Some extended Alpha models: properties and applications

Abstract

Generalizing distributions provide some advantages, allowing us to define new families, to extend well-known distributions and provide great flexibility in modeling real data, which can be applied in several fields. The Alpha distribution was studied for the first time to analyze tool wear problems by Katsev (1968) and Wager and Barash (1971). Salvia (1985) provided its characterization. In this thesis, we discuss the Alpha distribution, we present a simulation study to verify the performance of its maximum likelihood estimators and four real data sets are used to evaluate the Alpha model when compared to some distributions well-known in literature. Furthermore, we developed new distributions considering this model as the baseline distribution applied to Exponentiated class (Gompertz, 1825; Verhulst, 1838, 1845, 1847) and Kumaraswamy class, proposed by Cordeiro and de Castro (2011). We also propose a new family of distributions, called Exponentiated Generalized Exponentiated-Generated (EG-Exp-G), which is an extension of the exponentiated generalized class proposed by Cordeiro et al. (2013). Some new distributions are proposed as submodels of this family, including the EG-Exp-Alpha distribution. We study some mathematical properties, such as quantile function, moments, moment generating function, mean deviations and order statistics. In addition, we use the maximum likelihood method to estimate the parameters of the proposed models. We perform Monte Carlo simulation studies to analyze the asymptotic properties of the maximum likelihood estimators and we illustrate the flexibility of the new models through applications to real data set in order to show their competitiveness compared to well-known distributions in the literature.