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.     

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.     

Saddlepoint density and distribution functions for the ratio of two linear functions and the product of generalized gamma variates

The generalized gamma distribution is a flexible and attractive distribution because it incorporates several well-known distributions, i.e., gamma, Weibull, Rayleigh, and Maxwell. This article derives saddlepoint density and distribution functions for the ratio of two linear functions of generalized gamma variables and the product of n independent generalized gamma variables. Simulation studies are used to evaluate the accuracy of the saddlepoint approximations. The saddlepoint approximations are fast, easy, and very accurate.     

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.     

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.     

The compound class of linear failure rate-power series distributions: Model,properties, and applications

In this article, a new class of distributions is introduced, which generalizes the linear failure rate distribution and is obtained by compounding this distribution and power series class of distributions. This new class of distributions is called the linear failure rate-power series distributions and contains some new distributions such as linear failure rate-geometric, linear failure rate-Poisson, linear failure rate-logarithmic, linear failure rate-binomial distributions, and Rayleigh-power series class of distributions. Some former works such as exponential-power series class of distributions, exponential-geometric, exponential-Poisson, and exponential-logarithmic distributions are special cases of the new proposed model. The ability of the linear failure rate-power series class of distributions is in covering five possible hazard rate function, that is, increasing, decreasing, upside-down bathtub (unimodal), bathtub and increasing-decreasing-increasing shaped. Several properties of this class of distributions such as moments, maximum likelihood estimation procedure via an EM-algorithm and inference for a large sample, are discussed in this article. In order to show the flexibility and potentiality, the fitted results of the new class of distributions and some of its submodels are compared 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.     

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.     

A new three-parameter ageing distribution

We propose a new three-parameter ageing distribution called the Weibull-Poisson (WP) distribution, which generalizes the exponential-Poisson (EP) distribution introduced by Kus (2007). This new distribution has a more general form of failure rate (hazard rate) function. With appropriate choice of parameter values, it is able to model three ageing classes of life distributions including decreasing failure rate (DFR), increasing failure rate (IFR), and modified upside-down-bathtub (MUBT)-shaped failure rate. It thus provides an alternative to many existing life distributions. Various properties of this distribution are discussed and the estimation of the parameters is considered by the expectation maximization (EM) algorithm. Also, the asymptotic variance-covariance matrices of these estimates are obtained. Furthermore, some expressions for the Rènyi and Shannon entropies are given. Simulation studies are performed and experimental results are illustrated based on a real data set.     

A COM-Poisson-type generalization of the negative binomial distribution

ABSTRACT

This paper introduces a generalization of the negative binomial (NB) distribution in analogy with the COM-Poisson distribution. Many well-known distributions are particular and limiting distributions. The proposed distribution belongs to the modified power series, generalized hypergeometric and exponential families, and also arises as weighted NB and COM-Poisson distributions. Probability and moment recurrence formulae, and probabilistic and reliability properties have been derived. With the flexibility to model under-, equi- and over-dispersion, and its various interesting properties, this NB generalization will be a useful model for count data. An application to empirical modeling is illustrated with a real data set.     

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.     

Discriminating between the generalized Rayleigh and Weibull distributions: Some comparative studies

The generalized Rayleigh distribution was introduced and studied quite effectively in the literature. The closeness and separation between the distributions are extremely important for analyzing any lifetime data. In this spirit, both the generalized Rayleigh and Weibull distributions can be used for analyzing skewed datasets. In this article, we compare these two distributions based on the Fisher information measures and use it for discrimination purposes. It is evident that the Fisher information measures play an important role in separating between the distributions. The total information measures and the variances of the different percentile estimators are computed and presented. A real life dataset is analyzed for illustration purposes and a numerical comparison study is performed to assess our procedures in separating between these two distributions.     

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.     

A new class of weighted exponential distributions

Introducing a shape parameter to an exponential model is nothing new. There are many ways to introduce a shape parameter to an exponential distribution. The different methods may result in variety of weighted exponential (WE) distributions. In this article, we have introduced a shape parameter to an exponential model using the idea of Azzalini, which results in a new class of WE distributions. This new WE model has the probability density function (PDF) whose shape is very close to the shape of the PDFS of Weibull, gamma or generalized exponential distributions. Therefore, this model can be used as an alternative to any of these distributions. It is observed that this model can also be obtained as a hidden truncation model. Different properties of this new model have been discussed and compared with the corresponding properties of well-known distributions. Two data sets have been analysed for illustrative purposes and it is observed that in both the cases it fits better than Weibull, gamma or generalized exponential distributions.     

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.     

Modified Sarhan–Balakrishnan singular bivariate distribution

Recently Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527] introduced a new bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Unfortunately, they did not discuss any estimation procedure of the unknown parameters. In this paper using the similar idea as of Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527], we have proposed a singular bivariate distribution, which has an extra shape parameter. It is observed that the marginal distributions of the proposed bivariate distribution are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential distribution, Sarhan–Balakrishnan's bivariate distribution or the bivariate generalized exponential distribution. Different properties of this new distribution have been discussed. We provide the maximum likelihood estimators of the unknown parameters using EM algorithm. We reported some simulation results and performed two data analysis for illustrative purposes. Finally we propose some generalizations of this bivariate model.     

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.     

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.     

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.     

Slashed generalized Rayleigh distribution

We introduce a new family of distributions suitable for fitting positive data sets with high kurtosis which is called the Slashed Generalized Rayleigh Distribution. This distribution arises as the quotient of two independent random variables, one being a generalized Rayleigh distribution in the numerator and the other a power of the uniform distribution in the denominator. We present properties and carry out estimation of the model parameters by moment and maximum likelihood (ML) methods. Finally, we conduct a small simulation study to evaluate the performance of ML estimators and analyze real data sets to illustrate the usefulness of the new model.