Extended Weibull Distribution Research Articles

A New Inverse Extended Weibull Distribution for Modelling Insurance Loss Data

This paper proposes a new three-parameter inverse Weibull-type distribution via a similar way of constructing a skew normal distribution. The additional shape parameter increases the distribution’s modelling capability as other three-parameter inverse Weibull-type distributions. The mathematical properties of this inverse extended Weibull distribution are studied and proven. In the empirical study of four insurance datasets from Australia, Denmark and the USA, we show that the proposed distribution performs very well compared with other three-parameter competitors.

A New Extended Weibull Distribution with Application to Influenza and Hepatitis Data

The Weibull is a popular distribution that models monotonous failure rate data. In this work, we introduce the four-parameter Weibull extended Weibull distribution that presents greater flexibility, thus modeling data with bathtub-shaped and unimodal failure rate. Some of its mathematical properties such as quantile function, linear representation and moments are provided. The maximum likelihood estimation is adopted to estimate its parameters, and the log-Weibull extended Weibull regression model is presented. In addition, some simulations are carried out to show the consistency of the estimators. We prove the greater flexibility and performance of this distribution and the regression model through applications to influenza and hepatitis data. The new models perform much better than some of their competitors.

Three mixed-effects regression models using an extended Weibull with applications on games in differential and integral calculus

Three new mixed-effects regressions models using an extended Weibull distribution are defined for repeated measures, and their parameters are estimated by maximum likelihood. Monte Carlo simulations report the accuracy of the maximum likelihood estimators and the distribution of the quantile residuals in these regressions. The usefulness of the proposed regressions is illustrated in differential and integral class from the Exact Sciences Department at the University of São Paulo (Brazil) with the objective of showing a pedagogical alternative of learning diagnostic methodology as a game approach. The results indicate that the questions correctly answered by the students took less time to be solved than those incorrectly answered. In addition, the algebraic application and multiple representation questions has the lowest percentages of correct answers and, in general, the longest time to be solved. So, it is possible to note that the used game approach enables the identification of possible difficult points in a class and provides the teacher with the opportunity of search for different strategies to reduce these difficulties faced by differential and integral calculus students when entering higher education, which often result from basic education.

Marshall Olkin Alpha Power Extended Weibull Distribution: Different Methods of Estimation based on Type I and Type II Censoring

In this paper, we insert and study a novel five-parameter extended Weibull distribution denominated as the Marshall–Olkin alpha power extended Weibull (MOAPEW) distribution. This distribution's statistical properties are discussed. Maximum likelihood estimations (MLE), maximum product spacing (MPS), and Bayesian estimation for the MOAPEW distribution parameters are obtained using Type I and Type II censored samples. A numerical analysis using Monte-Carlo simulation and real data sets are realized to compare various estimation methods. The supremacy of this novel model upon some famous distributions is explicated using different real datasets as it appears the MOAPEW model achieves a good fit for these applications.

The extended Weibull distribution with its properties, estimation and modeling skewed data

The present article introduces a new distribution called the odd log–logistic Lindley-Weibull (OLLLW) distribution that provides greater flexibility in modeling data in applied areas such as medicine and engineering. The OLLLW model provides left-skewed, symmetric, right-skewed, and reversed-J shaped densities. Its hazard function can be bathtub, unimodal, increasing or decreasing. The OLLLW density was expressed as a linear mixture of Weibull densities. Some distributional properties of the introduced model are derived. Its parameters are estimated using five classical estimators called, the maximum likelihood, Anderson–Darling, least-squares, Cramér-von Mises, and weighted least squares estimators. The performance of the proposed estimators is explored by detailed simulation results. The flexibility of the OLLLW distribution is studied by two real data sets from medicine and engineering sciences, showing that its capability to fit the data effectively than the Weibull, Fréchet Weibull, transmuted Weibull, gamma Weibull, transmuted exponentiated Weibull, and modified Weibull distributions.

A New Flexible Statistical Model: Simulating and Modeling the Survival Times of COVID-19 Patients in China

The spread of the COVID-19 epidemic, since December 2019, has caused much damage around the world, disturbed every aspect of daily life, and has become a serious health threat. The COVID-19 epidemic impacted nearly 150 countries around the globe between December 2019 and March 2020. Since December 2019, researchers have been trying to develop new suitable statistical models to adequately describe the behavior of this deadly pandemic. In this paper, a flexible statistical model has been proposed that can be used to model the lifetime events associated with this deadly pandemic. The new distribution is derived from the combination of an extended Weibull distribution and a trigonometric strategy referred to as the arcsine-X approach. Hence, the new model may be referred to as the arcsine new flexible extended Weibull model. The proposed model is capable of capturing five different behaviors of the hazard rate function. The model parameters are estimated via the maximum likelihood approach. Furthermore, a Monte Carlo study is conducted to assess the behavior of the estimators. Finally, the applicability of the new model is demonstrated using the data of fifty-three patients taken from a hospital in China.

A Study on Various Transformation Method of Weibull Distribution: A Review

In this article a brief summary of some recent developments of Weibull lifetime models has been presented for a quick overview. Various extensions of the Weibull models and the properties of the extended Weibull distribution have been discussed. A brief discussion about the characteristics and shape behaviour has been presented in the tabular form. Finally, some future research topics have been given.

The Mixture of the Marshall–Olkin Extended Weibull Distribution under Type-II Censoring and Different Loss Functions

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 as compared to simple models. This paper considers a mixture of the Marshall–Olkin extended Weibull distribution for efficient modeling of failure, survival, and COVID-19 data under classical and Bayesian perspectives based on type-II censored data. We derive several properties of the new distribution such as moments, incomplete moments, mean deviation, average lifetime, mean residual lifetime, Rényi entropy, Shannon entropy, and order statistics of the proposed distribution. Maximum likelihood and Bayes procedure are used to derive both point and interval estimates of the parameters involved in the model. Bayes estimators of the unknown parameters of the model are obtained under symmetric (squared error) and asymmetric (linear exponential (LINEX)) loss functions using gamma priors for both the shape and the scale parameters. Furthermore, approximate confidence intervals and Bayes credible intervals (CIs) are also obtained. Monte Carlo simulation study is carried out to assess the performance of the maximum likelihood estimators and Bayes estimators with respect to their estimated risk. The flexibility and importance of the proposed distribution are illustrated by means of four real datasets.

Generalized extended Weibull power series family of distributions

In this study, we introduce a new family of models for lifetime data called generalized extended Weibull power series family of distributions by compounding generalized extended Weibull distributions and power series distributions. The compounding procedure follows the same setup carried out by Adamidis (1998). The proposed family contains all types of combinations between truncated discrete with generalized and non-generalized Weibull distributions. Some existing power series and subclasses of mixed lifetime distributions become special cases of the proposed family, such as the compound class of extended Weibull power series distributions proposed by Silva et al. (2013) and the generalized exponential power series distributions introduced by Mahmoudi and Jafari (2012). Some mathematical properties of the new class are studied, including the cumulative distribution function, density function, survival function, and hazard rate function. The method of maximum likelihood is used for obtaining a general setup for estimating the parameters of any distribution in this class. An expectation-maximization algorithm is introduced for estimating maximum likelihood estimates. Special subclasses and applications for some models in a real dataset are introduced to demonstrate the flexibility and the benefit of this new family.

A Modification of the Gompertz Distribution Based on the Class of Extended-Weibull Distributions

This paper introduces a new four-parameter extension of the generalized Gompertz distributions. This distribution involves some well-known distributions such as extension of generalized exponential, generalized exponential, and generalized Gompertz distributions. In addition, it can have a decreasing, increasing, upside-down bathtub, and bathtub-shaped hazard rate function depending on its parameters. Some mathematical properties of this new distribution, such as moments, quantiles, hazard rate function, and reversible hazard rate function are obtained. In addition, the density function and the moments of the ordered statistics of this new distribution is provided. The parameters of model are estimated using the maximum likelihood method. Also, a real data set was used to illustrate the validity of the proposed distribution.

Inference of extended Weibull distribution using ramp stress accelerated life testing under censoring

In this paper, the expectation-maximization (EM) algorithm and adaptive type-II progressive censoring (A-II-PC) scheme are discussed for the first time under ramp-stress accelerated life testing (RS-ALT) experiment based on extended Weibull (EW) distribution. Under the assumptions of cumulative exposure (CE) model and inverse power law relationship, the maximum likelihood estimators (MLEs) of the unknown parameters and acceleration factors are obtained using the EM algorithm, then compared with the scoring algorithm. Furthermore, the observed information matrix based on the missing value principle is computed and used to construct the asymptotic confidence intervals for the parameters and acceleration factors. Bootstrap techniques are also considered in the interval estimation. Moreover, a numerical example is presented to illustrate the application of both the scoring and EM algorithms. Finally, a Monte Carlo simulation study is carried out to assess the performance of the suggested algorithms.

A New Extended-F Family: Properties and Applications to Lifetime Data

In this article, a new approach is used to introduce an additional parameter to a continuous class of distributions. The new class is referred to as a new extended-F family of distributions. The new extended-Weibull distribution, as a special submodel of this family, is discussed. General expressions for some mathematical properties of the proposed family are derived, and maximum likelihood estimators of the model parameters are obtained. Furthermore, a simulation study is provided to evaluate the validity of the maximum likelihood estimators. Finally, the flexibility of the proposed method is illustrated via two applications to real data, and the comparison is made with the Weibull and some of its well-known extensions such as Marshall–Olkin Weibull, alpha power-transformed Weibull, and Kumaraswamy Weibull distributions.

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

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.

A New Flexible Bathtub-Shaped Modification of the Weibull Model: Properties and Applications

Many studies have suggested the modifications and generalizations of the Weibull distribution to model the nonmonotone hazards. In this paper, we combine the logarithms of two cumulative hazard rate functions and propose a new modified form of the Weibull distribution. The newly proposed distribution may be called a new flexible extended Weibull distribution. Corresponding hazard rate function of the proposed distribution shows flexible (monotone and nonmonotone) shapes. Three different characterizations along with some mathematical properties are provided. We also consider the maximum likelihood estimation procedure to estimate the model parameters. For the 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.

Inference about the bivariate new extended Weibull distribution based on complete and censored data

In this article, we have discussed the problem of point estimation of the three unknown parameters of a bivariate new extended Weibull distribution under complete and randomly right-censored samples. The expectation-maximization algorithm is used to estimate the unknown parameters. Simulation experiments are performed to see the effectiveness of the estimators for complete and censored data. One dataset has been considered to illustrate the practical utility of the article.

Discrete Bivariate Distributions Generated By Copulas: DBEEW Distribution

Copulas are the most powerful tools in constructing continuous multivariate distributions given marginals and certain dependence structures. In this paper, we shall propose a general and novel method of generating discrete bivariate distributions using copulas. The advantage of our method is that, contrary to the standard methods, we do not need to have the joint distribution of the base variables, but we only need the marginal distributions. In particular, we shall concentrate on generating a new family of discrete bivariate exponentiated extended Weibull (DBEEW) distribution by a Cuadras–Auge copula. We shall study several mathematical properties of a DBEEW distribution. Three special submodels of our new proposed distribution are studied in more detail. Finally, estimation of parameters for these three submodels is investigated and their potential and flexibility are examined using a real data set.

A New Discrete Extended Weibull Distribution

In this paper, the discrete extended Weibull distribution is introduced by discretizing a new extended the continuous Weibull distribution. The new model has decreasing, increasing, constant and upside-down bathtub shaped hazard rates. Some of basic distributional and reliability properties are studied. The maximum likelihood method is used to estimate the parameters of the model. The performance of the estimation method is evaluated by a Monte-Carlo simulation. Four-real life data sets are considered for illustrating the advantages of the proposed distribution over some other well-known discrete distributions.

On a Generalization of the Weibull Distribution and Its Application in Quality Control

AbstractIn this article, a generalization of the Weibull distribution called Harris extended Weibull distribution is studied, and its properties are discussed. We fit the distribution to a real-life data set to show the applicability of this distribution in reliability modeling. Also, we derive a reliability test plan for acceptance or rejection of a lot of products submitted for inspection with lifetimes following this distribution. The operating characteristic functions of the sampling plans are obtained. The producer’s risk, minimum sample sizes and associated characteristics are computed and presented in tables. The results are illustrated using two data sets on ordered failure times of products as well as failure times of ball bearings.

New Extended Weibull Distribution

This article considers a new function to propose a new lifetime model. The new model is introduced by utilizing the linear scheme of the two logarithms of cumulative hazard functions. The new model is named as new extended Weibull distribution, and is able to model data with unimodal or modified unimodal shaped failure rates. A brief explanation of the mathematical properties of the proposed model is provided. The model parameters will be estimated by deploying the maximum likelihood method. To illustrate the usefulness of the proposed model, an example will be discussed.

On Marshall-Olkin Extended Weibull Distribution

In this paper, we introduce a new form of distributions called “Marshall-Olkin Extended Weibull distribution” MOEW. We study some of its structural properties and the shape of its hazard rate function. We provide some methods of point estimation for the unknown parameters and discuss its asymptotic properties. Finally, numerical methods and simulations are used to compare between different methods of estimation, then we study the behavior of parameters under different sample sizes and different values of parameters.