Abstract We define two new flexible families of continuous distributions to fit real data by compoun-ding the Marshall–Olkin class and the power series distribution. These families are very competitive to the popular beta and Kumaraswamy generators. Their densities have linear representations of exponentiated densities. In fact, as the main properties of thirty five exponentiated distributions are well-known, we can easily obtain several properties of about three hundred fifty distributions using the references of this article and five special cases of the power series distribution. We provide a package implemented in R software that shows numerically the precision of one of the linear representations. This package is useful to calculate numerical values for some statistical measurements of the generated distributions. We estimate the parameters by maximum likelihood. We define a regression based on one of the two families. The usefulness of a generated distribution and the associated regression is proved empirically.

Abstract We investigate the use of the Probabilistic Incremental Programming Evolution (PIPE) algorithm as a tool to construct continuous cumulative distribution functions to model given data sets. The PIPE algorithm can generate several candidate functions to fit the empirical distribution of data. These candidates are generated by following a set of probability rules. The set of rules is then evolved over a number of iterations to generate better candidates regarding some optimality criteria. This approach rivals that of generated distribution, obtained by adding parameters to existing probability distributions. There are two main advantages for this method. The first is that it is possible to explicitly control the complexity of the candidate functions, by specifying which mathematical functions and operators can be used and how lengthy the mathematical expression of the candidate can be. The second advantage is that this approach deals with model selection and estimation at the same time. The overall performance in both simulated and real data was very satisfying. For the real data applications, the PIPE algorithm obtained better likelihoods for the data when compared to existing models, but with remarkably simpler mathematical expressions.

Abstract We study a five-parameter model called the Weibull Burr XII (WBXII) distribution, which extends several models, including new ones. This model is quite flexible in terms of the hazard function, which exhibits increasing, decreasing, upside-down bathtub, and bathtub shapes. Its density function allows different forms such as left-skewed, right-skewed, reversed-J, and bimodal. We aim to provide some general mathematical quantities for the proposed distribution, which can be useful to real data analysis. We develop a shiny application to provide interactive illustrations of the WBXII density and hazard functions. We estimate the model parameters using maximum likelihood and derive a profile log-likelihood for all members of the Weibull-G family. The survival analysis application reveals that the WBXII model is suitable to accommodate left-skewed tails, which are very common when the variable of interest is the time to failure of a product. The income application is related to player salaries within a professional sports league and it is peculiar because the mean of the player’s salaries is much higher than for most professions. Both applications illustrate that the new distribution provides much better fits than other models with the same and less number of parameters.

Abstract The transmuted family of distributions has been receiving increased attention over the last few years. In this paper, we generalize the Marshall-Olkin extended Lomax distribution using the quadratic rank transmutation map to obtain the transmuted Marshall-Olkin extended Lomax distribution. Several properties of the new distribution are discussed including the hazard rate function, ordinary and incomplete moments, characteristic function and order statistics. We provide an estimation procedure by the maximum likelihood method and a simulation study to assess the performance of the new distribution. We prove empirically the flexibility of the new model by means of an application to a real data set. It is superior to other three and four parameter lifetime distributions.

Abstract: We introduce a new class of continuous distributions called the generalized odd Lindley-G family. Four special models of the new family are provided. Some explicit expressions for the quantile and generating functions, ordinary and incomplete moments, order statistics and Rényi and Shannon entropies are derived. The maximum likelihood method is used for estimating the model parameters. The flexibility of the generated family is illustrated by means of two applications to real data sets.

Abstract: The normal distribution has a central place in distribution theory and statistics. We propose the log-odd normal generalized (LONG) family of distributions based on log-odds and obtain some of its mathematical properties including a useful linear representation for the new family. We investigate, as a special model, the log-odd normal power-Cauchy (LONPC) distribution. Some structural properties of LONPC distribution are obtained including quantile function, ordinary and incomplete moments, generating function and some asymptotics. We estimate the model parameters using the maximum likelihood method. The usefulness of the proposed family is proved empirically by means of a real air pollution data set.

Abstract: In this paper, we introduce a new three-parameter distribution by compounding the Nadarajah-Haghighi and geometric distributions, which can be interpreted as a truncated Marshall-Olkin extended Weibull. The compounding procedure is based on the work by Marshall and Olkin 1997. We prove that the new distribution can be obtained as a compound model with mixing exponential distribution. It can have decreasing, increasing, upside-down bathtub, bathtub-shaped, constant and decreasing-increasing-decreasing failure rate functions depending on the values of the parameters. Some mathematical properties of the new distribution are studied including moments and quantile function. The maximum likelihood estimation procedure is discussed and a particle swarm optimization algorithm is provided for estimating the model parameters. The flexibility of the new model is illustrated with an application to a real data set.

Abstract: We define a new lifetime model based on compounding the Lindley and Nadarajah-Haghighi distributions. The proposed distribution is very competitive to other lifetime models. Some of its mathematical properties are investigated including generating function, mean residual life, moments, Bonferroni and Lorenz curves and mean deviations. We discuss the estimation of the model parameters by maximum likelihood. We provide a simulation study and two applications to real data for illustrative purposes. We prove empirically that the new distribution yields good fits to both data sets, and it can be a useful alternative for other classical lifetime models.

Abstract We propose a new lifetime model called the exponentiated power generalized Weibull (EPGW) distribution, which is obtained from the exponentiated family applied to the power generalized Weibull (PGW) distribution. It can also be derived from a power transform on an exponentiated Nadarajah-Haghighi random variable. Since several structural properties of the PGW distribution have not been studied, they can be obtained from those of the EPGW distribution. The model is very flexible for modeling all common types of hazard rate functions. It is a very competitive model to the well-known Weibull, exponentiated exponential and exponentiated Weibull distributions, among others. We also give a physical motivation for the new distribution if the power parameter is an integer. Some of its mathematical properties are investigated. We discuss estimation of the model parameters by maximum likelihood and provide two applications to real data. A simulation study is performed in order to examine the accuracy of the maximum likelihood estimators of the model parameters.

Abstract We introduce a new three-parameter lifetime model called the Lindley Weibull distribution, which accommodates unimodal and bathtub, and a broad variety of monotone failure rates. We provide a comprehensive account of some of its mathematical properties including ordinary and incomplete moments, quantile and generating functions and order statistics. The new density function can be expressed as a linear combination of exponentiated Weibull densities. The maximum likelihood method is used to estimate the model parameters. We present simulation results to assess the performance of the maximum likelihood estimation. We prove empirically the importance and flexibility of the new distribution in modeling two data sets.

A insuficiência renal crônica é um grave problema clínico e de saúde pública no mundo inteiro. Recentemente, apesar de muitas pesquisas já realizadas para identificar os fatores relacionados à evolução dos pacientes renais crônicos, os resultados conflitantes entre diversos estudos sugerem a necessidade de pesquisas por região. Portanto, o estudo busca predizer e identificar os fatores associados à evolução dos pacientes com insuficiência renal crônica (IRC) na área metropolitana de Maringá, Estado do Paraná, Brasil, com base nos modelos aditivos generalizados para localização, escala e forma (GAMLSS). Os dados utilizados neste estudo foram coletados no Instituto do Rim de Maringá e incluem 177 pacientes (classificados com IRC, a maioria tratada no Sistema Único de Saúde) inclusos no programa de hemodiálise entre 1978 e 2010. Através dessa abordagem, concluímos que em outras regiões, gênero, indicação para transplante renal e anticorpos aos vírus das hepatites B e C são fatores significativos que afetam a sobrevivência esperada.

Renal insufficiency is a serious medical and public health problem worldwide. Recently, although many surveys have been developed to identify factors related to the lifetime of patients with renal insufficiency, controversial results from several studies suggest that researches should be conducted by region. Thus, in this study we aim to predict and identify factors associated with the lifetime of patients with chronic renal failure (CRF) in the metropolitan area of Maringá, Paraná State, Brazil, based on the generalized additive models for location, scale and shape (GAMLSS) framework. Data used in this study were collected from the Maringá Kidney Institute and comprehends 177 patients (classified with CRF and mostly being treated under the Brazilian Unified National Health System) enrolled in a hemodialysis program from 1978 up to 2010. By using this approach, we concluded that in other regions, gender, kidney transplant indicator, antibodies to hepatitis B and antibodies to hepatitis C are significant factors that affect the expected lifetime.

La insuficiencia renal representa un problema médico y de salud pública serio en todo el mundo. Recientemente, pese a los muchos estudios que se han desarrollado para identificar factores relacionados con la esperanza de vida de los pacientes con insuficiencia renal, los resultados controvertidos de algunos estudios sugieren que las investigaciones deberían realizarse por regiones. No obstante, en este trabajo pretendemos predecir e identificar los factores asociados con la esperanza de vida de pacientes con fallo renal crónico (FRC) en el área metropolitana de Maringá, Estado de Paraná, Brasil, basado en el marco de modelos aditivos generalizados para ubicación, escala y forma (GAMLSS por sus siglas en inglés). Los datos usados en este estudio fueron recogidos del Instituto del riñón de Maringá y comprende a 177 pacientes (clasificados con FRC y en su mayoría siendo tratados en el Sistema Único de Salud), inscritos en un programa de hemodiálisis desde 1978 hasta 2010. Usando este enfoque, concluimos que en otras regiones, género, indicador de trasplante de riñón, anticuerpos a la hepatitis B y anticuerpos a la hepatitis C son factores significativos que afectan a la esperanza de vida.

A insuficiência renal crônica é um grave problema clínico e de saúde pública no mundo inteiro. Recentemente, apesar de muitas pesquisas já realizadas para identificar os fatores relacionados à evolução dos pacientes renais crônicos, os resultados conflitantes entre diversos estudos sugerem a necessidade de pesquisas por região. Portanto, o estudo busca predizer e identificar os fatores associados à evolução dos pacientes com insuficiência renal crônica (IRC) na área metropolitana de Maringá, Estado do Paraná, Brasil, com base nos modelos aditivos generalizados para localização, escala e forma (GAMLSS). Os dados utilizados neste estudo foram coletados no Instituto do Rim de Maringá e incluem 177 pacientes (classificados com IRC, a maioria tratada no Sistema Único de Saúde) inclusos no programa de hemodiálise entre 1978 e 2010. Através dessa abordagem, concluímos que em outras regiões, gênero, indicação para transplante renal e anticorpos aos vírus das hepatites B e C são fatores significativos que afetam a sobrevivência esperada.

Renal insufficiency is a serious medical and public health problem worldwide. Recently, although many surveys have been developed to identify factors related to the lifetime of patients with renal insufficiency, controversial results from several studies suggest that researches should be conducted by region. Thus, in this study we aim to predict and identify factors associated with the lifetime of patients with chronic renal failure (CRF) in the metropolitan area of Maringá, Paraná State, Brazil, based on the generalized additive models for location, scale and shape (GAMLSS) framework. Data used in this study were collected from the Maringá Kidney Institute and comprehends 177 patients (classified with CRF and mostly being treated under the Brazilian Unified National Health System) enrolled in a hemodialysis program from 1978 up to 2010. By using this approach, we concluded that in other regions, gender, kidney transplant indicator, antibodies to hepatitis B and antibodies to hepatitis C are significant factors that affect the expected lifetime.

La insuficiencia renal representa un problema médico y de salud pública serio en todo el mundo. Recientemente, pese a los muchos estudios que se han desarrollado para identificar factores relacionados con la esperanza de vida de los pacientes con insuficiencia renal, los resultados controvertidos de algunos estudios sugieren que las investigaciones deberían realizarse por regiones. No obstante, en este trabajo pretendemos predecir e identificar los factores asociados con la esperanza de vida de pacientes con fallo renal crónico (FRC) en el área metropolitana de Maringá, Estado de Paraná, Brasil, basado en el marco de modelos aditivos generalizados para ubicación, escala y forma (GAMLSS por sus siglas en inglés). Los datos usados en este estudio fueron recogidos del Instituto del riñón de Maringá y comprende a 177 pacientes (clasificados con FRC y en su mayoría siendo tratados en el Sistema Único de Salud), inscritos en un programa de hemodiálisis desde 1978 hasta 2010. Usando este enfoque, concluimos que en otras regiones, género, indicador de trasplante de riñón, anticuerpos a la hepatitis B y anticuerpos a la hepatitis C son factores significativos que afectan a la esperanza de vida.

New generators are required to define wider distributions for modeling real data in survival analysis. To that end we introduce the four-parameter generalized beta-generated Lindley distribution. It has explicit expressions for the ordinary and incomplete moments, mean deviations, generating and quantile functions. We propose a maximum likelihood procedure to estimate the model parameters, which is assessed through a Monte Carlo simulation study. We also derive an additional estimation scheme by means of least square between percentiles. The usefulness of the proposed distribution to describe remission times of cancer patients is illustrated by means of an application to real data.

Abstract We propose a new four-parameter lifetime model, called the extended log-logistic distribution, to generalize the two-parameter log-logistic model. The new model is quite flexible to analyze positive data. We provide some mathematical properties including explicit expressions for the ordinary and incomplete moments, probability weighted moments, mean deviations, quantile function and entropy measure. The estimation of the model parameters is performed by maximum likelihood using the BFGS algorithm. The flexibility of the new model is illustrated by means of an application to a real data set. We hope that the new distribution will serve as an alternative model to other useful distributions for modeling positive real data in many areas.

Marshall and Olkin (1997)introduziram um método interessante de adicionar um parâmetro a uma distribuição especificada para obter uma família ampla de distribuições. Entretanto, eles não investigaram propriedades matemáticas gerais dessa família. Nós deduzimos para essa classe de distribuições, expansões gerais para a função densidade, expressões explícitas para os momentos e momentos das estatísticas de ordem. Vários modelos especiais são investigados. Nós discutimos a estimação dos parâmetros do modelo. Uma aplicação de um conjunto de dados reais é apresentada para fins ilustrativos.

Abstract Marshall and Olkin (1997) introduced an interesting method of adding a parameter to a well-established distribution. However, they did not investigate general mathematical properties of their family of distributions. We provide for this family of distributions general expansions for the density function, explicit expressions for the moments and moments of the order statistics. Several especial models are investigated. We discuss estimation of the model parameters. An application to a real data set is presented for illustrative purposes.