull Distribution 125

7.5 Model II(b)-2: Exponentiated Weibull Distribution 127

7.6 Model II(b)-3: Modified Weibull Distribution 134

7.7 Models II(b)4–6: Generalized Weibull Family 138

7.8 Model II(b)-7: Three-Parameter Generalized Gamma 140

7.9 Model II(b)-8: Extended Generalized Gamma 143

7.10 Models II(b)9–10: Four- and Five-Parameter Weibulls 145

7.11 Model II(b)-11: Truncated Weibull Distribution 146

7.12 Model II(b)-12: Slymen–Lachenbruch Distributions 148

7.13 Model II(b)-13: Weibull Extension 151

Exercises 154

PART D TYPE III MODELS 157

Chapter 8 Type III(a) Weibull Models 159

8.1 Introduction 159

8.2 Model III(a)-1: Weibull Mixture Model 160

8.3 Model III(a)-2: Inverse Weibull Mixture Model 176

8.4 Model III(a)-3: Hybrid Weibull Mixture Models 179

8.5 Notes 179

Exercises 180

Chapter 9 Type III(b) Weibull Models 182

9.1 Introduction 182

9.2 Model III(b)-1: Weibull Competing Risk Model 183

9.3 Model III(b)-2: Inverse Weibull Competing Risk Model 190

9.4 Model III(b)-3: Hybrid Weibull Competing Risk Model 191

9.5 Model III(b)-4: Generalized Competing Risk Model 192

Exercises 195

Chapter 10 Type III(c) Weibull Models 197

10.1 Introduction 197

10.2 Model III(c)-1: Multiplicative Weibull Model 198

10.3 Model III(c)-2: Inverse Weibull Multiplicative Model 203

Exercises 206

Chapter 11 Type III(d) Weibull Models 208

11.1 Introduction 208

11.2 Analysis of Weibull Sectional Models 210

11.3 Parameter Estimation 216

11.4 Modeling Data Set 219

11.5 Applications 219

Exercises 220

PART E TYPES IV TO VII MODELS 221

Chapter 12 Type IV Weibull Models 223

12.1 Introduction 223

12.2 Type IV(a) Models 224

12.3 Type IV(b) Models: Accelerated Failure Time (AFT) Models 225

12.4 Type IV(c) Models: Proportional Hazard (PH) Models 229

12.5 Model IV(d)-1 231

12.6 Type IV(e) Models: Random Parameters 232

12.7 Bayesian Approach to Parameter Estimation 236

Exercises 236

Chapter 13 Type V Weibull Models 238

13.1 Introduction 238

13.2 Concepts and Notation 238

13.3 Model V-1 239

13.4 Model V-2 242

13.5 Model V-3 243

13.6 Model V-4 244

Exercises 245

Chapter 14 Type VI Weibull Models (Multivariate Models) 247

14.1 Introduction 247

14.2 Some Preliminaries and Model Classification 248

14.3 Bivariate Models 250

14.4 Multivariate Models 256

14.5 Other Models 258

Exercises 258

Chapter 15 Type VII Weibull Models 261

15.1 Introduction 261

15.2 Model Formulations 261

15.3 Model VII(a)-1: Power Law Process 265

15.4 Model VII(a)-2: Modulated Power Law Process 272

15.5 Model VII(a)-3: Proportional Intensity Model 273

15.6 Model VII(b)-1: Ordinary Weibull Renewal Process 274

15.7 Model VII(b)-2: Delayed Renewal Process 277

15.8 Model VII(b)-3: Alternating Renewal Process 278

15.9 Model VII(c): Power Law–Weibull Renewal Process 278

Exercises 278

PART F WEIBULL MODELING OF DATA 281

Chapter 16 Weibull Modeling of Data 283

16.1 Introduction 283

16.2 Data-Related Issues 284

16.3 Preliminary Model Selection and Parameter Estimation 285

16.4 Final Model Selection Parameter Estimation and Model Validation 287

16.5 Case Studies 290

16.6 Conclusions 299

Exercises 299

PART G APPLICATIONS IN RELIABILITY 301

Chapter 17 Modeling Product Failures 303

17.1 Introduction 303

17.2 Some Basic Concepts 304

17.3 Product Structure 306

17.4 Modeling Failures 306

17.5 Component-Level Modeling (Black-Box Approach) 306

17.6 Component-Level Modeling (White-Box Approach) 308

17.7 Component-Level Modeling (Gray-Box Approach) 312

17.8 System-Level Modeling (Black-Box Approach) 313

17.9 System-Level Modeling (White-Box Approach) 316

Chapter 18 Product Reliability and Weibull Models 324

18.1 Introduction 324

18.2 Premanufacturing Phase 325

18.3 Manufacturing Phase 332

18.4 Postsale Phase 336

18.5 Decision Models Involving Weibull Failure Models 341

References 348

Index 377

""…valuable to readers seeking an overview of Weibull models and…a valuable contribution to libraries…"" (Journal of the American Statistical Association, September 2005)

“…may serve as a handbook in research and teaching…” (Zentralblatt Math, Vol.1047, No.22, 2004)

""The book is generally well written and easy to read. It could serve as a useful reference to practitioners…and researchers."" (Technometrics, November 2004)

""The literature of Weibull models is vast, disjointed, and scattered across many different journals. There are a couple of books devoted solely to the Weibull distribution, but these are oriented toward training and/or consulting purposes. There is no book that deals with the different Weibull models in an integrated manner. This book fills that gap."" (Mathematical Reviews, Issue 2004h)