On bivariate discrete Weibull distribution

Recently, Lee and Cha proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall–Olkin bivariate Weibull distribution. We study various properties of the proposed distribution and discuss its interesting physical interpretations. The proposed model has four parameters, and because of that it is a very flexible distribution. The maximum likelihood estimators of the parameters cannot be obtained in closed forms, and we have proposed a very efficient nested EM algorithm which works quite well for discrete data. We have also proposed augmented Gibbs sampling procedure to compute Bayes estimates of the unknown parameters based on a very flexible set of priors. Two data sets have been analyzed to show how the proposed model and the method work in practice. We will see that the performances are quite satisfactory. Finally, we conclude the paper.     

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.     

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.     

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.     

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.     

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.     

Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes

In this paper, we introduce a bivariate Kumaraswamy (BVK) distribution whose marginals are Kumaraswamy distributions. The cumulative distribution function of this bivariate model has absolutely continuous and singular parts. Representations for the cumulative and density functions are presented and properties such as marginal and conditional distributions, product moments and conditional moments are obtained. We show that the BVK model can be obtained from the Marshall and Olkin survival copula and obtain a tail dependence measure. The estimation of the parameters by maximum likelihood is discussed and the Fisher information matrix is determined. We propose an EM algorithm to estimate the parameters. Some simulations are presented to verify the performance of the direct maximum-likelihood estimation and the proposed EM algorithm. We also present a method to generate bivariate distributions from our proposed BVK distribution. Furthermore, we introduce a BVK distribution which has only an absolutely continuous part and discuss some of its properties. Finally, a real data set is analysed for illustrative purposes.     

Modified slashed generalized exponential distribution

Abstract

In this article, we introduce a new distribution for modeling positive data sets with high kurtosis, the modified slashed generalized exponential distribution. The new model can be seen as a modified version of the slashed generalized exponential distribution. It arises as a quotient of two independent random variables, one being a generalized exponential distribution in the numerator and a power of the exponential distribution in the denominator. We studied various structural properties (such as the stochastic representation, density function, hazard rate function and moments) and discuss moment and maximum likelihood estimating approaches. Two real data sets are considered in which the utility of the new model in the analysis with high kurtosis is illustrated.     

On Two General Classes of Discrete Bivariate Distributions

In this article, we develop two general classes of discrete bivariate distributions. We derive general formulas for the joint distributions belonging to the classes. The obtained formulas for the joint distributions are very general in the sense that new families of distributions can be generated just by specifying the “baseline seed distributions.” The dependence structures of the bivariate distributions belonging to the proposed classes, along with basic statistical properties, are also discussed. New families of discrete bivariate distributions are generated from the classes. Furthermore, to assess the usefulness of the proposed classes, two discrete bivariate distributions generated from the classes are applied to analyze a real dataset and the results are compared with those obtained from conventional models.     

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.     

The generalized exponential cure rate model with covariates

In this article, we consider a parametric survival model that is appropriate when the population of interest contains long-term survivors or immunes. The model referred to as the cure rate model was introduced by Boag 1 Boag, J. W. 1949. Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J. R. Stat. Soc. Ser. B, 11: 15–53.  [Google Scholar] in terms of a mixture model that included a component representing the proportion of immunes and a distribution representing the life times of the susceptible population. We propose a cure rate model based on the generalized exponential distribution that incorporates the effects of risk factors or covariates on the probability of an individual being a long-time survivor. Maximum likelihood estimators of the model parameters are obtained using the the expectation-maximisation (EM) algorithm. A graphical method is also provided for assessing the goodness-of-fit of the model. We present an example to illustrate the fit of this model to data that examines the effects of different risk factors on relapse time for drug addicts.     

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.     

Reliability estimation of generalized inverted exponential distribution

A generalized version of inverted exponential distribution (IED) is introduced in this paper. This lifetime distribution is capable of modelling various shapes of failure rates, and hence various shapes of ageing criteria. The model can be considered as another useful two-parameter generalization of the IED. Statistical and reliability properties of the generalized inverted exponential distribution are derived. Maximum likelihood estimation and least square estimation are used to evaluate the parameters and the reliability of the distribution. Properties of the estimates are also studied.     

Discriminating between generalized exponential,geometric extreme exponential and Weibull distributions

Generalized exponential, geometric extreme exponential and Weibull distributions are three non-negative skewed distributions that are suitable for analysing lifetime data. We present diagnostic tools based on the likelihood ratio test (LRT) and the minimum Kolmogorov distance (KD) method to discriminate between these models. Probability of correct selection has been calculated for each model and for several combinations of shape parameters and sample sizes using Monte Carlo simulation. Application of LRT and KD discrimination methods to some real data sets has also been studied.     

An extension of the generalized exponential distribution

The two-parameter generalized exponential distribution has been used recently quite extensively to analyze lifetime data. In this paper the two-parameter generalized exponential distribution has been embedded in a larger class of distributions obtained by introducing another shape parameter. Because of the additional shape parameter, more flexibility has been introduced in the family. It is observed that the new family is positively skewed, and has increasing, decreasing, unimodal and bathtub shaped hazard functions. It can be observed as a proportional reversed hazard family of distributions. This new family of distributions is analytically quite tractable and it can be used quite effectively to analyze censored data also. Analyses of two data sets are performed and the results are quite satisfactory.     

Slashed generalized exponential distribution

In this paper, we introduce an extension of the generalized exponential (GE) distribution, making it more robust against possible influential observations. The new model is defined as the quotient between a GE random variable and a beta-distributed random variable with one unknown parameter. The resulting distribution is a distribution with greater kurtosis than the GE distribution. Probability properties of the distribution such as moments and asymmetry and kurtosis are studied. Likewise, statistical properties are investigated using the method of moments and the maximum likelihood approach. Two real data analyses are reported illustrating better performance of the new model over the GE model.     

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.     

Generalized fiducial inference for generalized exponential distribution

This article mainly considers interval estimation of the scale and shape parameters of the generalized exponential (GE) distribution. We adopt the generalized fiducial method to construct a kind of new confidence intervals for the parameters of interest and compare them with the frequentist and Bayesian methods. In addition, we give the comparison of the point estimation based on the frequentist, generalized fiducial and Bayesian methods. Simulation results show that a new procedure based on generalized fiducial inference is more applicable than the non-fiducial methods for the point and interval estimation of the GE distribution. Finally, two lifetime data sets are used to illustrate the application of our new procedure.     

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.     

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.