Rodrigo B. Silva

The Weibull-G Family of Probability Distributions

The Weibull distribution is the most important distribution for problems in reliability. We study some mathematical properties of the new wider Weibull-G family of distributions. Some special models in the new family are discussed. The properties derived hold to any distribution in this family. We obtain general explicit expressions for the quantile function, ordinary and incomplete moments, generating function and order statistics. We discuss the estimation of the model parameters by maximum likelihood and illustrate the potentiality of the extended family with two applications to real data.

A new distribution with decreasing, increasing and upside-down bathtub failure rate

The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields. For the first time, the so-called generalized exponential geometric distribution is introduced. The new distribution can have a decreasing, increasing and upside-down bathtub failure rate function depending on its parameters. It includes the exponential geometric (Adamidis and Loukas, 1998), the generalized exponential (Gupta and Kundu, 1999) and the extended exponential geometric (Adamidis et al., 2005) distributions as special sub-models. We provide a comprehensive mathematical treatment of the distribution and derive expressions for the moment generating function, characteristic function and rth moment. An expression for Renyi entropy is obtained, and estimation of the stress-strength parameter is discussed. We estimate the parameters by maximum likelihood and obtain the Fisher information matrix. The flexibility of the new model is illustrated in an application to a real data set.

The compound class of extended Weibull power series distributions

We introduce a general method for obtaining more flexible new distributions by compounding the extended Weibull and power series distributions. The compounding procedure follows the same set-up carried out by Adamidis and Loukas (1998) and defines 68 new sub-models. The new class of generated distributions includes some well-known mixing distributions, such as the Weibull power series (Morais and Barreto-Souza, 2011) and exponential power series (Chahkandi and Ganjali, 2009) distributions. Some mathematical properties of the new class are studied including moments and the generating function. We provide the density function of the order statistics and their moments. The method of maximum likelihood is used for estimating the model parameters. Special distributions are investigated. We illustrate the usefulness of the new distributions by means of two applications to real data sets.

The Kumaraswamy Pareto distribution

The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields. For the first time, the called Kumaraswamy Pareto distribution is introduced and studied. The new distribution can have a decreasing and upside-down bathtub failure rate function depending on the values of its parameters. It includes as special sub-models the Pareto and exponentiated Pareto (Gupta et al. [12]) distributions. Some structural properties of the proposed distribution are studied including explicit expressions for the moments and generating function. We provide the density function of the order statistics and obtain their moments. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is derived. A real data set is used to compare the new model with widely known distributions.

The Burr XII power series distributions: A new compounding family

Generalizing lifetime distributions is always precious for applied statisticians. In this paper, we introduce a new family of distributions by compounding the Burr XII and power series distributions. The compounding procedure follows the key idea by Adamidis and Loukas (Statist. Probab. Lett. 39 (1998) 35–42) or, more generally, by Chahkandi and Ganjali (Comput. Statist. Data Anal. 53 (2009) 4433–4440) and Morais and Barreto-Souza (Comput. Statist. Data Anal. 55 (2011) 1410–1425). The proposed family includes as a basic exemplar the Burr XII distribution. We provide some mathematical properties including moments, quantile and generating functions, order statistics and their moments, Kullback–Leibler divergence and Shannon entropy. The estimation of the model parameters is performed by maximum likelihood and the inference under large sample. Two special models of the new family are investigated in details. We illustrate the potential of the new family by means of two applications to real data. It provides better fits to these data than other important lifetime models available in the literature.

A new class of fatigue life distributions

In this paper, we introduce the Birnbaum–Saunders () power series class of distributions which is obtained by compounding and power series distributions. The new class of distributions has as a particular case the two-parameter distribution. The hazard rate function of the proposed class can be increasing and upside-down bathtub shaped. We provide important mathematical properties such as moments, order statistics, estimation of the parameters and inference for large sample. Three special cases of the new class are investigated with some details. We illustrate the usefulness of the new distributions by means of two applications to real data sets.

A likelihood ratio test to discriminate exponential–Poisson and gamma distributions

The exponential–Poisson (EP) distribution with scale and shape parameters β>0 and λ∈ℝ, respectively, is a lifetime distribution obtained by mixing exponential and zero-truncated Poisson models. The EP distribution has been a good alternative to the gamma distribution for modelling lifetime, reliability and time intervals of successive natural disasters. Both EP and gamma distributions have some similarities and properties in common, for example, their densities may be strictly decreasing or unimodal, and their hazard rate functions may be decreasing, increasing or constant depending on their shape parameters. On the other hand, the EP distribution has several interesting applications based on stochastic representations involving maximum and minimum of iid exponential variables (with random sample size) which make it of distinguishable scientific importance from the gamma distribution. Given the similarities and different scientific relevance between these models, one question of interest is how to discriminate them. With this in mind, we propose a likelihood ratio test based on Coxs statistic to discriminate the EP and gamma distributions. The asymptotic distribution of the normalized logarithm of the ratio of the maximized likelihoods under two null hypotheses – data come from EP or gamma distributions – is provided. With this, we obtain the probabilities of correct selection. Hence, we propose to choose the model that maximizes the probability of correct selection (PCS). We also determinate the minimum sample size required to discriminate the EP and gamma distributions when the PCS and a given tolerance level based on some distance are before stated. A simulation study to evaluate the accuracy of the asymptotic probabilities of correct selection is also presented. The paper is motivated by two applications to real data sets.

The gamma extended Weibull family of distributions

We introduce a new family of distributions called the gamma extended Weibull family. The proposed family includes several well-known models as special cases and defines at least seventeen new special models. Structural properties of this family are studied. Additionally, the maximum likelihood method for estimating the model parameters is discussed. An application to real data illustrates the usefulness of the new family. The results provide evidence that the proposed family outperforms other classes of lifetime models.

Fréchet and Inverse Gamma Distributions: Correct Selection and Minimum Sample Size to Discriminate Them

In this article, we propose a likelihood ratio test to discriminate between the inverse gamma and Fréchet distributions. The asymptotic distribution of the logarithm of the ratio of the maximized likelihoods under the null hypothesis is provided for both cases; the data come from the Fréchet and inverse gamma models. We also provide the minimum sample size required to discriminate between the two distributions when the probability of correct selection is fixed. A simulation study is presented in order to compare the empirical and asymptotic probabilities of the correct selection. The article is motivated by two applications to real data sets.