On length and area-biased Maxwell distributions

This article addresses the various properties and different methods of estimation of the unknown parameter of length and area-biased Maxwell distributions. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of length and area-biased Maxwell distributions (such as moments, moment-generating function (mgf), hazard rate function, mean residual lifetime function, residual lifetime function, reversed residual life function, conditional moments and conditional mgf, stochastic ordering, and measures of uncertainty) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimator, moments estimator, least-square and weighted least-square estimators, maximum product of spacings estimator and compare them using extensive numerical simulations. Next we consider Bayes estimation under different types of loss function (symmetric and asymmetric loss functions) using inverted gamma prior for the scale parameter. Furthermore, Bayes estimators and their respective posterior risks are computed and compared using Markov chain Monte Carlo (MCMC) algorithm. Also, bootstrap confidence intervals using frequentist approaches are provided to compare with Bayes credible intervals. Finally, a real dataset has been analyzed for illustrative purposes.     

The odd log-logistic generalized half-normal lifetime distribution: Properties and applications

Cooray and Ananda (2008 Cooray, K., Ananda, M.M.A. (2008). A Generalization of the half-normal distribution with applications to lifetime data. Commun. Stat. - Theory Methods 37:1323–1337.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) pioneered a lifetime model commonly used in reliability studies. Based on this distribution, we propose a new model called the odd log-logistic generalized half-normal distribution for describing fatigue lifetime data. Various of its structural properties are derived. We discuss the method of maximum likelihood to fit the model parameters. For different parameter settings and sample sizes, some simulation studies compare the performance of the new lifetime model. It can be very useful, and its superiority is illustrated by means of a real dataset.     

Estimation in Maxwell distribution with randomly censored data

In many practical situations, complete data are not available in lifetime studies. Many of the available observations are right censored giving survival information up to a noted time and not the exact failure times. This constitutes randomly censored data. In this paper, we consider Maxwell distribution as a survival time model. The censoring time is also assumed to follow a Maxwell distribution with a different parameter. Maximum likelihood estimators and confidence intervals for the parameters are derived with randomly censored data. Bayes estimators are also developed with inverted gamma priors and generalized entropy loss function. A Monte Carlo simulation study is performed to compare the developed estimation procedures. A real data example is given at the end of the study.     

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.     

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.     

A new useful four-parameter extension of the Gumbel distribution: Properties,regression model and applications using the GAMLSS framework

In this paper we introduce a flexible extension of the Gumbel distribution called the odd log-logistic exponentiated Gumbel distribution. The new model was implemented in GAMLSS package of R software and a brief tutorial on how to use this package is presented throughout the paper. We provide a comprehensive treatment of its general mathematical properties. Further, we propose a new extended regression model considering four regression structures. We discuss estimation methods based on censored and uncensored data. Two simulation studies are presented and four real data sets are applied to illustrating the usefulness of the new model.     

The McDonald Gompertz distribution: Properties and applications

This article introduces a five-parameter lifetime model called the McDonald Gompertz (McG) distribution to extend the Gompertz, generalized Gompertz, generalized exponential, beta Gompertz, and Kumaraswamy Gompertz distributions among several other models. The hazard function of new distribution can be increasing, decreasing, upside-down bathtub, and bathtub shaped. We obtain several properties of the McG distribution including moments, entropies, quantile, and generating functions. We provide the density function of the order statistics and their moments. The parameter estimation is based on the usual maximum likelihood approach. We also provide the observed information matrix and discuss inferences issues. The flexibility and usefulness of the new distribution are illustrated by means of application to two real datasets.     

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 extended Gompertz-Makeham distribution with application to lifetime data

In this paper, we propose an extension of the Gompertz-Makeham distribution. This distribution is called the transmuted Gompertz-Makeham (TGM). The new model which can handle bathtub-shaped, increasing, increasing-constant and constant hazard rate functions. This property makes TGM is useful in survival analysis. Various statistical and reliability measures of the model are obtained, including hazard rate function, moments, moment generating function (mgf), quantile function, random number generating, skewness, kurtosis, conditional moments, mean deviations, Bonferroni curve, Lorenz curve, Gini index, mean inactivity time, mean residual lifetime and stochastic ordering; we also obtain the density of the ith order statistic. Estimation of the model parameters is justified by the method of maximum likelihood. An application to real data demonstrates that the TGM distribution can provides a better fit than some other very well known distributions.     

A new three-parameter extension of the log-logistic distribution with applications to survival data

In this article, a new three-parameter extension of the two-parameter log-logistic distribution is introduced. Several distributional properties such as moment-generating function, quantile function, mean residual lifetime, the Renyi and Shanon entropies, and order statistics are considered. The estimation of the model parameters for complete and right-censored cases is investigated competently by maximum likelihood estimation (MLE). A simulation study is conducted to show that these MLEs are consistent in moderate samples. Two real datasets are considered; one is a right-censored data to show that the proposed model has a superior performance over several existing popular models.     

An extended Poisson distribution

ABSTRACT

The Poisson distribution is extended over the set of all integers. The motivation comes from the many reflected versions of the gamma distribution, the continuous analog of the Poisson distribution, defined over the entire real line. Various mathematical properties of the extended Poisson distribution are derived. Estimation procedures by the methods of moments and maximum likelihood are also derived with their performance assessed by simulation. Finally, a real data application is illustrated.     

Tail behavior of the generalized Maxwell distribution

Abstract

In this article, we investigate the tail propertiesof the generalized Maxwell distribution and gain an asymptoticbehavior of its Mills-type ratio. Meanwhile, we show two applications. The first application thinks about the asymptotic property of the ratio of density functions and the ratio of the tails of the generalized Maxwell and classical Maxwell distributions. Another application obtains the asymptotic distribution of the partial maximum of an independent and identically distributed sequence from the distribution.     

A new flexible hazard rate distribution: Application and estimation of its parameters

We introduce a new class of flexible hazard rate distributions which have constant, increasing, decreasing, and bathtub-shaped hazard function. This class of distributions obtained by compounding the power and exponential hazard rate functions, which is called the power-exponential hazard rate distribution and contains several important lifetime distributions. We obtain some distributional properties of the new family of distributions. The estimation of parameters is obtained by using the maximum likelihood and the Bayesian methods under squared error, linear-exponential, and Stein’s loss functions. Also, approximate confidence intervals and HPD credible intervals of parameters are presented. An application to real dataset is provided to show that the new hazard rate distribution has a better fit than the other existing hazard rate distributions and some four-parameter distributions. Finally , to compare the performance of proposed estimators and confidence intervals, an extensive Monte Carlo simulation study is conducted.     

The Beta-Dagum Distribution: Definition and Properties

This article introduces a five-parameter Beta-Dagum distribution from which moments, hazard and entropy, and reliability measures are then derived. These properties show the high flexibility of the said distribution. The maximum likelihood estimators of the Beta-Dagum parameters are examined and the expected Fisher information matrix provided. Next, a simulation study is carried out which shows the good performance of maximum likelihood estimators for finite samples. Finally, the usefulness of the new distribution is illustrated through real data sets.     

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.     

The exponentiated reduced Kies distribution: Properties and applications

Here we consider an exponentiated version of the reduced Kies distribution and discuss some of its properties. The parameters of the distribution are estimated by the method of maximum likelihood and illustrated with the help of certain real-life data sets. Asymptotic behavior of the maximum likelihood estimators of the parameters of the distribution is also studied by using certain simulated data sets.     

A complementary generalized linear failure rate-geometric distribution

In this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.     

Alpha power Weibull distribution: Properties and applications

In this paper, a new lifetime distribution is defined and studied. We refer to the new distribution as alpha power Weibull distribution. The importance of the new distribution comes from its ability to model monotone and non monotone failure rate functions, which are quite common in reliability studies. Various properties of the proposed distribution are obtained including moments, quantiles, entropy, order statistics, mean residual life function, and stress-strength parameter. The maximum likelihood estimation method is used to estimate the parameters. Two real data sets are used to illustrate the importance of the proposed distribution.     

A new method for generating distributions with an application to exponential distribution

A new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility. A special case has been considered in detail, namely one-parameter exponential distribution. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, mode, moment-generating function, mean residual lifetime, stochastic orders, order statistics, and expression of the entropies, are derived. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non linear equations only. Further, we consider an extension of the two-parameter exponential distribution also, mainly for data analysis purposes. Two datasets have been analyzed to show how the proposed models work in practice.     

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.