The beta generalized exponential distribution

A new distribution called the beta generalized exponential distribution is proposed. It includes the beta exponential and generalized exponential (GE) distributions as special cases. We provide a comprehensive mathematical treatment of this distribution. The density function can be expressed as a mixture of generalized exponential densities. This is important to obtain some mathematical properties of the new distribution in terms of the corresponding properties of the GE distribution. We derive the moment generating function (mgf) and the moments, thus generalizing some results in the literature. Expressions for the density, mgf and moments of the order statistics are also obtained. We discuss estimation of the parameters by maximum likelihood and obtain the information matrix that is easily numerically determined. We observe in one application to a real skewed data set that this model is quite flexible and can be used effectively in analyzing positive data in place of the beta exponential and GE distributions.     

An extension of the generalized linear failure rate distribution

In this article, we introduce a new extension of the generalized linear failure rate (GLFR) distributions. It includes some well-known lifetime distributions such as extension of generalized exponential and GLFR distributions as special sub-models. In addition, it can have a constant, decreasing, increasing, upside-down bathtub (unimodal), and bathtub-shaped hazard rate function (hrf) depending on its parameters. We provide some of its statistical properties such as moments, quantiles, skewness, kurtosis, hrf, and reversible hrf. The maximum likelihood estimation of the parameters is also discussed. At the end, a real dataset is given to illustrate the usefulness of this new distribution in analyzing lifetime data.     

Generalized Linear Failure Rate Distribution

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately, the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non increasing hazard function also. This article introduces a new distribution, which generalizes linear failure rate distribution. This distribution generalizes the well-known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this article. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.     

The beta Weibull-geometric distribution

A new five-parameter distribution called the beta Weibull-geometric (BWG) distribution is proposed. The new distribution is generated from the logit of a beta random variable and includes the Weibull-geometric distribution of Barreto-Souza et al. [The Weibull-geometric distribution, J. Stat. Comput. Simul. 81 (2011), pp. 645–657], beta Weibull (BW), beta exponential, exponentiated Weibull, and some other lifetime distributions as special cases. A comprehensive mathematical treatment of this distribution is provided. The density function can be expressed as an infinite mixture of BW densities and then we derive some mathematical properties of the new distribution from the corresponding properties of the BW distribution. The density function of the order statistics and also estimation of the stress–strength parameter are obtained using two general expressions. To estimate the model parameters, we use the maximum likelihood method and the asymptotic distribution of the estimators is also discussed. The capacity of the new distribution are examined by various tools, using two real data sets.     

Truncated inverted generalized exponential distribution and its properties

The inverted generalized exponential distribution is defined as an alternative model for lifetime data. The existence of moments of this distribution is shown to hold under some restrictions. However, all the moments exist for the truncated inverted generalized exponential distribution and closed-form expressions for them are derived in this article. The distributional properties of this truncated distribution are studied. Maximum likelihood estimation method is discussed for the estimation of the parameters of the distribution both theoretically and empirically. In order to see the modeling performance of the distribution, two real datasets are analyzed.     

Transmuted Linear Exponential Distribution: A New Generalization of the Linear Exponential Distribution

In this article, a transmuted linear exponential distribution is developed that generalizes the linear exponential distribution with an additional parameter using the quadratic rank transmutation map which was studied by Shaw et al. Some statistical properties of the proposed distribution such as moments, quantiles, and the failure rate function are investigated. The maximum likelihood estimators of unknown parameters are also discussed and a real data analysis is carried out to illustrate the superiority of the proposed distribution.     

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.     

Inference for mixed generalized exponential distribution under progressively type-II censored samples

In industrial life tests, reliability analysis and clinical trials, the type-II progressive censoring methodology, which allows for random removals of the remaining survival units at each failure time, has become quite popular for analyzing lifetime data. Parameter estimation under progressively type-II censored samples for many common lifetime distributions has been investigated extensively. However, how to estimate unknown parameters of the mixed distribution models under progressive type-II censoring schemes is still a challenging and interesting problem. Based on progressively type-II censored samples, this paper addresses the estimation problem of mixed generalized exponential distributions. In addition, it is observed that the maximum-likelihood estimates (MLEs) cannot be easily obtained in closed form due to the complexity of the likelihood function. Thus, we make good use of the expectation-maximization algorithm to obtain the MLEs. Finally, some simulations are implemented in order to show the performance of the proposed method under finite samples and a case analysis is illustrated.     

A new four-parameter discrete distribution with bathtub and unimodal failure rate

In this paper, a discrete counterpart of the general class of continuous beta-G distributions is introduced. A discrete analog of the beta generalized exponential distribution of Barreto-Souza et al. [2], as an important special case of the proposed class, is studied. This new distribution contains some previously known discrete distributions as well as two new models. The hazard rate function of the new model can be increasing, decreasing, bathtub-shaped and upside-down bathtub. Some distributional and moment properties of the new distribution as well as its order statistics are discussed. Estimation of the parameters is illustrated using the maximum likelihood method and, finally, the model with a real data set is examined.     

Exact inferences of the two-parameter exponential distribution and Pareto distribution with censored data

We develop an exact inference for the location and the scale parameters of the two-exponential distribution and the Pareto distribution based on their maximum-likelihood estimators from the doubly Type-II and the progressive Type-II censored sample. Based on some pivotal quantities, exact confidence intervals and tests of hypotheses are constructed. Exact distributions of the pivotal quantities are expressed as mixtures of linear combinations and of ratios of linear combinations of standard exponential random variables, which facilitates the computation of quantiles of these pivotal quantities. We also provide a bootstrap method for constructing a confidence interval. Some simulation studies are carried out to assess their performances. Using the exact distribution of the scale parameter, we establish an acceptance sampling procedure based on the lifetime of the unit. Some numerical results are tabulated for the illustration. One biometrical example is also given to illustrate the proposed methods.     

An EM algorithm for the estimation of parameters of bivariate generalized exponential distribution under random left censoring

In this paper, we consider the four-parameter bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution, J. Multivariate Anal. 100 (2009), pp. 581–593] and propose an expectation–maximization algorithm to find the maximum-likelihood estimators of the four parameters under random left censoring. A numerical experiment is carried out to discuss the properties of the estimators obtained iteratively.     

Power binomial exponential distribution: Modeling,simulation and application

ABSTRACT

The binomial exponential 2 (BE2) distribution was proposed by Bakouch et al. as a distribution of a random sum of independent exponential random variables, when the sample size has a zero truncated binomial distribution. In this article, we introduce a generalization of BE2 distribution which offers a more flexible model for lifetime data than the BE2 distribution. The hazard rate function of the proposed distribution can be decreasing, increasing, decreasing–increasing–decreasing and unimodal, so it turns out to be quite flexible for analyzing non-negative real life data. Some statistical properties and parameters estimation of the distribution are investigated. Three different algorithms are proposed for generating random data from the new distribution. Two real data applications regarding the strength data and Proschan's air-conditioner data are used to show that the new distribution is better than the BE2 distribution and some other well-known distributions in modeling lifetime data.     

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.     

The gamma-exponentiated exponential distribution

In this paper, we introduce a new distribution generated by gamma random variables. We show that this distribution includes as a special case the distribution of the lower record value from a sequence of i.i.d. random variables from a population with the exponentiated (generalized) exponential distribution. The properties of this distribution are derived and the estimation of the model parameters is discussed. Some applications to real data sets are finally presented for illustration.     

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 generalization of the Weibull-geometric distribution with bathtub failure rate

In this paper, the researchers attempt to introduce a new generalization of the Weibull-geometric distribution. The failure rate function of the new model is found to be increasing, decreasing, upside-down bathtub, and bathtub-shaped. The researchers obtained the new model by compounding Weibull distribution and discrete generalized exponential distribution of a second type, which is a generalization of the geometric distribution. The new introduced model contains some previously known lifetime distributions as well as a new one. Some basic distributional properties and moments of the new model are discussed. Estimation of the parameters is illustrated and the model with two known real data sets is examined.     

Bivariate discrete generalized exponential distribution

In this paper, we develop a bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distribution as proposed by Nekoukhou, Alamatsaz and Bidram [Discrete generalized exponential distribution of a second type. Statistics. 2013;47:876–887]. It is observed that the proposed bivariate distribution is a very flexible distribution and the bivariate geometric distribution can be obtained as a special case of this distribution. The proposed distribution can be seen as a natural discrete analogue of the bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivariate Anal. 2009;100:581–593]. We study different properties of this distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum-likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of one data set has been performed to show the effectiveness of the proposed model. Finally, we propose some open problems and conclude the paper.     

On the bivariate weighted exponential distribution based on the generalized exponential distribution

In this paper, we introduce a bivariate weighted exponential distribution based on the generalized exponential distribution. This class of distributions generalizes the bivariate distribution with weighted exponential marginals (BWE). We derive different properties of this new distribution. It is a four-parameter distribution, and the maximum-likelihood estimator of unknown parameters cannot be obtained in explicit forms. One data set has been re-analyzed and it is observed that the proposed distribution provides better fit than the BWE distribution.     

The beta-Weibull geometric distribution

We propose a new distribution, the so-called beta-Weibull geometric distribution, whose failure rate function can be decreasing, increasing or an upside-down bathtub. This distribution contains special sub-models the exponential geometric [K. Adamidis and S. Loukas, A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42], beta exponential [S. Nadarajah and S. Kotz, The exponentiated type distributions, Acta Appl. Math. 92 (2006), pp. 97–111; The beta exponential distribution, Reliab. Eng. Syst. Saf. 91 (2006), pp. 689–697], Weibull geometric [W. Barreto-Souza, A.L. de Morais, and G.M. Cordeiro, The Weibull-geometric distribution, J. Stat. Comput. Simul. 81 (2011), pp. 645–657], generalized exponential geometric [R.B. Silva, W. Barreto-Souza, and G.M. Cordeiro, A new distribution with decreasing, increasing and upside-down bathtub failure rate, Comput. Statist. Data Anal. 54 (2010), pp. 935–944; G.O. Silva, E.M.M. Ortega, and G.M. Cordeiro, The beta modified Weibull distribution, Lifetime Data Anal. 16 (2010), pp. 409–430] and beta Weibull [S. Nadarajah, G.M. Cordeiro, and E.M.M. Ortega, General results for the Kumaraswamy-G distribution, J. Stat. Comput. Simul. (2011). DOI: 10.1080/00949655.2011.562504] distributions, among others. The density function can be expressed as a mixture of Weibull density functions. We derive expansions for the moments, generating function, mean deviations and Rénvy entropy. The parameters of the proposed model are estimated by maximum likelihood. The model fitting using envelops was conducted. The proposed distribution gives a good fit to the ozone level data in New York.