Antonio E. Gomes

The beta exponentiated Weibull distribution

The Weibull distribution is one of the most important distributions in reliability. For the first time, we introduce the beta exponentiated Weibull distribution which extends recent models by Lee et al. [Beta-Weibull distribution: some properties and applications to censored data, J. Mod. Appl. Statist. Meth. 6 (2007), pp. 173–186] and Barreto-Souza et al. [The beta generalized exponential distribution, J. Statist. Comput. Simul. 80 (2010), pp. 159–172]. The new distribution is an important competitive model to the Weibull, exponentiated exponential, exponentiated Weibull, beta exponential and beta Weibull distributions since it contains all these models as special cases. We demonstrate that the density of the new distribution can be expressed as a linear combination of Weibull densities. We provide the moments and two closed-form expressions for the moment-generating function. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The density of the order statistics can also be expressed as a linear combination of Weibull densities. We obtain the moments of the order statistics. The expected information matrix is derived. We define a log-beta exponentiated Weibull regression model to analyse censored data. The estimation of the parameters is approached by the method of maximum likelihood. The usefulness of the new distribution to analyse positive data is illustrated in two real data sets.

A new lifetime model: the Kumaraswamy generalized Rayleigh distribution

The generalized Rayleigh (GR) distribution [V.G. Vodǎ, Inferential procedures on a generalized Rayleigh variate, I, Appl. Math. 21 (1976), pp. 395–412; V.G. Vodǎ, Inferential procedures on a generalized Rayleigh variate, II, Appl. Math. 21 (1976), pp. 413–419] has been applied in several areas such as health, agriculture, biology and other sciences. For the first time, we propose the Kumaraswamy GR (KwGR) distribution for analysing lifetime data. The new density function can be expressed as a mixture of GR density functions. Explicit formulae are derived for some of its statistical quantities. The density function of the order statistics can be expressed as a mixture of GR density functions. We also propose a linear log-KwGR regression model for analysing data with real support to extend some known regression models. The estimation of parameters is approached by maximum likelihood. The importance of the new models is illustrated in two real data sets.

Two Extended Burr Models: Theory and Practice

We propose two new six-parameter distributions, called the McDonald Burr III and McDonald Burr XII models, which contain some recently published distributions as special models. We provide a comprehensive description of some of their mathematical properties with the hope that they will attract wider applications in lifetime analysis. The potentiality of both models to analyze positive data is illustrated by means of two real data sets.

Another extended Burr III model: some properties and applications

We introduce an extended Burr III distribution as an important model for problems in survival analysis and reliability. The new distribution can be expressed as a linear combination of Burr III distributions and then it has tractable properties for the ordinary and incomplete moments, generating and quantile functions, mean deviations and reliability. The density of its order statistics can be given in terms of an infinite linear combination of Burr III densities. The estimation of the model parameters is approached by maximum likelihood and the observed information matrix is derived. The proposed model is applied to a real data set to illustrate its potentiality.

An Item Response Theory approach to spatial cluster estimation and visualization

The scan statistic is widely used in spatial cluster detection applications of inhomogeneous Poisson processes. The most popular variant of the spatial scan is the circular scan. However, such approach has several limitations, in particular, the circular window is not suitable to make the correct description of irregularly shaped and/or unconnected clusters. Additionally, such methodology does not incorporate the tools needed for quantifying the uncertainty in the description of the most likely cluster in the analysis. In the present work we build upon the previously proposed methodology called intensity function a more efficient and accurate way of defining the uncertainty in the identification of spatial clusters using Item Response Theory ideas. Using simulated data we show that the proposed method can correctly identify primary, secondary and irregular clusters.

The Exponentiated G Poisson Model

We study a new family of distributions defined by the minimum of the Poisson random number of independent identically distributed random variables having a general exponentiated G distribution. Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics and their moments, reliability, and Shannon entropy are derived. Maximum likelihood estimation of the model parameters is investigated. Two special models of the new family are discussed. We perform an application to a real data set to show the potentiality of the proposed family.

Spatial Cluster Estimation and Visualization using Item Response Theory

In recent years Kulldorff’s circular scan statistic has become the most popular tool for detecting spatial clusters. However, window-imposed limitation may not be appropriate to detect the true cluster. To work around this problem we usually use complex tools that allow the detection of clusters with arbitrary format, but at the expense of an increase in computational effort. In this chapter we describe a methodology that assists the detection of unconnected and arbitrarily shaped clusters and that provides a measure of uncertainty in the design of such clusters.