Recent Advances in Lifetime and Reliability Models

This chapter presents mathematical and statistical background for understanding this book. Some results and formulae presented in this chapter are revisited in the next chapters. Initially, important survival analysis concepts are dened and issues with respect to inference and statistical methods. Subsequently, several special functions are presented. The chapter ends with some discussions on statistical elements that will be used in the rest of the book.

The exponentiation transform of cumulative distributions can furnish more exible models. Such procedure generates the exponentiated G (exp-G) distributions. This chapter presents a survey on the exp-G models and its mathematical properties. In particular, explicit expressions for the quantile function, ordinary and incomplete moments, generating function, income measures, order statistics and entropies are addressed.

The beta transformation gives a great variety of shapes which allow to model symmetric, skewed and bimodal shaped densities. Such procedure generates what we call the beta generalized (beta-G) family of distributions. This chapter presents a survey on the beta-G models and their mathematical properties. We present some explicit expressions for the ordinary and incomplete moments, probability weighted moments, cumulants and generating function, mean deviations, entropies and order statistics.

In this chapter, we provide theoretical essays about five special models in the beta family. For each model, its cdf, pdf and hrf have explicit forms, which can be utilized for determining some mathematical properties. Two applications are performed in order to illustrate the exibility of the densities under discussion.

This chapter addresses the Kumaraswamy's generalized (\Kw-G" for short) family of models. A physical motivation for the Kw-G family is presented and some of its special cases are discussed in detail. This family receives a baseline distribution as input and returns a new distribution with two additional parameters. The returned model is often more exible than the baseline one. Several structural properties are presented and discussed for the Kw-G family. Among them, useful expansions, mgf, moments and mean deviations. Additionally, estimation and generation procedures are presented.

In this chapter, three special cases of the Kumaraswamy generalized family of distributions are presented. Some mathematical characteristics are provided such as the moments and generating function. An expanded expression for the density function and some special cases are presented. For illustrative purposes, practical examples of the KwG models are reported by means of applications to empirical data.

This chapter presents the gamma generalized family of distributions proposed by Zofragos and Balakrishnan (2009). Several mathematical properties are provided such as representations for gamma-G density and cumulative functions, some generalized moments, quantile and generating functions and entropies. A bivariate generalization is presented. An application is performed in order to illustrate empirically the usefulness of this family.

In this chapter, we introduce a family of models dened by compounding two (a continuous and other discrete) distributions. The new family has as limiting case the adopted baseline distribution. The generated models are frequently more exible than the baseline distributions. Several mathematical properties such as moments, quantile and generating functions, among others, are provided. Further, the estimation procedure is approched by maximum likelihood. The potentiality of the family of models is illustrated by means of two applications to real data.