Modified Weibull Research Articles

Increasing Sorption Isotherms Accuracy: Weibull Modelling and Linear Regression

Relying on an adequate mathematical approach, two different mathematical procedures can be applied to the huge database produced during gas sorption isotherm experiments in order to obtain accurate data to be used in the industrial practice. To treat data determined from gas sorption isotherms without a careful mathematical support will produce inaccurate results, because all the determinations will be dependent on human decision. The minimum error reported since the first stage of a sorption isotherm determination, which corresponds to volume calibrations of reference and sample cells performed through the use of helium, will produce enormous inaccuracies on sorption isotherm behavior. These inaccurate behaviors may sometimes invalidate any Coalbed Methane recovery and CO2 injection programs. The study consisted on investigating gas sorption isotherm accuracies determined during the first part of the sorption process, which is mainly conducted by monitoring the pressure decline with time, in the reference and the sample cells (when both cells are not in contact), until the stabilization stage is achieved. Three samples from two different coals were selected in order to study their gas sorption behavior, in terms of a clear mathematical approach, when submitted to three different gas compositions, viz. 99.999% methane (CH4); 99.999% carbon dioxide (CO2); and a gas mixture containing 74.99% CH4 + 19.99% CO2 + 5.02% nitrogen (N2). Sorption experiments allow to conclude that the three samples present the same mathematical response during the first part of the sorption process. However, all gas sorption data (adsorption and desorption) collected from reference cell have a better fitting to a Modified Weibull Model, and all gas sorption data (adsorption and desorption) collected from sample cell respond in a trustworthy way to a Linear Regression Model. Confidence bands and prediction intervals (or bands) were also computed.

On the Exponentiated Generalized Modified Weibull Distribution

In this paper, we study a generalization of the modified Weibull distribution. The generalization follows the recent work of Cordeiro et al. (2013) and is based on a class of exponentiated generalized distributions that can be interpreted as a double construction of Lehmann. We introduce a class of exponentiated generalized modified Weibull (EGMW) distribution and provide a list of some well-known distributions embedded within the proposed distribution. We derive some mathematical properties of this class that include ordinary moments, generating function and order statistics. We propose a maximum likelihood method to estimate model parameters and provide simulation results to assess the model performance. Real data is used to illustrate the usefulness of the proposed distribution for modeling reliability data.

Prediction Intervals for Future Order Statistics from a Generalized Modified Weibull

Viewing the future order statistics as latent variables at each Gibbs sampling iteration, several Bayesian approaches to predict future order statistics based on type-II censored order statistics, X(1), X(2), …, X(r), of a size n( > r) random sample from a four-parameter generalized modified Weibull (GMW) distribution, are studied. Four parameters of the GMW distribution are first estimated via simulation study. Then various Bayesian approaches, which include the plug-in method, the Monte Carlo method, the Gibbs sampling scheme, and the MCMC procedure, are proposed to develop the prediction intervals of unobserved order statistics. Finally, four type-II censored samples are utilized to investigate the predictions.

A new generalization of lifetime distributions

In the current study, we set out to extend the three-parameter Modified Weibull (MW) distribution in an attempt to propose a four-parameter distribution named the Modified Weibull Poisson (MWP) distribution including such noticeable submodels as Exponential Poisson, Weibull Poisson, and Rayleigh Poisson known as the distributions subsumed under the umbrella term MWP. Depending on its parameter values, this overarching distribution was demonstrated by this work to exhibit some hazard rates like decreasing, increasing, bathtub, and upside-down bathtub ones. In addition to the hazard rates of the MWP, the mathematical properties as well as the properties of maximum likelihood estimations were brought to the forefront, and the very capability of the quantile measures to be explicitly expressed in terms of the Lambert W function was vigorously discussed. To shed light on the functioning of the maximum likelihood estimators and their asymptomatic results for the finite sample sizes, some numerical experiments were carried out leading to two data sets intended chiefly to illustrate or explicate the higher levels of importance and flexibility of the MWP in comparison with its standard counterparts, namely the Weibull, Gamma, and MW distributions.

A New Modified Weibull Distribution and Its Applications

In this paper we introduce a four-parameter generalized Weibull distribution. This new distribution has a more general form of failure rate function. It is more general for modeling than six ageing classes of life dis- tributions with appropriate choices of parameter values, so it can display decreasing, increasing, bathtub shaped, unimodal, increasing-decreasing in- creasing and decreasing-increasing-decreasing failure rates. The new distri- bution has also a bimodal density function. The moments are obtained and the method of maximum likelihood is used to estimate the model parame- ters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the advantage of the proposed distribution.

The Logistic-Uniform Distribution and Its Applications

In this article, we give a new family of univariate distributions generated by the Logistic random variable. A special case of this family is the Logistic-Uniform distribution. We show that the Logistic-Uniform distribution provides great flexibility in modeling for symmetric, negatively and positively skewed, bathtub-shaped, “J”-shaped, and reverse “J”-shaped distributions. We discuss simulation issues, estimation by the methods of moments, maximum likelihood, and the new method of minimum spacing distance estimator. We also derive Shannon entropy and asymptotic distribution of the extreme order statistics of this distribution. The new distribution can be used effectively in the analysis of survival data since the hazard function of the distribution can be “J,” bathtub, and concave-convex shaped. The usefulness of the new distribution is illustrated through two real datasets by showing that it is more flexible in analyzing the data than the Beta Generalized-Exponential, Beta-Exponential, Beta-Normal, Beta-Laplace, Beta Generalized half-Normal, β-Birnbaum-Saunders, Gamma-Uniform, Beta Generalized Pareto, Beta Modified Weibull, Beta-Pareto, Generalized Modified Weibull, Beta-Weibull, and Modified-Weibull distributions.

Transmuted Exponentiated Modified Weibull Distribution

A generalization of the exponentiated modified Weibull distribution so-called the transmuted exponentiated modified Weibull distribution is proposed and studied. Various structural properties including explicit expressions for the moments, quantiles, and moment generating function of the new distribution are derived. The estimation of the model parameters is performed by least square method. We hope that the new distribution proposed here will serve as an alternative model to the other models which are available in the literature for modeling positive real data in many areas.

The Kumaraswamy modified Weibull distribution: theory and applications

A five-parameter extension of the Weibull distribution capable of modelling a bathtub-shaped hazard rate function is introduced and studied. The beauty and importance of the new distribution lies in its ability to model both monotone and non-monotone failure rates that are quite common in lifetime problems and reliability. The proposed distribution has a number of well-known lifetime distributions as special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull (MW) distributions, among others. We obtain quantile and generating functions, mean deviations, Bonferroni and Lorenz curves and reliability. We provide explicit expressions for the density function of the order statistics and their moments. For the first time, we define the log-Kumaraswamy MW regression model to analyse censored data. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is determined. Two applications illustrate the potentiality of the proposed distribution.

949 - Parametric Effect Size Estimates from Sipuleucel-T Randomized Trials

ABSTRACT Introduction Median overall survival (OS) difference is frequently used as a measure of effect size because it is easily understood. Other effect size estimates can more fully represent the OS distribution, and are less subject to local variations. Analysis of the sipuleucel-T D9901 and IMPACT trials using non-parametric (median) or semi-parametric (hazard ratio [HR]) methods provide 4.5- and 4.1-month estimated median OS differences, respectively, with HRs of 0.586 and 0.752 (not stratified or adjusted). The OS curves from these trials exhibit delayed separation, suggesting a delayed treatment effect consistent with that shown for other immunotherapies. In this research, parametric statistical models are used to account for delayed effect and obtain alternative effect size estimates from D9901 and IMPACT. Methods Parametric models based on the Weibull distribution and a modification of the Weibull distribution that allows for delay of effect were applied. These models require specification of the general shape of the hazard function. Results The table shows the original OS and HR estimates in comparison to those obtained through parametric modeling. D9901 estimates from the Weibull model suggested a larger median OS difference vs the original estimates. The OS difference and the HR estimate using the delayed effect Weibull model were both indicative of greater sipuleucel-T effect. For IMPACT, the OS difference and HR estimates from the Weibull model indicated greater effect, but to a lesser degree than in D9901. The delayed effect modified Weibull model for IMPACT did not support a delayed effect. Discussion This statistical modeling illustrates alternative methods of obtaining effect size estimates that may be particularly applicable to clinical trials of cancer immunotherapies. For the sipuleucel-T trials, these models suggest a greater sipuleucel-T effect than obtained from non-parametric or semi-parametric methods. Study Estimate Type Median Difference (mos) Hazard Ratio D9901 Non-parametric/semi-parametric Weibull Modified Weibull 4.5 10.3 6.6 0.586 0.585 0.486* IMPACT Non-parametric/semi-parametric Weibull Modified Weibull 4.1 4.7 NS 0.752 0.749 NS NS, delayed effect not supported *Applicable after delayed efffect Disclosure All authors have declared no conflicts of interest.

ON CHARACTERIZATIONS OF CONTINUOUS DISTRIBUTIONS BY CONDITIONAL EXPECTATIONS OF UPPER RECORD VALUES

In this paper, general classes of continuous distributions are characterized by considering the conditional expectations of functions of upper record statistics. The specific distribution considered as a particular case of the general class of distribution are Exponential, Exponential Power(EP), Inverse Weibull, Beta Gumbel, Modified Weibull(MW), Weibull, Pareto, Power, Singh-Maddala, Gumbel, Rayleigh, Gompertz, Extream value 1, Beta of the first kind, Beta of the second kind and Lomax.

Genetic Algorithm Based Approaches for the Modified Weibull Renewal Approximation

Abstract The importance of renewal factors mainly in the reliability field is becoming an important subject because of the increasing sophistication of industrial plants. The conventional Weibull distribution have been often used to describe the life times of components and widely applied to characterize the breakdown statistics. Our aim in this paper is to propose optimization methods that allow the approximation of the modified Weibull renewal function. The proposed framework is elaborated based on Laplace transforms and genetic algorithms where the parameters of our model are tuned up by considering two minimization criteria. The obtained results confirm well the good accuracy of the analytical renewal function model when compared to numerical data.

Bayesian and Non Bayesian Estimations on the Exponentiated Modified Weibull Distribution for Progressive Censored Sample

The four-parameter Exponentiated Modified Weibull (EMW) is considered as an important lifetime distribution. Based on progressive Type-II censored sample, maximum likelihood and Bayesian estimators of the parameters, reliability function, and hazard rate function are derived. Two cases are considered: first, the case of one unknown exponent parameter of EMW and second, the case when two parameters of the EMW are both unknown. The Bayes estimators are studied under squared error and LINEX loss functions. The standard Bayes and importance sampling are considered for the estimation. Monte Carlo simulations are performed under different samples sizes and different censoring schemes for investigating and comparing the methods of estimation.

Software Reliability Growth Modeling with New Modified Weibull Testing–effort and Optimal Release Policy

development life cycle, software testing is one of the most important tasks; and in the testing, software reliably is very important aspect for any category of software systems. A number of testing-effort functions for software reliability growth model based on non-homogeneous Poisson process (NHPP) have been proposed in the past. Although these models are quite helpful for software developers and have been widely accepted and applied in the industries and research centers, we still need to put more testing-effort functions into software reliability growth model for accuracy on estimate of the parameters. In this paper, we will consider the case where the time dependent behaviors of testing- effort expenditures are described by New Modified Weibull Distribution (NMWD). Software Reliability Growth Models (SRGM) based on the NHPP are developed which incorporates the (NMWD) testing-effort expenditure during the software- testing phase. It is assumed that the error detection rate to the amount of testing-effort spent during the testing phase is proportional to the current error content. Model parameters are estimated by Least Square and Maximum Likelihood estimation techniques, and software measures are investigated through numerical experiments on real data from various software projects. The evaluation results are analyzed and compared with other existing models to show that the proposed SRGM with (NMWD) testing-effort has a fairly better faults prediction capability and it depicts the real-life situation more faithfully. This model can be applied to a wide range of software system. In addition, the optimal release policy for this model, based on reliability criterion is discussed.

Accounting for High Stress Gradient by a Modified Weibull Failure Theory

A high stress gradient occurs in a component when the stress, due to external loading, rises asymptotically. The Weibull failure theory overestimates the probability of failure for components with high stress gradients generated due to the geometric irregularities, material mismatch, thermal mismatch, and contact loading. A modified Weibull failure theory is proposed in this paper. The method is based on the weight function method. The modified Weibull failure theory was applied to two specimens, and the results showed the ability of the proposed theory to handle high stress gradients. The theory considers variable equivalent stress intensity factors along the faces of cracks; hence, it considers the strength of a specimen to be dependent on the stress field.

Statistical Simulation of Boxing Day Tsunami of The Indian Ocean and a Predictive Equation for Beach Run up Heights Based on Work-Energy Theorem

The tsunami similar to the one that has occurred in December 26, 2004 (Boxing Day Tsunami) in the Indian Ocean is simulated using the expression derived from Modified Weibull Distribution (for maximum wave height simulation) for extreme wave height predictions. The tuning coefficient plays a significant role in estimating the tsunami heights at various stages. It follows well defined mathematical laws at different stages. It is time dependent in the first three stages and depth dependent in the last two stages. The beach run-up heights estimated by the expression derived from the work-energy relation are comparable with observed values with reasonable accuracy.

Characterization of Nicalon fibres with varying diameters Part II Modified Weibull distribution

Diameters vary significantly in a tow of commercial NicalonTM fibres, which is one of the most attractive ceramic reinforcements for structural composites. It was found that the strength distribution of Nicalon fibres could not be adequately characterized using either single- or bi-modal Weibull distribution. A recently proposed modified Weibull distribution can account for the effect of varying diameter in the characterization of fibre strength. To verify the validity of the modified Weibull distribution, comprehensive mechanical testing and fractographic studies have been conducted on Nicalon SiC fibres with diameters varying from 8 to 22 μm. The experimental results have been reported in Part I. Part II of this paper further modifies the derivation of the modified Weibull distribution to yield a relationship which is similar in form, but soundly based on experimental findings. Factors considered in the modified Weibull distribution include the dependence of fracture toughness and flaw density on fibre diameter, both of which may vary with fibre diameter, as reported in Part I. Comparison with experimental data shows that the current modified Weibull distribution works very well, while both single-modal and bi-modal Weibull distributions are inadequate for describing Nicalon fibres with varying diameters. © 1998 Chapman & Hall

Models for estimating the risk of ulcers in the small intestine after localized single or fractionated irradiation.

Subacute and chronic ulcerations of the intestinal mucosa are important causes of serious complications following radiation therapy for abdominal or pelvic tumours. We describe dose-response models for estimating the risk of mucosal ulcers in the small intestine after uniform, localized single or fractionated (once-daily) X-ray exposure. The models were fitted to data for ulceration incidence, based on a 26 week post-irradiation follow-up of male Sprague-Dawley rats which received a wide range of single and fractionated once-daily 250 kV X-ray doses to a short loop (partial volume) of transposed, but functionally intact, small intestine. The models presented for single (Weibull (W)) and fractionated (modified Weibull (MW)) exposures of a partial volume of tissue allow estimation of the risk of radiation-induced injury. While the W model is not new, its adaptation to partial volume irradiation and the MW model are. Isoeffect relationships are presented for the uniform fractional dose Ds(i%) associated with an i% (e.g. 0%, 5%, 10%, 50%) risk of intestinal mucosal ulcers as a function of the number of once-daily dose fractions, where Ds(0%) represents the threshold fractional dose. Although the Ds(5%) and Ds(0%) estimates provided for intestinal mucosal ulcers are based on animal data, the ratio Ds(0%)/Ds(5%) and more generally ratios Ds(j%)/Ds(i%) (where i not equal to j), are presumed to apply to humans. The indicated ratios are predicted to be independent of the partial volume irradiated and the number of once-daily dose fractions, and may be independent of radiation quality. Isoeffect equations are also presented that apply to circumstances where different partial volumes within the same reference volume (i.e. the total volume of tissue considered) receive different doses, but the dose within a given partial volume is uniformly distributed. These isoeffect equations provide a means of converting non-uniform dose within a reference volume to uniform isoeffect dose to the total reference volume and may have applications outside the field of radiation therapy (e.g. evaluating effects of non-uniform exposure of the small intestine or skin by a hot particle).

On Some Nonregular Tests for a Modified Weibull Model

On Some Nonregular Tests for a Modified Weibull Model

Evaluation of Concurrent Flaw Populations in Silicon Carbide in Terms of a Modified Weibull Distribution Function

Concurrent surface‐ and edge‐flaw populations are often observed in strength testing of ceramics The overall strength distribution in such ceramic samples has been analyzed in terms of a modified Weibull distribution function. The junction provides an upper and lower strength limit and is characterized by two shape and location parameters. To evaluate the parameters of the distribution function, the type of flaw that causes failure need not be identified. The applicability of the function has been evaluated in terms of experimental results on SiC specimens. The analysis also shows that the relative values of shape parameters determine which flaw type dominates.

Estimating Log Yields with a Modified Weibull Function

(1984). Estimating Log Yields with a Modified Weibull Function. South African Forestry Journal: Vol. 131, No. 1, pp. 69-71.