Weibull Models

D. N. PRABHAKAR MURTHY, PhD, is a Professor of Engineering andOperations Management at the University of Queensland in Brisbane,Australia. He received his PhD in applied mathematics from HarvardUniversity. MIN XIE, PhD, is an Associate Professor of Industrial andSystems Engineering at the National University of Singapore in KentRidge Crescent, Singapore. He received his PhD in qualitytechnology from Linkoping University in Linkoping, Sweden. RENYAN JIANG, PhD, is a Professor of Engineering at the ChangshaUniversity of Science and Technology and is also affiliated withthe Department of Mechanical Industrial Engineering at theUniversity of Toronto in Toronto, Ontario, Canada. He received hisPhD in mechanical engineering from the University ofQueensland.

"...valuable to readers seeking an overview of Weibull modelsand...a valuable contribution to libraries..." (Journalof the American Statistical Association, September 2005) "...may serve as a handbook in research andteaching..." (Zentralblatt Math, Vol.1047, No.22,2004) "The book is generally well written and easy to read. It couldserve as a useful reference to practitioners...andresearchers." (Technometrics, November 2004) "The literature of Weibull models is vast, disjointed, andscattered across many different journals. There are a couple ofbooks devoted solely to the Weibull distribution, but these areoriented toward training and/or consulting purposes. There is nobook that deals with the different Weibull models in an integratedmanner. This book fills that gap." (Mathematical Reviews,Issue 2004h)

• Preface xiiiPART A OVERVIEW 1Chapter 1 Overview 31.1 Introduction 31.2 Illustrative Problems 51.3 Empirical Modeling Methodology 71.4 Weibull Models 91.5 Weibull Model Selection 111.6 Applications of Weibull Models 121.7 Outline of the Book 151.8 Notes 16Exercises 16Chapter 2 Taxonomy for Weibull Models 182.1 Introduction 182.2 Taxonomy for Weibull Models 182.3 Type I Models: Transformation of Weibull Variable 212.4 Type II Models: Modification/Generalization of Weibull Distribution 232.5 Type III Models: Models Involving Two or More Distributions 282.6 Type IV Models: Weibull Models with Varying Parameters 302.7 Type V Models: Discrete Weibull Models 332.8 Type VI Models: Multivariate Weibull Models 342.9 Type VII Models: Stochastic Point Process Models 37Exercises 39PART B BASIC WEIBULL MODEL 43Chapter 3 Model Analysis 453.1 Introduction 453.2 Basic Concepts 453.3 Standard Weibull Model 503.4 Three-Parameter Weibull Model 543.5 Notes 55Exercises 56Chapter 4 Parameter Estimation 584.1 Introduction 584.2 Data Types 584.3 Estimation: An Overview 604.4 Estimation Methods and Estimators 614.5 Two-Parameter Weibull Model: Graphical Methods 654.6 Standard Weibull Model: Statistical Methods 674.7 Three-Parameter Weibull Model 74Exercises 82Chapter 5 Model Selection and Validation 855.1 Introduction 855.2 Graphical Methods 865.3 Goodness-of-Fit Tests 895.4 Model Discrimination 935.5 Model Validation 945.6 Two-Parameter Weibull Model 955.7 Three-Parameter Weibull Model 99Exercises 100PART C TYPES I AND II MODELS 103Chapter 6 Type I Weibull Models 1056.1 Introduction 1056.2 Model I(a)-3: Reflected Weibull Distribution 1066.3 Model I(a)-4: Double Weibull Distribution 1086.4 Model I(b)-1: Power Law Transformation 1096.5 Model I(b)-2: Log Weibull Transformation 1116.6 Model I(b)-3: Inverse Weibull Distribution 114Exercises 119Chapter 7 Type II Weibull Models 1217.1 Introduction 1217.2 Model II(a)-1: Pseudo-Weibull Distribution 1227.3 Model II(a)-2: Stacy–Mihram Model 1247.4 Model II(b)-1: Extended Weibull Distribution 1257.5 Model II(b)-2: Exponentiated Weibull Distribution 1277.6 Model II(b)-3: Modified Weibull Distribution 1347.7 Models II(b)4–6: Generalized Weibull Family 1387.8 Model II(b)-7: Three-Parameter Generalized Gamma 1407.9 Model II(b)-8: Extended Generalized Gamma 1437.10 Models II(b)9–10: Four- and Five-Parameter Weibulls 1457.11 Model II(b)-11: Truncated Weibull Distribution 1467.12 Model II(b)-12: Slymen–Lachenbruch Distributions 1487.13 Model II(b)-13: Weibull Extension 151Exercises 154PART D TYPE III MODELS 157Chapter 8 Type III(a) Weibull Models 1598.1 Introduction 1598.2 Model III(a)-1: Weibull Mixture Model 1608.3 Model III(a)-2: Inverse Weibull Mixture Model 1768.4 Model III(a)-3: Hybrid Weibull Mixture Models 1798.5 Notes 179Exercises 180Chapter 9 Type III(b) Weibull Models 1829.1 Introduction 1829.2 Model III(b)-1: Weibull Competing Risk Model 1839.3 Model III(b)-2: Inverse Weibull Competing Risk Model 1909.4 Model III(b)-3: Hybrid Weibull Competing Risk Model 1919.5 Model III(b)-4: Generalized Competing Risk Model 192Exercises 195Chapter 10 Type III(c) Weibull Models 19710.1 Introduction 19710.2 Model III(c)-1: Multiplicative Weibull Model 19810.3 Model III(c)-2: Inverse Weibull Multiplicative Model 203Exercises 206Chapter 11 Type III(d) Weibull Models 20811.1 Introduction 20811.2 Analysis of Weibull Sectional Models 21011.3 Parameter Estimation 21611.4 Modeling Data Set 21911.5 Applications 219Exercises 220PART E TYPES IV TO VII MODELS 221Chapter 12 Type IV Weibull Models 22312.1 Introduction 22312.2 Type IV(a) Models 22412.3 Type IV(b) Models: Accelerated Failure Time (AFT) Models 22512.4 Type IV(c) Models: Proportional Hazard (PH) Models 22912.5 Model IV(d)-1 23112.6 Type IV(e) Models: Random Parameters 23212.7 Bayesian Approach to Parameter Estimation 236Exercises 236Chapter 13 Type V Weibull Models 23813.1 Introduction 23813.2 Concepts and Notation 23813.3 Model V-1 23913.4 Model V-2 24213.5 Model V-3 24313.6 Model V-4 244Exercises 245Chapter 14 Type VI Weibull Models (Multivariate Models) 24714.1 Introduction 24714.2 Some Preliminaries and Model Classification 24814.3 Bivariate Models 25014.4 Multivariate Models 25614.5 Other Models 258Exercises 258Chapter 15 Type VII Weibull Models 26115.1 Introduction 26115.2 Model Formulations 26115.3 Model VII(a)-1: Power Law Process 26515.4 Model VII(a)-2: Modulated Power Law Process 27215.5 Model VII(a)-3: Proportional Intensity Model 27315.6 Model VII(b)-1: Ordinary Weibull Renewal Process 27415.7 Model VII(b)-2: Delayed Renewal Process 27715.8 Model VII(b)-3: Alternating Renewal Process 27815.9 Model VII(c): Power Law–Weibull Renewal Process 278Exercises 278PART F WEIBULL MODELING OF DATA 281Chapter 16 Weibull Modeling of Data 28316.1 Introduction 28316.2 Data-Related Issues 28416.3 Preliminary Model Selection and Parameter Estimation 28516.4 Final Model Selection Parameter Estimation and Model Validation 28716.5 Case Studies 29016.6 Conclusions 299Exercises 299PART G APPLICATIONS IN RELIABILITY 301Chapter 17 Modeling Product Failures 30317.1 Introduction 30317.2 Some Basic Concepts 30417.3 Product Structure 30617.4 Modeling Failures 30617.5 Component-Level Modeling (Black-Box Approach) 30617.6 Component-Level Modeling (White-Box Approach) 30817.7 Component-Level Modeling (Gray-Box Approach) 31217.8 System-Level Modeling (Black-Box Approach) 31317.9 System-Level Modeling (White-Box Approach) 316Chapter 18 Product Reliability and Weibull Models 32418.1 Introduction 32418.2 Premanufacturing Phase 32518.3 Manufacturing Phase 33218.4 Postsale Phase 33618.5 Decision Models Involving Weibull Failure Models 341References 348Index 377