Modified Weibull Distribution Research Articles

A Generalized Class of Exponentiated Modified Weibull Distribution With Applications

Abstract: In this paper, a new class of five parameter gamma-exponentiated or generalized modified Weibull (GEMW) distribution which includes exponential, Rayleigh, Weibull, modified Weibull, exponentiated Weibull, exponentiated exponential, exponentiated modified Weibull, exponentiated modified exponential, gamma-exponentiated exponential, gamma exponentiated Rayleigh, gamma-modified Weibull, gamma-modified exponential, gamma-Weibull, gamma-Rayleigh and gamma-exponential distributions as special cases is proposed and studied. Mathematical properties of this new class of distributions including moments, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics and Renyi entropy are presented. Maximum likelihood estimation technique is used to estimate the model parameters and applications to real data sets presented in order to illustrate the usefulness of this new class of distributions and its sub-models.

Theoretical framework and inference for fitting extreme data through the modified Weibull distribution in a first-failure censored progressive approach

The importance of biomedical physical data is underscored by its crucial role in advancing our comprehension of human health, unraveling the mechanisms underlying diseases, and facilitating the development of innovative medical treatments and interventions. This data serves as a fundamental resource, empowering researchers, healthcare professionals, and scientists to make informed decisions, pioneer research, and ultimately enhance global healthcare quality and individual well-being. It forms a cornerstone in the ongoing pursuit of medical progress and improved healthcare outcomes. This article aims to tackle challenges in estimating unknown parameters and reliability measures related to the modified Weibull distribution when applied to censored progressive biomedical data from the initial failure occurrence. In this context, the article proposes both classical and Bayesian techniques to derive estimates for unknown parameters, survival, and failure rate functions. Bayesian estimates are computed considering both asymmetric and symmetric loss functions. The Markov chain Monte Carlo method is employed to obtain these Bayesian estimates and their corresponding highest posterior density credible intervals. Due to the inherent complexity of these estimators, which cannot be theoretically compared, a simulation study is conducted to evaluate the performance of various estimation procedures. Additionally, a range of optimization criteria is utilized to identify the most effective progressive control strategies. Lastly, the article presents a medical application to illustrate the effectiveness of the proposed estimators. Numerical findings indicate that Bayesian estimates outperform other estimation methods by achieving minimal root mean square errors and narrower interval lengths.

A modified Weibull distribution under competing risks

A modified Weibull distribution under competing risks

Emergent lifetime distribution from complex network systems aging

Most theoretical analysis for lifetime distribution explains origins of specific distribution based on independent failure. We develop one unified framework encompassing different kinds of lifetime distribution for failure coupling system. For typical complex networks, we found that three types of system lifetime distributions are emerged shaped by system size and failure coupling strength. When the failure coupling strength ϕ dominates, systems exhibit a cascade failure mode, the system lifetime following an exponential distribution as series systems due to long-range correlation. When the system size N dominates, systems exhibit wear-out failure mode, the system lifetime following the Gompertz model as parallel systems due to short-range correlation. When N and ϕ have comparable impact on systems, system lifetime follows a modified Weibull distribution. We find the critical failure coupling strength and critical system size which are helpful to identify the failure mode switch point. We provide rigorous theoretical analysis for emerged lifetime distribution. We reveal the microscopic mechanism of system lifetime distribution switch pattern by analyzing the competence between correlation length and network diameter. Finally, we verify our conclusions in real networks. Our study will help to understand the lifetime origin of complex systems and design highly reliable systems.

A New Modified-X family of distributions with applications in modeling biomedical data

In the present article, we derive a new Modified-X (NM-X) family of probability distributions by using a popular approach known as the T-X method. For the NM-X family, some of the main distributional (or statistical) properties such as quantile function, identifiability property, order statistics, residual life and reverse residual life, moments, and moments generating function are determined. A well-known method of estimation such as the Maximum Likelihood estimation method is used for estimating the model parameters of the introduced family of distributions. A Monte Carlo simulation method is incorporated to judge the performance of the estimators with different selected values of parameters. To check the tails behaviors of the proposed model, the risk measurements such as value-at-risk (VaR), tailed-value-at-risk (TVaR), tailed-variance (TV), and tailed-variance-premium (TVP) are investigated. For illustrative purposes, a special sub-model of the proposed class is derived and named a New Modified Weibull distribution (abbreviated as the NM-Weib). Furthermore, four applications from the field of medicine are analyzed to demonstrate the application, and effectiveness of the NM-Weib distribution. By adopting different diagnostic criteria, the NM-Weib distribution is compared with Weibull (Weib) distribution, Alpha Power Transformed-Weib (APTra-Weib) distribution, new Exponent-Weib (NExpo-Weib) distribution, Flexible Reduced Logarithmic-Weib (FRLog-Weib) distribution, Marshal Olkin-Weib (MO-Weib) distribution, and Kumaraswamy-Weib (K-Weib) distribution. Finally, based on the diagnostic criteria, it is observed that the NM-Weib distribution provides a suitable and best fit for each considered biomedical data set.

THE COMPARISON SOME EXTENDED WEIBULL DISTRIBUTION (WEIBULL (W), EXPONENTIATED WEIBULL (EW), EXPONENTIATED EXPONENTIAL WEIBULL (EEW) AND ADDITIVE WEIBULL (AW)) FOR THE WIND SPEED PROBABILITY MODELLING

Accurate wind speed modeling is critical in estimating wind energy potential for harnessing wind power effectively. One of the bases for assessment of wind energy potential for a specified region is the probability distribution of wind speed, therefore wind speed data is needed to produce statistical modeling, especially in determining the best probability distribution. For this purpose, several modified weibull distributions will be used and tested to determine the best model to describe wind speed in Pekanbaru. The main goal of this study is to find the best fitting distribution to the daily wind speed measured over Pekanbaru region for the years 1999-2020 by using the four modified weibull distributions, namely Weibull (W), Exponentiated Weibull (EW), Additive Weibull (AW) and Exponentiated Exponential Weibull (EEW). The maximum likelihood method will be used to get the estimated parameter value from the distribution used in this study. Furthermore, the graphical inspection (density-density plot and cumulative plot) and numerical criteria (Akaike’s information criterion (AIC), Bayesian Information Criteria (BIC), - log likelihood (- l) ) were used to determine the best fit model. In most cases, the results produced by the graphical inspection were similar, and differed from the numerical criteria . The best fit result was chosen as the distribution with the lowest values of AIC, BIC and - l. In general, the Exponentiated Exponential Weibull (EEW) distribution has been selected as the best model.

Estimation of the Parameters of the Modified Weibull Distribution with Bathtub-shaped Failure Rate Function

Estimation of the Parameters of the Modified Weibull Distribution with Bathtub-shaped Failure Rate Function

Transmuted Modified Weibull Distribution for Modeling Skewed Lifetime Dataset; Properties and Application

A new five parameter model called Generalized Modified Weibull distribution is proposed and studied. The new distribution generalizes the Modified Weibull Distribution introduced by Lai et al. (2003) using transformed – Transformer framework of Alzaatreh et al. (2014) which resulted to a distribution capable of modeling skewed data (positively or negatively skewed data), as well as, symmetric data. Property of a proper probability density function was used to ascertain that the resulting function is a proper probability density function. Statistical properties of the newly generated distribution were studied. Graph of probability density function of the distribution was used to show that it is capable of modeling skewed data. Graph of cumulative density functions of the distribution was plotted using varying parameter values. Monte Carlo simulation approach was used for the test of homogeneity of the distribution and it was observed that the parameters in the distribution approach the true values as sample size increases. Maximum likelihood Estimation method was used for estimation of the model parameters. Real life dataset was used for model comparison as well as demonstration of its application.

A two‐parameter parsimonious bathtub model for the analysis of system failure times with illustrations to reliability data

AbstractIn this study, a two‐parameter, upper‐bounded probability distribution called the tau distribution is introduced and its applications in reliability engineering are presented. Each of the parameters of the tau distribution has a clear semantic meaning. Namely, one of them determines the upper bound of the distribution, while the value of the other parameter influences the shape of the cumulative distribution function. A remarkable property of this new probability distribution is that its probability density function, survival function, hazard rate function (HRF), and quantile function can all be expressed in terms of its cumulative distribution function. The HRF of the proposed probability distribution can exhibit an increasing trend and various bathtub shapes with or without a low and long‐flat phase (useful time phase), which makes this new distribution suitable for modeling a wide range of real‐world problems. The constraint maximum likelihood estimation, percentile estimation, approximate Bayesian computation, and approximate quantile estimation computation are proposed to calculate the unknown parameters of the model. The suitability of the estimation methods is verified with the aid of simulation and real‐world data results. The modeling capability of the tau distribution was compared with that of some well‐known two‐ and three‐parameter probability distributions using two data sets known from the literature of reliability engineering: time between failures data of a machining center, and time to failure of data acquisition system cards. Based on empirical results, the new distribution may be viewed as a viable competitor to the Weibull, Gamma, Chen, and modified Weibull distributions.

A new extension of the modified Weibull distribution with applications for engineering data

In this study, a novel extension of the three-parameter modified Weibull distribution called the NMW3 distribution is proposed. The NMW3 distribution exhibits increasing or bathtub-shaped hazard rates, which can be beneficial in the context of reliability. The statistical characteristics of the NMW3 distribution are examined, including the difference between the change points of the hazard rate and the mean residual lifetime functions. Additionally, the sensitivities of the parameters and the flatness of the bathtub curve of the NMW3 distribution are discussed. The Weibull probability paper is utilized to characterize plots related to the NMW3 distribution. The parameters of the NMW3 distribution are estimated using both the graphical approach and the maximum likelihood method. Through the analysis of three real data sets, it is demonstrated that the NMW3 distribution outperforms existing extensions of the modified Weibull distributions, highlighting its practical importance. Finally, a reduced distribution of the NMW3 distribution is proposed, and it shows that the reduced distribution has almost the same advantageous characteristics as the NMW3 distribution.

Modified generalized Weibull distribution: theory and applications

This article presents and investigates a modified version of the Weibull distribution that incorporates four parameters and can effectively represent a hazard rate function with a shape resembling a bathtub. Its significance in the fields of lifetime and reliability stems from its ability to model both increasing and decreasing failure rates. The proposed distribution encompasses several well-known models such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh, and modified Weibull distributions. The paper derives key mathematical statistics of the proposed distribution, including the quantile function, moments, moment-generating function, and order statistics density. Various mathematical properties of the proposed model are established, and the unknown parameters of the distribution are estimated using different estimation techniques. Furthermore, the effectiveness of these estimators is assessed through numerical simulation studies. Finally, the paper applies the new model and compares it with various existing distributions by analyzing two real-life time data sets.

Data-driven identification for approximate analytical solution of first-passage problem

The first-passage problem plays a significant role in engineering performance evaluation and design optimization. To address general stochastic dynamical systems, a data-driven method is proposed to identify approximate analytical solutions for the first-passage problem which explicitly includes parameters of the system, excitation, and those related to the initial and boundary conditions. The method consists of two successive processes. First, the probability density of the first-passage time is assumed to satisfy the modified Weibull distribution and its expansion expression is constructed by using the rule of dimensional consistency. Second, by comparing the expansion with the probability density of the first-passage time estimated from random state data, the coefficients are determined by solving a set of overdetermined linear algebraic equations. Two representative examples, including the Duffing oscillator and a 2-DOF nonlinear dynamical system, are discussed in detail to illustrate the application and efficiency of the data-driven method. The efficacies of the approximate analytical solutions for the external parameters are also verified.

On Bivariate Nadarajah-Haghighi Distribution derived from Farlie-Gumbel-Morgenstern copula in the Presence of Covariates

An important alternative distribution to the Weibull, generalized exponen-tial and gamma distributions that is used in survival analysis is the Nadarajah-Haghighi exponential distribution. Similar to the Weibull, generalized exponen-tial and gamma distributions, the Nadarajah-Haghighi exponential distributionis an extension of the well known exponential distribution. In this paper, a copulafunction commonly used to model very weak linear dependence was used to intro-duced a bivariate Nadarajah-Haghighi distribution. The joint survival function,joint probability density function and joint cumulative distribution were givenin closed form. Bayesian method of estimation was used to estimate the modelparameters considering the presence of right censoring and covariates. Posteriorsummaries of interest were obtained via standard Markov Monte Carlo (MCMC )technique. Two real data sets were used to illustrate the importance and flexi-bility of the bivariate model in comparison with some competing models. It wasobserved that, the bivariate Nadarajah-Haghighi distribution provides a better fltthan bivariate exponential, bivariate Weibull, bivariate generalized exponentialand bivariate modified Weibull distributions.

An improved modified Weibull distribution applied to predict the reliability evolution of an aircraft lock mechanism

A new improved modified Weibull distribution (IMW), which consists of two tandem failure models is proposed in this paper. The order statistics, moment estimate and maximum likelihood estimate of the new distribution are investigated. The distribution is flexible for modeling the three phases of the modified bathtub-shaped hazard function. The modified distribution allows for enhanced statistical simulation. The new distribution is analyzed and applied to two sets of known lifetime data to demonstrate the advantages of the model. To verify the practical engineering applications of the model, reliability evolution test data for an aircraft door lock mechanism are analyzed and fitted. Based on the obtained trends of the lock hook angle and lock hook distance, we predict the evolutionary data for numerous mechanism samples, thus reducing the test cost. By considering the subdistribution and removing one parameter, the fitting effect is improved over that of the traditional distribution model.

Estimation of the Modified Weibull Additive Hazards Regression Model under Competing Risks

The additive hazard regression model plays an important role when the excess risk is the quantity of interest compared to the relative risks, where the proportional hazard model is better. This paper discusses parametric regression analysis of survival data using the additive hazards model with competing risks in the presence of independent right censoring. In this paper, the baseline hazard function is parameterized using a modified Weibull distribution as a lifetime model. The model parameters are estimated using maximum likelihood and Bayesian estimation methods. We also derive the asymptotic confidence interval and the Bayes credible interval of the unknown parameters. The finite sample behaviour of the proposed estimators is investigated through a Monte Carlo simulation study. The proposed model is applied to liver transplant data.

Increasing to Accuracy Reductions Heterogeneous Dose Distribution to All-Out Volume of the Irradiation at Models Modified Weibull Distribution

Purpose: To development of the methods of the estimation quality, systematized clinical information, which is described in the manner of probability of the arising the NTCP (complications in tissues) depending on volume of the irradiation and sharing the dose their uniform and heterogeneous irradiation.
 Material and methods: Used designed mathematical models for calculation of NTCP and mathematical models, which allow to calculate reduction values of the identical doses of the uniform irradiation tissues equivalent on probability of the arising the beam complication to heterogeneous sharing the doses.
 Results: Method of increasing to precision of the estimation to consensus systematized clinical information, founded on its presentation in the manner of differential histogram dose-volume, and on reductions heterogeneous doses distribution in identical dose all-out and uniform irradiation tissue.
 Conclusion: It is shown that designed method of the estimation of the heterogeneous sharing the dose by way of his adduction to equivalent, on probability of the arising the beam complications in tissues, all-out irradiations tissues, allows to raise accuracy of the calculation of the identical dose of the source lumpy distribution tolerants doses. She allows to raise precision of the estimation to consensus systematized clinical values of tolerant doses, received of specialist.

The New Bivariate Odd Generalized Exponential Modified Weibull Distribution with Application

The New Bivariate Odd Generalized Exponential Modified Weibull Distribution with Application

Efficiency and Wave Run-Up of Porous Breakwater with Sloping Deck

In order to protect fragile shoreline and coastal assets during extreme storms, a combined floating breakwater-windbreak has been proposed to reduce both wind and wave energies in the sheltered area. The 1 km-long breakwater has a porous hull with internal tubes to allow free passage of water; thereby further dissipating wave energy. The deck of the structure is designed to have a slope of 25 degrees facing the upstream side, and arrays of cylindrical tubes are placed on the sloping deck to form a windbreak. A reduced-scale (1:50) model test was carried out in a wave flume to examine wave sheltering performance under significant wave heights Hs = 3.0 m to 7.5 m and peak wave periods Tp = 9.4 s to 14 s sea states. Both regular and random wave conditions with different wave heights were considered. It is found that transmission coefficients ranging from 0.4 to 0.6 can be achieved under tested wave conditions. Porous breakwater hull increases the wave dissipation coefficients and is effective in reducing the wave reflection at the upstream side. The wave run-up length is dependent on the Iribarren number if the reduction induced by vertical freeboard is considered. Based on experimental data, empirical formulae have been proposed to predict the wave run-up responses in regular waves, probability of non-zero wave run-up occurrence, modified Weibull distribution of the wave run-up peaks and extreme wave run-up in random waves.

Topp–Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data

In this article, a four parameter lifetime model called the Topp–Leone modified Weibull distribution is proposed. The suggested distribution can be considered as an alternative to Kumaraswamy Weibull, generalized modified Weibull, extend odd Weibull Lomax, Weibull-Lomax, Marshall-Olkin alpha power extended Weibull and exponentiated generalized alpha power exponential distributions, etc. The suggested model includes the Topp-Leone Weibull, Topp-Leone Linear failure rate, Topp-Leone exponential and Topp-Leone Rayleigh distributions as a special case. Several characteristics of the new suggested model including quantile function, moments, moment generating function, central moments, mean, variance, coefficient of skewness, coefficient of kurtosis, incomplete moments, the mean residual life and the mean inactive time are derived. The probability density function of the Topp–Leone modified Weibull distribution can be right skewed and uni-modal shaped but, the hazard rate function may be decreasing, increasing, J-shaped, U-shaped and bathtub on its parameters. Three different methods of estimation as; maximum likelihood, maximum product spacing and Bayesian methods are used to estimate the model parameters. For illustrative reasons, applications of the Topp–Leone modified Weibull model to four real data sets related to medical and engineering sciences are provided and contrasted with the fit reached by several other well-known distributions.

Statistical Inference for Multi State Systems under the Generalized Modified Weibull Class

Multi state systems can be seen as semi-Markov processes by considering an arbitrary distribution function for sojourn times. Especially, in this work, the Modified Weibull distribution is employed to be the distribution of sojourn times with a shape parameter λ such that is member of a distributions family that is closed under minima. Parameters estimators are provided and the proposed methodology is evaluated using a detailed simulation procedure.