A new lifetime distribution

Many if not most lifetime distributions are motivated only by mathematical interest. Here, a new three-parameter distribution motivated mainly by lifetime issues is introduced. Some properties of the new distribution including estimation procedures, univariate generalizations and bivariate generalizations are derived. A real data application is described to show its superior performance versus at least that of 15 of the known lifetime models.     

A new three-parameter lifetime distribution

Many if not most lifetime distributions are motivated only by mathematical interest. Here, a new three-parameter distribution motivated mainly by lifetime issues is introduced. Some properties of the new distribution including estimation procedures are derived. Three real-data applications are described to show superior performance versus at least five of the known lifetime models.     

Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data

In this article, we investigate the potential usefulness of the three-parameter transmuted generalized exponential distribution for analyzing lifetime data. We compare it with various generalizations of the two-parameter exponential distribution using maximum likelihood estimation. Some mathematical properties of the new extended model including expressions for the quantile and moments are investigated. We propose a location-scale regression model, based on the log-transmuted generalized exponential distribution. Two applications with real data are given to illustrate the proposed family of lifetime distributions.     

A new lifetime model with decreasing failure rate

In this paper, we introduce a new lifetime distribution by compounding exponential and Poisson–Lindley distributions, named the exponential Poisson–Lindley (EPL) distribution. A practical situation where the EPL distribution is most appropriate for modelling lifetime data than exponential–geometric, exponential–Poisson and exponential–logarithmic distributions is presented. We obtain the density and failure rate of the EPL distribution and properties such as mean lifetime, moments, order statistics and Rényi entropy. Furthermore, estimation by maximum likelihood and inference for large samples are discussed. The paper is motivated by two applications to real data sets and we hope that this model will be able to attract wider applicability in survival and reliability.     

The inverse power Lindley distribution in the presence of left-censored data

In this study, classical and Bayesian inference methods are introduced to analyze lifetime data sets in the presence of left censoring considering two generalizations of the Lindley distribution: a first generalization proposed by Ghitany et al. [Power Lindley distribution and associated inference, Comput. Statist. Data Anal. 64 (2013), pp. 20–33], denoted as a power Lindley distribution and a second generalization proposed by Sharma et al. [The inverse Lindley distribution: A stress–strength reliability model with application to head and neck cancer data, J. Ind. Prod. Eng. 32 (2015), pp. 162–173], denoted as an inverse Lindley distribution. In our approach, we have used a distribution obtained from these two generalizations denoted as an inverse power Lindley distribution. A numerical illustration is presented considering a dataset of thyroglobulin levels present in a group of individuals with differentiated cancer of thyroid.     

On generalized dynamic cumulative past entropy measure

Recently, the concept of dynamic cumulative residual entropy and its generalizations has gained much attention among researchers. In this work, a new generalized dynamic cumulative measure in the past lifetime is proposed. Further, some characterization results connecting this new generalized dynamic entropy measure and other reversed measures are obtained.     

The exponentiated G geometric family of distributions

We propose a new class of distributions called the exponentiated G geometric family motivated mainly by lifetime issues which can generate several lifetime models discussed in the literature. Some mathematical properties of the new family including asymptotes and shapes, moments, quantile and generating functions, extreme values and order statistics are fully investigated. We propose the log-exponentiated Weibull geometric and log-exponentiated log-logistic geometric regression models to cope with censored data. The model parameters are estimated by maximum likelihood. Three examples with real data expose quite well the new family.     

Some gamma distributions

Three new generalizations of the standard gamma distribution introduced by the author are reviewed. Various properties are derived for each distribution, including its hazard rate function and moments. An application is illustrated to drought data.     

The exponentiated Weibull distribution: a survey

A review is given of the exponentiated Weibull distribution, the first generalization of the two-parameter Weibull distribution to accommodate nonmonotone hazard rates. The properties reviewed include: moments, order statistics, characterizations, generalizations and related distributions, transformations, graphical estimation, maximum likelihood estimation, Bayes estimation, other estimation, discrimination, goodness of fit tests, regression models, applications, multivariate generalizations, and computer software. Some of the results given are new and hitherto unknown. It is hoped that this review could serve as an important reference and encourage developments of further generalizations of the two-parameter Weibull distribution.     

Further results on discrete mean past lifetime

Abstract

The present paper aims at studying the mean past lifetime of a discrete random variable. The notion of discrete mean past lifetime is studied in relation to the concepts of reversed hazard rate, reversed lack of memory property, and cumulative past entropy. New classes of distributions characterized by particular forms of discrete mean past life are also investigated. Implications of an increasing mean past lifetime on other reliability notions are studied and finally some bivariate generalizations are discussed.     

A new absolute continuous bivariate generalized exponential distribution

The generalized exponential is the most commonly used distribution for analyzing lifetime data. This distribution has several desirable properties and it can be used quite effectively to analyse several skewed life time data. The main aim of this paper is to introduce absolutely continuous bivariate generalized exponential distribution using the method of Block and Basu (1974). In fact, the Block and Basu exponential distribution will be extended to the generalized exponential distribution. We call the new proposed model as the Block and Basu bivariate generalized exponential distribution, then, discuss its different properties. In this case the joint probability distribution function and the joint cumulative distribution function can be expressed in compact forms. The model has four unknown parameters and the maximum likelihood estimators cannot be obtained in explicit form. To compute the maximum likelihood estimators directly, one needs to solve a four dimensional optimization problem. The EM algorithm has been proposed to compute the maximum likelihood estimations of the unknown parameters. One data analysis is provided for illustrative purposes. Finally, we propose some generalizations of the proposed model and compare their models with each other.     

A new long-term lifetime distribution induced by a latent complementary risk framework

In this paper, we proposed a new three-parameter long-term lifetime distribution induced by a latent complementary risk framework with decreasing, increasing and unimodal hazard function, the long-term complementary exponential geometric distribution. The new distribution arises from latent competing risk scenarios, where the lifetime associated scenario, with a particular risk, is not observable, rather we observe only the maximum lifetime value among all risks, and the presence of long-term survival. The properties of the proposed distribution are discussed, including its probability density function and explicit algebraic formulas for its reliability, hazard and quantile functions and order statistics. The parameter estimation is based on the usual maximum-likelihood approach. A simulation study assesses the performance of the estimation procedure. We compare the new distribution with its particular cases, as well as with the long-term Weibull distribution on three real data sets, observing its potential and competitiveness in comparison with some usual long-term lifetime distributions.     

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 GEV distribution

The popular generalized extreme value (GEV) distribution has not been a flexible model for extreme values in many areas. We propose a generalization – referred to as the Kumaraswamy GEV distribution – and provide a comprehensive treatment of its mathematical properties. We estimate its parameters by the method of maximum likelihood and provide the observed information matrix. An application to some real data illustrates flexibility of the new model. Finally, some bivariate generalizations of the model are proposed.     

Modelling accelerated life test data by using a Bayesian approach

Summary. Because of the high reliability of many modern products, accelerated life tests are becoming widely used to obtain timely information about their time-to-failure distributions. We propose a general class of accelerated life testing models which are motivated by the actual failure process of units from a limited failure population with a positive probability of not failing during the technological lifetime. We demonstrate a Bayesian approach to this problem, using a new class of models with non-monotone hazard rates, the hazard model with potential scope for use far beyond accelerated life testing. Our methods are illustrated with the modelling and analysis of a data set on lifetimes of printed circuit boards under humidity accelerated life testing.     

The exponentiated exponential–geometric distribution: a distribution with decreasing,increasing and unimodal failure rate

In this paper, we proposed a new family of distributions namely exponentiated exponential–geometric (E2G) distribution. The E2G distribution is a straightforwardly generalization of the exponential–geometric (EG) distribution proposed by Adamidis and Loukas [A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42], which accommodates increasing, decreasing and unimodal hazard functions. It arises on a latent competing risk scenarios, where the lifetime associated with a particular risk is not observable but only the minimum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments, rth moment of the ith order statistic, mean residual lifetime and modal value. Maximum-likelihood inference is implemented straightforwardly. From a mis-specification simulation study performed in order to assess the extent of the mis-specification errors when testing the EG distribution against the E2G, and we observed that it is usually possible to discriminate between both distributions even for moderate samples with presence of censoring. The practical importance of the new distribution was demonstrated in three applications where we compare the E2G distribution with several lifetime distributions.     

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.     

Power-normal distribution

Recently, Gupta and Gupta [Analyzing skewed data by power-normal model, Test 17 (2008), pp. 197–210] proposed the power-normal distribution for which normal distribution is a special case. The power-normal distribution is a skewed distribution, whose support is the whole real line. Our main aim of this paper is to consider bivariate power-normal distribution, whose marginals are power-normal distributions. We obtain the proposed bivariate power-normal distribution from Clayton copula, and by making a suitable transformation in both the marginals. Lindley–Singpurwalla distribution also can be used to obtain the same distribution. Different properties of this new distribution have been investigated in detail. Two different estimators are proposed. One data analysis has been performed for illustrative purposes. Finally, we propose some generalizations to multivariate case also along the same line and discuss some of its properties.     

A new four-parameter lifetime distribution

A new four-parameter distribution is introduced. It appears to be a distribution allowing for and only allowing for monotonically increasing, bathtub-shaped and upside down bathtub-shaped hazard rates. It contains as particular cases many of the known lifetime distributions. Some mathematical properties of the new distribution, including estimation procedures by the method of maximum likelihood are derived. Some simulations are run to assess the performance of the maximum-likelihood estimators. Finally, the flexibility of the new distribution is illustrated using a real data set.     

A new lifetime distribution for a series-parallel system: properties,applications and estimations under progressive type-II censoring

A compound class of zero truncated Poisson and lifetime distributions is introduced. A specialization is paved to a new three-parameter distribution, called doubly Poisson-exponential distribution, which may represent the lifetime of units connected in a series-parallel system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an exponential distribution. Among its motivations is that its hazard rate function can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods [maximum likelihood, moments, least squares, weighted least squares and Bayes (under linear-exponential and general entropy loss functions) estimations] are used to estimate the involved parameters. The performance of these methods is investigated through a simulation study. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. In addition, confidence intervals, symmetric credible intervals and highest posterior density credible intervals of the parameters are obtained. Finally, an application to a real data set is used to compare the new distribution with other five distributions.