Marshall–Olkin bivariate Weibull distributions and processes

In this paper we introduce a new probability model known as type 2 Marshall–Olkin bivariate Weibull distribution as an extension of type 1 Marshall–Olkin bivariate Weibull distribution of Marshall–Olkin (J Am Stat Assoc 62:30–44, 1967). Various properties of the new distribution are considered. Bivariate minification processes with the two types of Weibull distributions as marginals are constructed and their properties are considered. It is shown that the processes are strictly stationary. The unknown parameters of the type 1 process are estimated and their properties are discussed. Some numerical results of the estimates are also given.     

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.     

Marshall–Olkin gamma–Weibull distribution with applications

Abstract

A Marshall–Olkin variant of the Provost type gamma–Weibull probability distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed. This density function is utilized to model two real data sets. The new distribution provides a better fit than related distributions as measured by the Anderson–Darling and Cramér–von Mises statistics. The proposed distribution could find applications for instance in the physical and biological sciences, hydrology, medicine, meteorology, engineering, etc.     

Marshall–Olkin extended two-parameter bathtub-shaped lifetime distribution

ABSTRACT

The Marshall–Olkin extended two-parameter bathtub distribution is introduced and its structural properties are investigated, including the compounding representation of the distribution, the shapes of the density and the hazard rate function, the moments and quantiles. Estimation of the model parameters by maximum likelihood is discussed. Applications to some real data sets which motivate the usefulness of the model are provided. Comparison between the proposed model and other commonly used distributions is performed using real data sets. A simulation study is presented to investigate the accuracy of the estimates of the model's parameters.     

On the Marshall–Olkin extended Weibull distribution

We study some mathematical properties of the Marshall–Olkin extended Weibull distribution introduced by Marshall and Olkin (Biometrika 84:641–652, 1997). We provide explicit expressions for the moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves, reliability and Rényi entropy. We determine the moments of the order statistics. We also discuss the estimation of the model parameters by maximum likelihood and obtain the observed information matrix. We provide an application to real data which illustrates the usefulness of the model.     

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

In this article, we define and study a new three-parameter model called the Marshall–Olkin extended generalized Lindley distribution. We derive various structural properties of the proposed model including expansions for the density function, ordinary moments, moment generating function, quantile function, mean deviations, Bonferroni and Lorenz curves, order statistics and their moments, Rényi entropy and reliability. We estimate the model parameters using the maximum likelihood technique of estimation. We assess the performance of the maximum likelihood estimators in a simulation study. Finally, by means of two real datasets, we illustrate the usefulness of the new model.     

Marshall–Olkin distribution: parameter estimation and application to cancer data

In this study, as alternatives to the maximum likelihood (ML) and the frequency estimators, we propose robust estimators for the parameters of Zipf and Marshall–Olkin Zipf distributions. A small simulation study is given to illustrate the performance of the proposed estimators. We apply the proposed estimators to a real data set from cancer research to illustrate the performance of the proposed estimators over the ML, moments and frequency estimators. We observe that the robust estimators have superiority over the frequency estimators based on classical sample mean.     

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.     

Testing for the Marshall–Olkin extended form of the Weibull distribution

Marshall–Olkin extended distributions offer a wider range of behaviour than the basic distributions from which they are derived and therefore may find applications in modeling lifetime data, especially within proportional odds models, and elsewhere. The present paper carries out a simulation study of likelihood ratio, Wald and score tests for the parameter that distinguishes the extended distribution from the basic one, for the Weibull and exponential cases, allowing for right censored data. The likelihood ratio test is found to perform better than the others. The test is shown to have sufficient power to detect alternatives that correspond to interesting departures from the basic model and can be useful in modeling.     

Comparison of estimation methods for the Marshall–Olkin extended Lindley distribution

The aim of this paper is to compare the parameters' estimations of the Marshall–Olkin extended Lindley distribution obtained by six estimation methods: maximum likelihood, ordinary least-squares, weighted least-squares, maximum product of spacings, Cramér–von Mises and Anderson–Darling. The bias, root mean-squared error, average absolute difference between the true and estimate distributions' functions and the maximum absolute difference between the true and estimate distributions' functions are used as comparison criteria. Although the maximum product of spacings method is not widely used, the simulation study concludes that it is highly competitive with the maximum likelihood method.     

On a generalization to Marshall–Olkin scheme and its application to Burr type XII distribution

Marshall–Olkin semi-Burr and Marshall–Olkin Burr distributions are introduced and studied. Their various characteristics in reliability analysis are derived. Applications in time series analysis are discussed.     

Estimation and prediction of Marshall–Olkin extended exponential distribution under progressively type-II censored data

In this paper, we consider Marshall–Olkin extended exponential (MOEE) distribution which is capable of modelling various shapes of failure rates and aging criteria. The purpose of this paper is three fold. First, we derive the maximum likelihood estimators of the unknown parameters and the observed the Fisher information matrix from progressively type-II censored data. Next, the Bayes estimates are evaluated by applying Lindley’s approximation method and Markov Chain Monte Carlo method under the squared error loss function. We have performed a simulation study in order to compare the proposed Bayes estimators with the maximum likelihood estimators. We also compute 95% asymptotic confidence interval and symmetric credible interval along with the coverage probability. Third, we consider one-sample and two-sample prediction problems based on the observed sample and provide appropriate predictive intervals under classical as well as Bayesian framework. Finally, we analyse a real data set to illustrate the results derived.     

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.     

Modified proportional hazard rates and proportional reversed hazard rates models via Marshall–Olkin distribution and some stochastic comparisons

Adding parameters to a known distribution is a useful way of constructing flexible families of distributions. Marshall and Olkin (1997) introduced a general method of adding a shape parameter to a family of distributions. In this paper, based on the Marshall–Olkin extension of a specified distribution, we introduce two new models, referred to as modified proportional hazard rates (MPHR) and modified proportional reversed hazard rates (MPRHR) models, which include as special cases the well-known proportional hazard rates and proportional reversed hazard rates models, respectively. Next, when two sets of random variables follow either the MPHR or the MPRHR model, we establish some stochastic comparisons between the corresponding order statistics based on majorization theory. The results established here extend some well-known results in the literature.     

Marshall–Olkin Extended Lomax Distribution and Its Application to Censored Data

The problem of making statistical inference about θ =P(X > Y) has been under great investigation in the literature using simple random sampling (SRS) data. This problem arises naturally in the area of reliability for a system with strength X and stress Y. In this study, we will consider making statistical inference about θ using ranked set sampling (RSS) data. Several estimators are proposed to estimate θ using RSS. The properties of these estimators are investigated and compared with known estimators based on simple random sample (SRS) data. The proposed estimators based on RSS dominate those based on SRS. A motivated example using real data set is given to illustrate the computation of the newly suggested estimators.     

Comparisons of ten estimation methods for the parameters of Marshall–Olkin extended exponential distribution

The aim of this article is to compare via Monte Carlo simulations the finite sample properties of the parameter estimates of the Marshall–Olkin extended exponential distribution obtained by ten estimation methods: maximum likelihood, modified moments, L-moments, maximum product of spacings, ordinary least-squares, weighted least-squares, percentile, Crámer–von-Mises, Anderson–Darling, and Right-tail Anderson–Darling. The bias, root mean-squared error, absolute and maximum absolute difference between the true and estimated distribution functions are used as criterion of comparison. The simulation study reveals that the L-moments and maximum products of spacings methods are highly competitive with the maximum likelihood method in small as well as in large-sized samples.     

Applications of Marshall–Olkin Fréchet Distribution

In this article, we consider the applications of Marshall–Olkin Fréchet distribution. The reliability of a system when both stress and strength follows the new distribution is discussed and related characteristics are computed for simulated data. The model is applied to a real data set on failure times of air-conditioning systems in jet planes and reliability is estimated. We also develop acceptance sampling plan for the acceptance of a lot whose lifetime follows this distribution. Four different autoregressive time series models of order 1 are developed with minification structure as well as max-min structure having these stationary marginal distributions. Some properties of the models are also established.     

Duality of the Multivariate Distributions of Marshall–Olkin Type and Tail Dependence

A dual class of the multivariate distributions of Marshall–Olkin type is introduced, and their copulas are presented and utilized to derive explicit expressions of the distributional tail dependencies, which describe the amount of dependence in the upper-orthant tail or lower-orthant tail of a multivariate distribution and can be used in the study of dependence among extreme values. A sufficient condition under which tail dependencies of two such distributions can be compared are obtained. Some examples are also presented to illustrate our results.     

Estimation of reliability from Marshall–Olkin extended Lomax distributions

In this paper, we are mainly interested in estimating the reliability R=P(X>Y) in the Marshall–Olkin extended Lomax distribution, recently proposed by Ghitany et al. [Marshall–Olkin extended Lomax distribution and its application, Commun. Statist. Theory Methods 36 (2007), pp. 1855–1866]. The model arises as a proportional odds model where the covariate effect is replaced by an additional parameter. Maximum likelihood estimators of the parameters are developed and an asymptotic confidence interval for R is obtained. Extensive simulation studies are carried out to investigate the performance of these intervals. Using real data we illustrate the procedure.