Marshall-Olkin Family Research Articles

A new Marshall-Olkin generalized-k family of distributions with applications

Abstract In this article, we describe a new Marshall-Olkin family of distributions based on the Marshall-Olkin family of distributions by employing a different generator. We have derived several statistical properties of the new Marshall-Olkin family, including survival function, hazard rate function, quantile function, moments, and linear representation. We have taken the Weibull distribution as a baseline distribution and found that the new Marshall-Olkin Weibull distribution exhibits a variety of shapes for its hazard rates and densities. We have tested the performance of various estimation methods for the new Marshall-Olkin family of distributions, including eight different estimation methods. Additionally, we conducted extensive simulations to assess the robustness of the estimation methods across various scenarios and determine which methods are most effective for varying sample sizes. To further validate the new Marshall-Olkin Weibull distribution, we have used two different data sets to compare the fitted new Marshall-Olkin Weibull distribution with the well-known comparative models. By testing the performance of different estimation methods and using multiple data sets, we can demonstrate the versatility and usefulness of the new Marshall-Olkin family of distributions for modeling a wide range of phenomena.

New results for the Marshall-Olkin family of distributions

Abstract The Marshall-Olkin family of probability distributions has been the inspiration of numerous research publications in the field of probability distributions. In this paper, we present several new properties of this family. In particular, we focus on stochastic orders, stress-strength reliability, Lorenz and the Leimkhuler curves, compounding, and integrated tail distribution. Two applications related to Lorenz curves and ruin theory are finally presented.

An extended Rayleigh Weibull model with actuarial measures and applications

In this article, we discuss a new extension of the Rayleigh-Weibull model using the Marshall-Olkin family of distributions. The proposed model is called the Marshall-Olkin-Rayleigh-Weibull (MORW) model. Various statistical properties of the MORW distribution are discussed, including explicit expressions for quantiles, moments, incomplete and conditional moments, some inequality measures, moment generating function, moments of the residual and reversed residual life, the Rényi entropy, and order statistics. Six different estimation methods are considered to investigate the behavior of the model parameters within the MORW model. A Monte Carlo simulation study is conducted to evaluate the performance of these different estimators. In addition, some actuarial measures, such as the Value-at-Risk, the expected shortfall, the tail Value-at-Risk, the tail variance, the tail variance premium, the tail conditional expectation, and the tail standard deviation premium, are calculated. Finally, applying the model to real data sets illustrates the applicability and usefulness of the MORW distribution.

Stochastic comparisons of largest claim amounts from heterogeneous portfolios

This paper investigates stochastic comparisons of largest claim amounts of two sets of independent or interdependent portfolios in the sense of some stochastic orders. Let random variable () with distribution function , represents the claim amount for ith risk of a portfolio. Here two largest claim amounts are compared considering that the claim variables follow a general semiparametric family of distributions having the property that the survival function is increasing in or is increasing and convex/concave in . The results obtained in this paper apply to a large class of well‐known distributions including the family of exponentiated/generalized distributions (e.g., exponentiated exponential, Weibull, gamma and Pareto family), Rayleigh distribution and Marshall–Olkin family of distributions. As a direct consequence of some main theorems, we also obtained the results for scale family of distributions. Several numerical examples are provided to illustrate the results.

Stochastic comparisons of largest claim and aggregate claim amounts

AbstractIn this paper, we establish some stochastic comparison results for largest claim amounts of two sets of independent and also for interdependent portfolios under the setup of the proportional odds model. We also establish stochastic comparison results for aggregate claim amounts of two sets of independent portfolios. Further, stochastic comparisons for largest claim amounts from two sets of independent multiple-outlier claims have also been studied. The results we obtained apply to the whole family of extended distributions, also known as the Marshall–Olkin family of distributions. We have given many numerical examples to illustrate the results obtained.

Generalized extended Marshall-Olkin family of lifetime distributions

Abstract We introduce a new generalized family of nonnegative continuous distributions by adding two extra parameters to a lifetime distribution, called the baseline distribution, by twice compounding a power series distribution. The new family, called the lifetime power series-power series family, has a serial arrangement of parallel structures, which extends the Marshall and Olkin structure. Four special models are discussed. A mathematical treatment of the new distributions is provided, including ordinary and incomplete moments, quantile, moment generating and mean residual functions. The maximum likelihood estimation technique is used to estimate the model parameters and a simulation study is conducted to investigate the performance of the maximum likelihood estimates. Its applicability is also illustrated by means of two real data sets.

Marshall-Olkin family of distribution: additional properties and comparative studies

In this paper we will study the relationship between the baseline pdf and the corresponding MO pdf, we will show the effectiveness on the percentile points, also we will give a closed form for the MLE estimator of the new parameter and will study some examples.

Harmonic Mixture-G Family of Distributions: Survival Regression, Simulation by Likelihood, Bootstrap and Bayesian Discussion with MCMC Algorithm

To study the heterogeneous nature of lifetimes of certain mechanical or engineering processes, a mixture model of some suitable lifetime distributions may be more appropriate and appealing than simpler models. In this paper, a new mixture family of the lifetime distributions is introduced via harmonic weighted mean of an underlying distribution and the distribution of the proportional hazard model corresponding to the baseline model.The proposed class of distributions includes the general Marshall-Olkin family of distributions as a special case. Some important properties of the proposed model such as survival function, hazard function, order statistics and some results on stochastic ordering are obtained in a general setting. A special case of this new family is considered by employingWeibull distribution as the parent distribution. We derive several properties of the special distribution such as moments,hazard function survival regression and certain characterizations results. Moreover, we estimate the parameters of the model by using frequentist and Bayesian approaches. For Bayesian analysis, five loss functions, namely the squared error loss function (SELF), weighted squared error loss function (WSELF), modified squared error loss function (MSELF), precautionary loss function (PLF), and K-loss function (KLF) are considered. The beta prior as well as the gamma prior are used to obtain the Bayes estimators and posterior risk of the unknown parameters of the model. Furthermore, credible intervals (CIs) and highest posterior density (HPD) intervals are also obtained. A simulation study is presented via Monte Carlo to investigate the bias and mean square error of the maximum likelihood estimators. For illustrative purposes, two real-life applications of the proposed distribution to Kidney and cancer patients are provided.

An Alternative to the Marshall-Olkin Family of Distributions: Bootstrap, Regression and Applications

An Alternative to the Marshall-Olkin Family of Distributions: Bootstrap, Regression and Applications

Generalization of the Ratio of Minimized Kullback- Leibler Divergence Discrimination Technique to Bivariate Marshall-Olkin Family

Generalization of the Ratio of Minimized Kullback- Leibler Divergence Discrimination Technique to Bivariate Marshall-Olkin Family

The new discrete distribution with application to COVID-19 Data

This research aims to model the COVID-19 in different countries, including Italy, Puerto Rico, and Singapore. Due to the great applicability of the discrete distributions in analyzing count data, we model a new novel discrete distribution by using the survival discretization method. Because of importance Marshall–Olkin family and the inverse Toppe–Leone distribution, both of them were used to introduce a new discrete distribution called Marshall–Olkin inverse Toppe–Leone distribution, this new distribution namely the new discrete distribution called discrete Marshall–Olkin Inverse Toppe–Leone (DMOITL). This new model possesses only two parameters, also many properties have been obtained such as reliability measures and moment functions. The classical method as likelihood method and Bayesian estimation methods are applied to estimate the unknown parameters of DMOITL distributions. The Monte-Carlo simulation procedure is carried out to compare the maximum likelihood and Bayesian estimation methods. The highest posterior density (HPD) confidence intervals are used to discuss credible confidence intervals of parameters of new discrete distribution for the results of the Markov Chain Monte Carlo technique (MCMC).

On the Discrete Weibull Marshall–Olkin Family of Distributions: Properties, Characterizations, and Applications

In this article, we introduce a new flexible discrete family of distributions, which accommodates wide collection of monotone failure rates. A sub-model of geometric distribution or a discrete generalization of the exponential model is proposed as a special case of the derived family. Besides, we point out a comprehensive record of some of its mathematical properties. Two distinct estimation methods for parameters estimation and two different methods for constructing confidence intervals are explored for the proposed distribution. In addition, three extensive Monte Carlo simulations studies are conducted to assess the advantages between estimation methods. Finally, the utility of the new model is embellished by dint of two real datasets.

Marshall-Olkin Log-Logistic Extended Weibull Distribution : Theory, Properties and Applications

Marshall and Olkin (1997) introduced a general method for obtaining more flexible distributions by adding a new parameter to an existing one, called the Marshall-Olkin family of distributions. We introduce a new class of distributions called the Marshall - Olkin Log-Logistic Extended Weibull (MOLLEW) family of distributions. Its mathematical and statistical properties including the quantile function hazard rate functions, moments, conditional moments, moment generating function are presented. Mean deviations, Lorenz and Bonferroni curves, R´enyi entropy and the distribution of the order statistics are given. The Maximum likelihood estimation technique is used to estimate the model parameters and a special distribution called the Marshall-Olkin Log Logistic Weibull (MOLLW) distribution is studied, and its mathematical and statistical properties explored. Applications and usefulness of the proposed distribution is illustrated by real datasets.

A Generalization Of Inverse Marshall-Olkin Family Of Distributions

We introduce a new family of distributions namely inverse truncated discrete Linnik G family of distributions. This family is a generalization of inverse Marshall-Olkin family of distributions, inverse family of distributions generated through truncated negative binomial distribution and inverse family of distributions generated through truncated discrete Mittag-Leffler distribution. A particular member of the family, inverse truncated negative binomial Weibull distribution is studied in detail. The shape properties of the probability density function and hazard rate, model identifiability, moments, median, mean deviation, entropy, distribution of order statistics, stochastic ordering property, mean residual life function and stress-strength properties of the new generalized inverse Weibull distribution are studied. The unknown parameters of the distribution are estimated using maximum likelihood method, product spacing method and least square method. The existence and uniqueness of the maximum likelihood estimates are proved. Simulation is carried out to illustrate the performance of maximum likelihood estimates of model parameters. An AR(1) minification model with this distribution as marginal is developed. The inverse truncated negative binomial Weibull distribution is fitted to a real data set and it is shown that the distribution is more appropriate for modeling in comparison with some other competitive models.

A sustainable generalization of inverse Lindley distribution for wind speed analysis in certain regions of Pakistan

Information on wind speed and wind power distribution is significant for a few reasons, for example, surveying wind assets, arranging wind cultivates, and limiting the liabilities for wind power improvement. This study provided an application of a new generalization of two-parameter generalized inverse Lindley distribution using the Marshall–Olkin family for analyzing wind speed and wind power characteristics. Some mathematical properties of the new distribution were studied. We had observed the suitability of new distribution as compared to the other well-known wind speed distributions such as Weibull, inverse exponential, inverted Kumaraswamy, inverse Weibull, inverse Lindley, and generalized inverse Lindley distribution. For this purpose, the time-based wind speed data is taken from the four stations of Pakistan as a case study. We conclude based on certain goodness of fit criteria that the newly developed distribution has a better fit as compared to the other wind speed distributions. Therefore, the new model can be used as an alternative distribution for the assessment of wind speed energy potential.

On the Genesis of the Marshall-Olkin Family of Distributions via the T-X Family Approach: Statistical Modeling

In the last couple of years, there Has been an increased interest among the statisticians to define new families of distributions by adding one or more additional parameter(s) to the baseline distribution. In this regard, a number of families have been introduced and studied. One such example is the Marshall-Olkin family of distributions that is one of the most prominent approaches used to generalize the existing distributions. Whenever, we see a new method, the natural questions come in to mind are (i) what are the genesis of the newly proposed method and (ii) how did the proposed method is obtained. No doubt, the Marshall-Olkin family is a very useful method and has attracted the researchers. But, unfortunately, the authors failed to provide the explanation about the genesis of the method that how this family of distributions is obtained. To address this issue, in this article, an attempt Has been made to provide a straight forward computation about the genesis of the Marshall-Olkin family that somehow completes its derivation. The genesis of the Marshall-Olkin family is based on the T-<i>X</i> family approach. Furthermore, we have showed that other extensions of the Marshall-Olkin family can also be obtained via the T-<i>X</i> family method. Finally, a real-life application form insurance science is presented to illustrate the newly proposed extension of the Marshall-Olkin family.

Multivariate Inverted Kumaraswamy Distribution: Derivation and Estimation

Industrial revolution leads to the manufacturing of multicomponent products; to guarantee the sufficiency of the product and consumer satisfaction, the producer has to study the lifetime of the products. This leads to the use of bivariate and multivariate lifetime distributions in reliability engineering. The most popular and applicable is Marshall–Olkin family of distributions. In this paper, a new bivariate lifetime distribution which is the bivariate inverted Kumaraswamy (BIK) distribution is found and its properties are illustrated. Estimation using both maximum likelihood and Bayesian approaches is accomplished. Using different selection criteria, it is found that BIK provides the best performance compared with other bivariate distributions like bivariate exponential and bivariate inverse Weibull distributions. As a generalization, the multivariate inverted Kumaraswamy (MIK) distribution is derived. Few studies have been conducted on the multivariate Marshall–Olkin lifetime distributions. To the best of our knowledge, none of them handle estimation process. In this paper, we developed an algorithm to show how to estimate the unknown parameters of MIK using both maximum likelihood and Bayesian approaches. This algorithm could be applied in estimating other Marshall–Olkin multivariate lifetime distributions.

Min-infinite divisibility of the bivariate Marshall–Olkin copulas

There are many well-known bivariate distributions, such as the normal distribution, for which the question of whether they are max- or min-infinite divisible was settled a long time ago. However, despite its popularity, the bivariate Marshall–Olkin family of copulas was never the target of such an investigation, presumably due to the deterrent character of its density. Herein, we show that the challenges faced can be overcome with ease thanks to a convenient factorization.

A new flexible bivariate discrete Rayleigh distribution generated by the Marshall-Olkin family

In the statistical analysis of bivariate data, it is possible to have discrete observations instead of continuous data, as observed in many studies with survival data. In this study it is introduced a new bivariate discrete distribution derived from two Rayleigh distributions using a method propose d by Marshall and Olkin (1997) where an additional parameter is is introduced to a family of distributions related to the dependence structure of two discrete random variables X1 and X2. The study results show that this new bivariate distribution has good statistical properties and a simple mathematical expression for its correlation coefficient. The usual classical and Bayesian estimators for the parameters of the new distribution are also presented. A simulation study is carried out in order to evaluate some frequentist properties of the proposed model. The usefulness of the proposed model is illustrated using a real medical dataset introduced in the literature in presence of censoring and covariates.

Modified generalized marshall-olkin family of distributions

In this article, we propose a new family of distributions using the T-X family named as modified generalized Marshall-Olkin family of distributions. Comprehensive mathematical and statistical properties of this family of distributions are provided. The model parameters are estimated by maximum likelihood method. The maximum likelihood estimation under Type-II censoring is also discussed. Two lifetime data sets are used to show the suitability and applicability of the new family of distributions. For comparison purposes, different goodness of fit tests are used. Â