Preface

The theory of distributions with support in the positive real numbers has grown and matured in the last two decades, becoming one of the main statistical tools for the analysis of lifetime (survival) data. In fact, 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.

This book provides a comprehensive account of models and methods for lifetime models. Building from primary definitions such as density and hazard rate functions, the book presents the distribution theory in survival analysis. This framework covers classical methods, such as the exponentiated method, and also the most recent developments in lifetime distributions, such as the beta family and compounding models. Additionally, there is a detailed discussion of mathematical and statistical properties of each family, such as mixture representations, asymptotes, some types of moments, order statistics, quantile and generating functions and estimation. There is also a brief exploration of regression models for the beta generalized family of distributions. Throughout the text, we focus not only on the theoretical arguments but also on issues that arise in implementing the statistical methods in practice. The most recent parametric models in lifetime data analysis are covered without concentrating exclusively on any specific field of application, and most of the examples are drawn from engineering and biomedical sciences. It is important to emphasize that even with omission of some models, the great amount of models available has forced us to be very selective for inclusion in this work. To keep the book at a reasonable length we have had to omit or merely outline certain models that might have been included.

To help readers, lists of notation, terminology, and some probability distributions are given at the beginning of the book. All notational conventions are the same or very similar to the articles from which the models are based. Readers are assumed to have a good knowledge in advanced calculus. A course in real analysis is also recommended. If this book is used with a statistics textbook that does not include probability theory, then knowledge in probability theory is required.

The main five generators of new distributions are grouped into seven sections corresponding to those to which they give names. Chapter 1 contains introductory material with mathematical and statistical background for understanding this book. Chapter 2 deals with the exponentiated method. Explicit expressions for the quantile function, ordinary and incomplete moments, probability weighted moments, cumulants and generating functions are presented for the exponentiated-G family. Chapter 3 discusses the procedure that generates what we call the beta generalized family. Further, useful expansions and several statistical properties are presented. Chapter 4 provides theoretical essays about five special models in the beta family. For each model, its cumulative, density and hazard rate functions have explicit forms and important linear representations, which can be used to obtain some mathematical properties. Two applications are performed in order to illustrate the flexibility of the densities under discussion. Chapter 5 introduces the Kumaraswamy generalized family. In addition, several structural properties are presented and discussed for this family. Among them, useful expansions, quantile and generating functions, moments and mean deviations. Additionally, estimation and generation procedures are investigated. Chapter 6 presents three special cases of the Kumaraswamy generalized family. Some mathematical properties are provided such as the moments and generating function. Useful expansions for the density function and some special cases are presented. Chapter 7 discusses the gamma generalized family proposed by Zografos and Balakrishnan (2009). Several mathematical properties are provided such as expansions for the density and cumulative functions, quantile function, moments, generating function and entropies. A bivariate generalization is presented. Chapter 8 introduces a family of models defined by compounding two (a continuous and other discrete) distributions. We provide important mathematical properties such as moments and order statistics. We discuss the estimation of the model parameters by maximum likelihood and prove empirically the potentiality of the family by means of two applications to real data.