The exponentiated generalized gamma distribution with application to lifetime data

A four-parameter extension of the generalized gamma distribution capable of modelling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in lifetime data analysis and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma and generalized Rayleigh, among others. We derive two infinite sum representations for its moments. We calculate the density of the order statistics and two expansions for their moments. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is obtained. Finally, a real data set from the medical area is analysed.     

The log-exponentiated generalized gamma regression model for censored data

For the first time, we introduce a generalized form of the exponentiated generalized gamma distribution [Cordeiro et al. The exponentiated generalized gamma distribution with application to lifetime data, J. Statist. Comput. Simul. 81 (2011), pp. 827–842.] that is the baseline for the log-exponentiated generalized gamma regression model. The new distribution can accommodate increasing, decreasing, bathtub- and unimodal-shaped hazard functions. A second advantage is that it includes classical distributions reported in the lifetime literature as special cases. We obtain explicit expressions for the moments of the baseline distribution of the new regression model. The proposed model can be applied to censored data since it includes as sub-models several widely known regression models. It therefore can be used more effectively in the analysis of survival data. We obtain maximum likelihood estimates for the model parameters by considering censored data. We show that our extended regression model is very useful by means of two applications to real data.     

A new four-parameter lifetime distribution

Generalizing lifetime distributions is always precious for applied statisticians. In this paper, we introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley geometric (EPLG) distribution, obtained by compounding EPL and geometric distributions. The new distribution arises in a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the maximum lifetime value among all risks. The distribution exhibits decreasing, increasing, unimodal and bathtub-shaped hazard rate functions, depending on its parameters. It contains several lifetime distributions as particular cases: EPL, new generalized Lindley, generalized Lindley, power Lindley and Lindley geometric distributions. We derive several properties of the new distribution such as closed-form expressions for the density, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Simulation studies are also provided. Finally, two real data applications are given for showing the flexibility and potentiality of the new distribution.     

The Kumaraswamy modified Weibull distribution: theory and applications

A five-parameter extension of the Weibull distribution capable of modelling a bathtub-shaped hazard rate function is introduced and studied. The beauty and importance of the new distribution lies in its ability to model both monotone and non-monotone failure rates that are quite common in lifetime problems and reliability. The proposed distribution has a number of well-known lifetime distributions as special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull (MW) distributions, among others. We obtain quantile and generating functions, mean deviations, Bonferroni and Lorenz curves and reliability. We provide explicit expressions for the density function of the order statistics and their moments. For the first time, we define the log-Kumaraswamy MW regression model to analyse censored data. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is determined. Two applications illustrate the potentiality of the proposed distribution.     

Exponentiated power Lindley power series class of distributions: Theory and applications

ABSTRACT

We introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley power series (EPLPS) distribution. The new distribution arises on a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the minimum lifetime value among all risks. The distribution exhibits a variety of bathtub-shaped hazard rate functions. It contains as particular cases several lifetime distributions. Various properties of the distribution are investigated including closed-form expressions for the density function, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Expressions for the Rényi and Shannon entropies are also given. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Finally, two data applications are given showing flexibility and potentiality of the EPLPS distribution.     

A generalization of discrete Weibull distribution

In this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.     

Generalized exponential distribution: Existing results and some recent developments

Mudholkar and Srivastava [1993. Exponentiated Weibull family for analyzing bathtub failure data. IEEE Trans. Reliability 42, 299–302] introduced three-parameter exponentiated Weibull distribution. Two-parameter exponentiated exponential or generalized exponential distribution is a particular member of the exponentiated Weibull distribution. Generalized exponential distribution has a right skewed unimodal density function and monotone hazard function similar to the density functions and hazard functions of the gamma and Weibull distributions. It is observed that it can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. The genesis of this model, several properties, different estimation procedures and their properties, estimation of the stress-strength parameter, closeness of this distribution to some of the well-known distribution functions are discussed in this article.     

The exponentiated-log-logistic geometric distribution: Dual activation

ABSTRACT

The log-logistic distribution is commonly used to model lifetime data. We propose a wider distribution, named the exponentiated log-logistic geometric distribution, based on a double activation approach. We obtain the quantile function, ordinary moments, and generating function. The method of maximum likelihood is used to estimate the model parameters. We propose a new extended regression model based on the logarithm of the exponentiated log-logistic geometric distribution. This regression model can be very useful in the analysis of real data and could provide better fits than other special regression models. The potentiality of the new models is illustrated by means of two applications to real lifetime data sets.     

The McDonald Weibull model

For the first time, we propose a five-parameter lifetime model called the McDonald Weibull distribution to extend the Weibull, exponentiated Weibull, beta Weibull and Kumaraswamy Weibull distributions, among several other models. We obtain explicit expressions for the ordinary moments, quantile and generating functions, mean deviations and moments of the order statistics. We use the method of maximum likelihood to fit the new distribution and determine the observed information matrix. We define the log-McDonald Weibull regression model for censored data. The potentiality of the new model is illustrated by means of two real data sets.     

On an Extension of the Exponentiated Weibull Distribution

In this article, the exponentiated Weibull distribution is extended by the Marshall-Olkin family. Our new four-parameter family has a hazard rate function with various desired shapes depending on the choice of its parameters and, thus, it is very flexible in data modeling. It also contains two mixed distributions with applications to series and parallel systems in reliability and also contains several previously known lifetime distributions. We shall study some basic distributional properties of the new distribution. Some closed forms are derived for its moment generating function and moments as well as moments of its order statistics. The model parameters are estimated by the maximum likelihood method. The stress–strength parameter and its estimation are also investigated. Finally, an application of the new model is illustrated using two real datasets.     

The Kumaraswamy-transmuted exponentiated modified Weibull distribution

This article introduces a new generalization of the transmuted exponentiated modified Weibull distribution introduced by Eltehiwy and Ashour in 2013, using Kumaraswamy distribution introduced by Cordeiro and de Castro in 2011. We refer to the new distribution as Kumaraswamy-transmuted exponentiated modified Weibull (Kw-TEMW) distribution. The new model contains 54 lifetime distributions as special cases such as the KumaraswamyWeibull, exponentiated modified Weibull, exponentiated Weibull, exponentiated exponential, transmuted Weibull, Rayleigh, linear failure rate, and exponential distributions, among others. The properties of the new model are discussed and the maximum likelihood estimation is used to evaluate the parameters. Explicit expressions are derived for the moments and examine the order statistics. This model is capable of modeling various shapes of aging and failure criteria.     

The generalized odd log-logistic family of distributions: properties,regression models and applications

We propose a new class of continuous distributions with two extra shape parameters named the generalized odd log-logistic family of distributions. The proposed family contains as special cases the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, two entropy measures and order statistics are obtained. We derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behaviour of the estimators by means of Monte Carlo simulations. We introduce the log-odd log-logistic Weibull regression model with censored data based on the odd log-logistic-Weibull distribution. The importance of the new family is illustrated using three real data sets. These applications indicate that this family can provide better fits than other well-known classes of distributions. The beauty and importance of the proposed family lies in its ability to model different types of real data.     

A New Weibull–Pareto Distribution: Properties and Applications

Many distributions have been used as lifetime models. In this article, we propose a new three-parameter Weibull–Pareto distribution, which can produce the most important hazard rate shapes, namely, constant, increasing, decreasing, bathtub, and upsidedown bathtub. Various structural properties of the new distribution are derived including explicit expressions for the moments and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, and generating and quantile functions. The Rényi and q entropies are also derived. We obtain the density function of the order statistics and their moments. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of two real datasets on Wheaton river flood and bladder cancer. In the two applications, the new model provides better fits than the Kumaraswamy–Pareto, beta-exponentiated Pareto, beta-Pareto, exponentiated Pareto, and Pareto models.     

The exponentiated transmuted Weibull geometric distribution with application in survival analysis

This article introduces a new generalization of the transmuted Weibull distribution introduced by Aryal and Tsokos in 2011. We refer to the new distribution as exponentiated transmuted Weibull geometric (ETWG) distribution. The new model contains 22 lifetime distributions as special cases such as the exponentiated Weibull geometric, complementary Weibull geometric, exponentiated transmuted Weibull, exponentiated Weibull, and Weibull distributions, among others. The properties of the new model are discussed and the maximum likelihood estimation is used to evaluate the parameters. Explicit expressions are derived for the moments and examine the order statistics. To examine the performance of our new model in fitting several data we use two real sets of data, censored and uncensored, and then compare the fitting of the new model with some nested and nonnested models, which provides the best fit to all of the data. A simulation has been performed to assess the behavior of the maximum likelihood estimates of the parameters under the finite samples. This model is capable of modeling various shapes of aging and failure criteria.     

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 generalized odd log-logistic family of distributions

We introduce and study general mathematical properties of a new generator of continuous distributions with three extra parameters called the new generalized odd log-logistic family of distributions. The proposed family contains several important classes discussed in the literature as submodels such as the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a mixture of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, entropy measures, and order statistics, which hold for any baseline model, are presented. We also present certain characterization of the proposed distribution and derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behavior of the maximum likelihood estimator via simulation. The importance of the new family is illustrated by means of two real data sets. These applications indicate that the new family can provide better fits than other well-known classes of distributions. The beauty and importance of the new family lies in its ability to model real data.     

The beta generalized linear exponential distribution

In this paper, a new five-parameter lifetime distribution called beta generalized linear exponential distribution (BGLED) is introduced. It includes at least 17 popular sub-models as special cases such as the beta linear exponential, the beta generalized exponential, and the exponentiated generalized linear distributions. Mathematical and statistical properties of the proposed distribution are discussed in details. In particular, explicit expression for the density function, moments, asymptotics distributions for sample extreme statistics, and other statistical measures are obtained. The estimation of the parameters by the method of maximum-likelihood is discussed and the finite sample properties of the maximum-likelihood estimators (MLEs) are investigated numerically. A real data set is used to demonstrate its superior performance fit over several existing popular lifetime models.     

The Poisson Generalized Linear Failure Rate Model

We formulate a new cure rate survival model by assuming that the number of competing causes of the event of interest has the Poisson distribution, and the time to this event has the generalized linear failure rate distribution. A new distribution to analyze lifetime data is defined from the proposed cure rate model, and its quantile function as well as a general expansion for the moments is derived. We estimate the parameters of the model with cure rate in the presence of covariates for censored observations using maximum likelihood and derive the observed information matrix. We obtain the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. The usefulness of the proposed cure rate survival model is illustrated in an application to real data.     

Generalized exponential distributions

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.     

The gamma-linear failure rate distribution: theory and applications

For any continuous baseline G distribution, Zografos and Balakrishnan [On families of beta- and generalized gamma-generated distributions and associated inference. Statist Methodol. 2009;6:344–362] introduced the generalized gamma-generated distribution with an extra positive parameter. A new three-parameter continuous model called the gamma-linear failure rate (LFR) distribution, which extends the LFR model, is proposed and studied. Various structural properties of the new distribution are derived, including some explicit expressions for ordinary and incomplete moments, generating function, probability-weighted moments, mean deviations and Rényi and Shannon entropies. We estimate the model parameters by maximum likelihood and obtain the observed information matrix. The new model is modified to cope with possible long-term survivors in lifetime data. We illustrate the usefulness of the proposed model by means of two applications to real data.