Mathematicians and statisticians have made significant academic progress on the subject of distribution theory in the last two decades, and this area of study is becoming one of the main statistical tools for the analysis of lifetime (survival) data. In many ways, lifetime distributions are the common language of survival dialogue because the framework subsumes many statistical properties of interest, such as reliability, entropy and maximum likelihood.
Recent Advances in Lifetime and Reliability Models provides a comprehensive account of models and methods for lifetime models. Building from primary definitions such as density and hazard rate functions, this book presents readers a broad framework on distribution theory in survival analysis. This framework covers classical methods - such as the exponentiated distribution method – as well as recent models explaining lifetime distributions, such as the beta family and compounding models. Additionally, a detailed discussion of mathematical and statistical properties of each family, such as mixture representations, asymptotes, types of moments, order statistics, quantile functions, generating functions and estimation is presented in the book.
Key Features:
- Presents information about classical and modern lifetime methods
- Covers key properties of different models in detail
- Explores regression models for the beta generalized family of distributions
- Focuses information on both theoretical fundamentals and practical aspects of implementing different models
- Features examples relevant to business engineering and biomedical sciences
Recent Advances in Lifetime and Reliability Models will equip students, researchers and working professionals with the information to make extensive use of observational data in a variety of fields to create inferential models that make sense of lifetime data.
Table of Contents
1 Introduction
1.1 Primary Definitions
1.2 Censoring Kinds
1.2.1 First Censoring
1.2.2 Second Censoring
1.2.3 Parametric Estimation in Failure Data
1.3 Survival Regression Model
1.3.1 Cox Proportional Hazards Model
1.3.2 Accelerated Failure Time Model
1.4 Special Functions
1.5 Statistical Functions
2 Exponentiated Models
2.1 Introduction
2.2 Special Cases
2.2.1 The EE Distribution
2.2.2 The EW Distribution
2.3 Ordinary Moments
2.3.1 The EE Distribution
2.3.2 The EW Distribution
2.4 Other Moments
2.4.1 The EE Distribution
2.4.2 The EW Distribution
2.5 Income Measures
2.5.1 the EE Distribution
2.5.2 the EW Distribution
2.6 Order Statistics
2.6.1 For the EE Distribution
2.6.2 For the EW Distribution
2.7 Entropy
2.7.1 the EE Distribution
2.7.2 the EW Distribution
2.8 Estimation
2.8.1 For the EE Distribution
2.8.2 For the EW Distribution
2.9 Application
2.10 Conclusions
3 Beta Generalized Models
3.1 Introduction
3.2 Some Special Models
3.3 Quantile Function
3.4 Useful Expansions
3.5 Moments
3.6 Some Baseline PWMS
3.6.1 PWMS of the Beta Gamma
3.6.2 PWMS of the Beta Normal
3.6.3 PWMS of the Beta Beta
3.6.4 PWMS of the Beta Student T
3.7 PWMS Based on Quantiles
3.7.1 Moments of the Beta Gamma
3.7.2 Moments of the Beta Student T
3.7.3 Moments of the Beta Beta
3.8 Generating Function
3.9 Mean Deviations
3.10 Order Statistics
3.11 Reliability
3.12 Entropy
3.13 Estimation
3.14 Beta-G Regression Model
3.14.1 An Extended Weibull Distribution
3.14.2 The Log-Extended Weibull Distribution
3.14.3 The Log-Extended Weibull Regression Model
3.15 Conclusions
4 Special Generalized Beta Models
4.1 Beta Generalized Exponential
4.2 Beta Weibull
4.3 Beta Fr Echet
4.4 Beta Modified Weibull
4.5 Beta Birnbaum-Saunders
4.6 Applications
4.7 Conclusions
5 the Kumaraswamy's Generalized Family of Models
5.1 Introduction
5.2 Physical Motivation
5.3 Special Kw-G Distributions
5.3.1 Kw-Normal (Kwn)
5.3.2 Kw-Weibull (Kww)
5.3.3 Kw-Gamma (Kwg)
5.3.4 Kw-Gumbel (Kwgu)
5.3.5 Kw-Inverse Gaussian (Kwig)
5.3.6 Kw-Chen (Kwchen)
5.3.7 Kw-Xtg (Kwxtg)
5.3.8 Kw-Flexible Weibull (Kwfw)
5.4 Asymptotes and Shapes
5.5 Simulation
5.6 Useful Expansions
5.7 Moments
5.8 Generating Function
5.9 Mean Deviations
5.10 Relation with the Beta-G
5.11 Estimation
5.12 Conclusions
6 Special Kumaraswamy Generalized Models
6.1 Kumaraswamy Weibull
6.1.1 Linear Representation
6.1.2 Moments
6.1.3 Generating Function
6.1.4 Maximum Likelihood Estimation
6.1.5 Applications
6.2 Kumaraswamy Burr Xii
6.2.1 Expansion for the Density Function
6.2.2 Moments
6.2.3 Generating Function
6.2.4 Estimation
6.2.5 Simulation Studies
6.2.6 Applications
6.3 Kumaraswamy Gumbel
6.3.1 Distribution and Density Functions
6.3.2 Shapes
6.3.3 Moments
6.3.4 Generating Function
6.3.5 Maximum Likelihood Estimation
6.3.6 Bootstrap Re-Sampling Methods
6.3.7 a Bayesian Analysis
6.3.8 Application: Minimum Flow Data
6.4 Conclusions
7 the Gamma-G Family of Distributions
7.1 Introduction
7.2 Special Gamma-G Models
7.2.1 The Gamma-Weibull Distribution
7.2.2 The Gamma-Normal Distribution
7.2.3 The Gamma-Gumbel Distribution
7.2.4 The Gamma-Lognormal Distribution
7.2.5 The Gamma-Log-Logistic Distribution
7.3 Linear Representations
7.4 Asymptotes and Shapes
7.5 Quantile Function
7.6 Moments
7.7 Generating Function
7.8 Mean Deviations
7.9 Entropies
7.10 Order Statistics
7.11 Likelihood Estimation
7.12 A Bivariate Generalization
7.13 Application
7.14 The Risti C and Balakrishnan Family
7.15 Estimation and Application
7.16 Conclusions
8 Recent Compounding Models
8.1 Introduction
8.2 Quantile Function
8.3 Useful Expansions
8.4 Other Quantities
8.5 Order Statistics
8.6 Estimation
8.7 Applications
8.8 Conclusions
9 Conclusions and Recent Advances
Subject Index
Samples
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Author
- Abraão D. C. Nascimento
- Gauss M. Cordeiro
- Rodrigo B. Silva
