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.     

A new family of lifetime models

We introduce a new family of distributions by adding a parameter to the Marshall–Olkin family of distributions. Some properties of the new family of distributions are derived. A particular case of the family, a three-parameter generalization of the exponential distribution, is given special attention. The shape properties, moments, distributions of the order statistics, entropies and estimation procedures are derived. An application to a real data set is discussed.     

Reliability study of proportional odds family of discrete distributions

The proportional odds model gives a method of generating new family of distributions by adding a parameter, called tilt parameter, to expand an existing family of distributions. The new family of distributions so obtained is known as Marshall–Olkin family of distributions or Marshall–Olkin extended distributions. In this paper, we consider Marshall–Olkin family of distributions in discrete case with fixed tilt parameter. We study different ageing properties, as well as different stochastic orderings of this family of distributions. All the results of this paper are supported by several examples.     

Statistical analysis of the Lomax–Logarithmic distribution

In this paper we introduce a three-parameter lifetime distribution following the Marshall and Olkin [New method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika. 1997;84(3):641–652] approach. The proposed distribution is a compound of the Lomax and Logarithmic distributions (LLD). We provide a comprehensive study of the mathematical properties of the LLD. In particular, the density function, the shape of the hazard rate function, a general expansion for moments, the density of the rth order statistics, and the mean and median deviations of the LLD are derived and studied in detail. The maximum likelihood estimators of the three unknown parameters of LLD are obtained. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance–covariance matrix. Finally, a real data set is analysed to show the potential of the new proposed distribution.     

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 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.     

The Marshall–Olkin extended generalized Rayleigh distribution: Properties and applications

In this paper, a new extension for the generalized Rayleigh distribution is introduced. The proposed model, called Marshall–Olkin extended generalized Rayleigh distribution, arises based on the scheme introduced by Marshall and Olkin (1997) Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652.[Crossref], [Web of Science ®] , [Google Scholar]. A comprehensive account of the mathematical properties of the new distribution is provided. We discuss about the estimation of the model parameters based on two estimation methods. Empirical applications of the new model to real data are presented for illustrative purposes.     

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.     

An extension of the exponential distribution

A generalization of the exponential distribution is presented. The generalization always has its mode at zero and yet allows for increasing, decreasing and constant hazard rates. It can be used as an alternative to the gamma, Weibull and exponentiated exponential distributions. A comprehensive account of the mathematical properties of the generalization is presented. A real data example is discussed to illustrate its applicability.     

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.     

General results for the beta-modified Weibull distribution

We study in detail the so-called beta-modified Weibull distribution, motivated by the wide use of the Weibull distribution in practice, and also for the fact that the generalization provides a continuous crossover towards cases with different shapes. The new distribution is important since it contains as special sub-models some widely-known distributions, such as the generalized modified Weibull, beta Weibull, exponentiated Weibull, beta exponential, modified Weibull and Weibull distributions, among several others. It also provides more flexibility to analyse complex real data. Various mathematical properties of this distribution are derived, including its moments and moment generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are also derived for the chf, mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The estimation of parameters is approached by two methods: moments and maximum likelihood. We compare by simulation the performances of the estimates from these methods. We obtain the expected information matrix. Two applications are presented to illustrate the proposed distribution.     

On the Marshall–Olkin extended distributions

ABSTRACT

A general method of introducing a new parameter to a well-established continuous baseline cumulative function G to obtain more flexible distributions was proposed by Marshall and Olkin (1997 Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652.[Crossref], [Web of Science ®] , [Google Scholar]). This new family is known as Marshall–Olkin extended G family of distributions. In this article, we characterize this family as mixtures of the distributions of the minimum and maximum of random variables with cumulative function G. We demonstrate that the coefficients of the mixtures are probabilities of random variables with geometric distributions. Additionally, we present new representations for the density and cumulative functions of this class of distributions. Further, we introduce a new three-parameter continuous model for modeling rates and proportions based on the Marshall–Olkin's method. 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 a real dataset.     

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.     

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.     

Weighted Marshall–Olkin bivariate exponential distribution

Recently, Gupta and Kundu [R.D. Gupta and D. Kundu, A new class of weighted exponential distributions, Statistics 43 (2009), pp. 621–634] have introduced a new class of weighted exponential (WE) distributions, and this can be used quite effectively to model lifetime data. In this paper, we introduce a new class of weighted Marshall–Olkin bivariate exponential distributions. This new singular distribution has univariate WE marginals. We study different properties of the proposed model. There are four parameters in this model and the maximum-likelihood estimators (MLEs) of the unknown parameters cannot be obtained in explicit forms. We need to solve a four-dimensional optimization problem to compute the MLEs. One data set has been analysed for illustrative purposes and finally we propose some generalization of the proposed model.     

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.     

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.     

Recurrence relations for single and product moments of generalized order statistics from Marshall–Olkin extended Pareto distributions

Marshall and Olkin (1997 Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84(3):641–652.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new method of adding parameter to expand a family of distributions. Using this concept, in this article, the Marshall–Olkin extended Pareto distribution is introduced and some recurrence relations for single and product moments of generalized order statistics are studied. Also the results are deduced for record values and order statistics.     

Multivariate discrete exponential family of distributions and their properties

The paper generalizes the univariate discrete exponential family of distributions to the multivariate situation, and this generalization includes the multivariate power series distributions, the multivariate Lagrangian distributions, and the modified multivariate power-series distributions. This provides a unified approach for the study of these three classes of distributions. We obtain recurrence relations for moments and cumulants, and the maximum likelihood estimation for the discrete exponential family. These results are applied to some multivariate discrete distributions like the Lagrangian Poisson, Lagrangian (negative) multinomial, logarithmic series distributions and multivariate Lagrangian negative binomial distribution.