On a New Modification of the Weibull Model with Classical and Bayesian Analysis

Modelling data in applied areas particularly in reliability engineering is a prominent research topic. Statistical models play a vital role in modelling reliability data and are useful for further decision-making policies. In this paper, we study a new class of distributions with one additional shape parameter, called a new generalized exponential-X family. Some of its properties are taken into account. The maximum likelihood approach is adopted to obtain the estimates of the model parameters. For assessing the performance of these estimators, a comprehensive Monte Carlo simulation study is carried out. The usefulness of the proposed family is demonstrated by means of a real-life application representing the failure times of electronic components. The fitted results show that the new generalized exponential-X family provides a close fit to data. Finally, considering the failure times data, the Bayesian analysis and performance of Gibbs sampling are discussed. The diagnostics measures such as the Raftery–Lewis, Geweke, and Gelman–Rubin are applied to check the convergence of the algorithm.

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New generalized-X family: Modeling the reliability engineering applications

PLoS ONE ◽

10.1371/journal.pone.0248312 ◽

2021 ◽

Vol 16 (3) ◽

pp. e0248312

Author(s):

Wanting Wang ◽

Zubair Ahmad ◽

Omid Kharazmi ◽

Clement Boateng Ampadu ◽

E. H. Hafez ◽

...

Keyword(s):

Failure Time ◽

Weibull Model ◽

Failure Time Data ◽

Reliability Engineering ◽

Time Data ◽

Monte Carlo Simulation Study ◽

New Family ◽

Machine Failure ◽

Modeling Data ◽

And Performance

As is already known, statistical models are very important for modeling data in applied fields, particularly in engineering, medicine, and many other disciplines. In this paper, we propose a new family to introduce new distributions suitable for modeling reliability engineering data. We called our proposed family a new generalized-X family of distributions. For the practical illustration, we introduced a new special sub-model, called the new generalized-Weibull distribution, to describe the new family’s significance. For the proposed family, we introduced some mathematical reliability properties. The maximum likelihood estimators for the parameters of the new generalized-X distributions are derived. For assessing the performance of these estimators, a comprehensive Monte Carlo simulation study is carried out. To assess the efficiency of the proposed model, the new generalized-Weibull model is applied to the coating machine failure time data. Finally, Bayesian analysis and performance of Gibbs sampling for the coating machine failure time data are also carried out. Furthermore, the measures such as Gelman-Rubin, Geweke and Raftery-Lewis are used to track algorithm convergence.

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Estimation procedures on Type-II censored data from a scaled Muth distribution

Sigma Journal of Engineering and Natural Sciences – Sigma Mühendislik ve Fen Bilimleri Dergisi ◽

10.14744/sigma.2021.00004 ◽

2021 ◽

pp. 148-158

Author(s):

Hayrinisa Demirci BIÇER

Keyword(s):

Censored Data ◽

Mean Squared Error ◽

Real Life ◽

Model Parameters ◽

Type Ii ◽

Monte Carlo Simulation Study ◽

Anal Ysis ◽

Squared Error ◽

Type Ii Censored Data ◽

Censoring Scheme

In the present paper, we consider the estimation problem for the scaled Muth distribution under Type-II censoring scheme. In order to estimate the model parameters α and β, the maximum likelihood, the least-squares, and the maximum spacing estimators are derived. To show estimation efficiencies of the estimators obtained with this paper, we present an exten- sive Monte-Carlo simulation study in which the estimators are compared according to bias and mean squared error criteria. Furthermore, we evaluate the applicability of the scaled Muth distribution by taking into account both full and Type-II censored data situations by an anal- ysis conducted on a real-life dataset.

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A New Extended-X Family of Distributions: Properties and Applications

Computational and Mathematical Methods in Medicine ◽

10.1155/2020/4650520 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Mi Zichuan ◽

Saddam Hussain ◽

Anum Iftikhar ◽

Muhammad Ilyas ◽

Zubair Ahmad ◽

...

Keyword(s):

Real Data ◽

Model Parameters ◽

Data Sets ◽

Statistical Distributions ◽

Reliability Engineering ◽

Monte Carlo Simulation Study ◽

Diverse Range ◽

Heavy Tailed ◽

Log Normal ◽

Extended Weibull Distribution

During the past couple of years, statistical distributions have been widely used in applied areas such as reliability engineering, medical, and financial sciences. In this context, we come across a diverse range of statistical distributions for modeling heavy tailed data sets. Well-known distributions are log-normal, log-t, various versions of Pareto, log-logistic, Weibull, gamma, exponential, Rayleigh and its variants, and generalized beta of the second kind distributions, among others. In this paper, we try to supplement the distribution theory literature by incorporating a new model, called a new extended Weibull distribution. The proposed distribution is very flexible and exhibits desirable properties. Maximum likelihood estimators of the model parameters are obtained, and a Monte Carlo simulation study is conducted to assess the behavior of these estimators. Finally, we provide a comparative study of the newly proposed and some other existing methods via analyzing three real data sets from different disciplines such as reliability engineering, medical, and financial sciences. It has been observed that the proposed method outclasses well-known distributions on the basis of model selection criteria.

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A New Lifetime Exponential-X Family of Distributions with Applications to Reliability Data

Mathematical Problems in Engineering ◽

10.1155/2020/1316345 ◽

2020 ◽

Vol 2020 ◽

pp. 1-16

Author(s):

Xiaoyan Huo ◽

Saima K. Khosa ◽

Zubair Ahmad ◽

Zahra Almaspoor ◽

Muhammad Ilyas ◽

...

Keyword(s):

Weibull Distribution ◽

Model Parameters ◽

Reliability Engineering ◽

Reliability Data ◽

Monte Carlo Simulation Study ◽

Hazard Functions ◽

Practical Applications ◽

New Family ◽

Nonnested Models ◽

Family Of Distributions

Modeling reliability data with nonmonotone hazards is a prominent research topic that is quite rich and still growing rapidly. Many studies have suggested introducing new families of distributions to modify the Weibull distribution to model the nonmonotone hazards. In the present study, we propose a new family of distributions called a new lifetime exponential-X family. A special submodel of the proposed family called a new lifetime exponential-Weibull distribution suitable for modeling reliability data with bathtub-shaped hazard rates is discussed. The maximum-likelihood estimators of the model parameters are obtained. A brief Monte Carlo simulation study is conducted to evaluate the performance of these estimators. For illustrative purposes, two real applications from reliability engineering with bathtub-shaped hazard functions are analyzed. The practical applications show that the proposed model provides better fits than the other nonnested models.

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On a modified Burr XII distribution having flexible hazard rate shapes

Mathematica Slovaca ◽

10.1515/ms-2017-0344 ◽

2020 ◽

Vol 70 (1) ◽

pp. 193-212

Author(s):

Farrukh Jamal ◽

Christophe Chesneau ◽

M. Arslan Nasir ◽

Abdus Saboor ◽

Emrah Altun ◽

...

Keyword(s):

Hazard Rate ◽

Real Life ◽

Stochastic Ordering ◽

Maximum Likelihood Estimates ◽

Model Parameters ◽

Data Sets ◽

Monte Carlo Simulation Study ◽

Burr Xii Distribution ◽

Life Data ◽

Real Life Data

AbstractIn this paper, we propose a new three-parameter modified Burr XII distribution based on the standard Burr XII distribution and the composition technique developed by [14]. Among others, we show that this technique has the ability to significantly increase the flexibility of the former Burr XII distribution, with respect to the density and hazard rate shapes. Also, complementary theoretical aspects are studied as shapes, asymptotes, quantiles, useful expansion, moments, skewness, kurtosis, incomplete moments, moments generating function, stochastic ordering, reliability parameter and order statistics. Then, a Monte Carlo simulation study is carried out to assess the performance of the maximum likelihood estimates of the modified Burr XII model parameters. Finally, three applications to real-life data sets are presented, with models comparisons. The results are favorable for the new modified Burr XII model.

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A New Extended Burr XII Distribution

Austrian Journal of Statistics ◽

10.17713/ajs.v46i1.139 ◽

2017 ◽

Vol 46 (1) ◽

pp. 33-39 ◽

Cited By ~ 3

Author(s):

Indranil Ghosh ◽

Marcelo Bourguinon

Keyword(s):

Real Life ◽

Maximum Likelihood Estimates ◽

Model Parameters ◽

Data Sets ◽

Monte Carlo Simulation Study ◽

Burr Xii Distribution ◽

Life Data ◽

Real Life Data ◽

Proposed Model ◽

New Distribution

In this paper, we propose a new lifetime distribution, namely the extended Burr XII distribution (using the technique as mentioned in Cordeiro et al. (2015)). We derive some basic properties of the new distribution and provide a Monte Carlo simulation study to evaluate the maximum likelihood estimates of model parameters. For illustrative purposes, two real life data sets have been considered as an application of the proposed model.

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The Odd Gamma Weibull-Geometric Model: Theory and Applications

Mathematics ◽

10.3390/math7050399 ◽

2019 ◽

Vol 7 (5) ◽

pp. 399 ◽

Cited By ~ 1

Author(s):

Rana Muhammad Imran Arshad ◽

Christophe Chesneau ◽

Farrukh Jamal

Keyword(s):

Geometric Distribution ◽

Geometric Model ◽

Real Life ◽

Estimation Method ◽

Likelihood Estimation ◽

Stochastic Ordering ◽

Parameter Distribution ◽

Model Parameters ◽

Monte Carlo Simulation Study ◽

New Distribution

In this paper, we study a new four-parameter distribution called the odd gamma Weibull-geometric distribution. Having the qualities suggested by its name, the new distribution is a special member of the odd-gamma-G family of distributions, defined with the Weibull-geometric distribution as baseline, benefiting of their respective merits. Firstly, we present a comprehensive account of its mathematical properties, including shapes, asymptotes, quantile function, quantile density function, skewness, kurtosis, moments, moment generating function and stochastic ordering. Then, we focus our attention on the statistical inference of the corresponding model. The maximum likelihood estimation method is used to estimate the model parameters. The performance of this method is assessed by a Monte Carlo simulation study. An empirical illustration of the new distribution is presented by the analyses two real-life data sets. The results of the proposed model reveal to be better as compared to those of the useful beta-Weibull, gamma-Weibull and Weibull-geometric models.

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Critically evaluating the theory and performance of Bayesian analysis of macroevolutionary mixtures

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1518659113 ◽

2016 ◽

Vol 113 (34) ◽

pp. 9569-9574 ◽

Cited By ~ 167

Author(s):

Brian R. Moore ◽

Sebastian Höhna ◽

Michael R. May ◽

Bruce Rannala ◽

John P. Huelsenbeck

Keyword(s):

Bayesian Analysis ◽

Prior Distribution ◽

Likelihood Function ◽

Compound Poisson Process ◽

Diversification Rate ◽

Model Parameters ◽

Bayesian Approaches ◽

And Performance ◽

Theoretical Issues ◽

Lineage Diversification

Bayesian analysis of macroevolutionary mixtures (BAMM) has recently taken the study of lineage diversification by storm. BAMM estimates the diversification-rate parameters (speciation and extinction) for every branch of a study phylogeny and infers the number and location of diversification-rate shifts across branches of a tree. Our evaluation of BAMM reveals two major theoretical errors: (i) the likelihood function (which estimates the model parameters from the data) is incorrect, and (ii) the compound Poisson process prior model (which describes the prior distribution of diversification-rate shifts across branches) is incoherent. Using simulation, we demonstrate that these theoretical issues cause statistical pathologies; posterior estimates of the number of diversification-rate shifts are strongly influenced by the assumed prior, and estimates of diversification-rate parameters are unreliable. Moreover, the inability to correctly compute the likelihood or to correctly specify the prior for rate-variable trees precludes the use of Bayesian approaches for testing hypotheses regarding the number and location of diversification-rate shifts using BAMM.

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Beyond the Sin-G family: The transformed Sin-G family

PLoS ONE ◽

10.1371/journal.pone.0250790 ◽

2021 ◽

Vol 16 (5) ◽

pp. e0250790

Author(s):

Farrukh Jamal ◽

Christophe Chesneau ◽

Dalal Lala Bouali ◽

Mahmood Ul Hassan

Keyword(s):

Real Life ◽

Stochastic Ordering ◽

Theoretical Work ◽

Weibull Model ◽

Theory And Practice ◽

Likelihood Method ◽

Model Parameters ◽

Data Sets ◽

Real Life Data ◽

Convex Property

In recent years, the trigonometric families of continuous distributions have found a place of choice in the theory and practice of statistics, with the Sin-G family as leader. In this paper, we provide some contributions to the subject by introducing a flexible extension of the Sin-G family, called the transformed Sin-G family. It is constructed from a new polynomial-trigonometric function presenting a desirable “versatile concave/convex” property, among others. The modelling possibilities of the former Sin-G family are thus multiplied. This potential is also highlighted by a complete theoretical work, showing stochastic ordering results, studying the analytical properties of the main functions, deriving several kinds of moments, and discussing the reliability parameter as well. Then, the applied side of the proposed family is investigated, with numerical results and applications on the related models. In particular, the estimation of the unknown model parameters is performed through the use of the maximum likelihood method. Then, two real life data sets are analyzed by a new extended Weibull model derived to the considered trigonometric mechanism. We show that it performs the best among seven comparable models, illustrating the importance of the findings.

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Weighted Power Lomax Distribution and its Length Biased Version: Properties and Estimation Based on Censored Samples

Pakistan Journal of Statistics and Operation Research ◽

10.18187/pjsor.v17i2.3360 ◽

2021 ◽

pp. 343-356

Author(s):

Amal Soliman Hassan ◽

Ehab M. Almetwally ◽

Mundher Abdullah Khaleel ◽

Heba Fathy Nagy

Keyword(s):

Real Life ◽

Model Parameters ◽

Weighted Version ◽

Data Set ◽

Monte Carlo Simulation Study ◽

Lifetime Distributions ◽

Lomax Distribution ◽

Censored Samples ◽

Real Life Data ◽

Asymptotic Confidence Intervals

In this paper, a weighted version of the power Lomax distribution referred to the weighted power Lomax distribution, is introduced. The new distribution comprises the length biased and the area biased of the power Lomax distribution as new models as well as containing an existing model as the length biased Lomax distribution as special model. Essential distributional properties of the weighted power Lomax distribution are studied. Maximum likelihood and maximum product spacing methods are proposed for estimating the population parameters in cases of complete and Type-II censored samples. Asymptotic confidence intervals of the model parameters are obtained. A sample generation algorithm along with a Monte Carlo simulation study is provided to demonstrate the pattern of the estimates for different sample sizes. Finally, a real-life data set is analyzed as an illustration and its length biased distribution is compared with some other lifetime distributions.

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