Application of a 2 Parameter Weibull Distribution in Modeling of State Holding Time in HIV/AIDS Transition Dynamics

This study investigates the application of the two-parameter Weibull distribution in modeling state holding times within HIV/AIDS progression dynamics. By comparing the performance of the Weibull-based Accelerated Failure Time (AFT) model, Cox Proportional Hazards model, and Survival model, we assess the effectiveness of these models in capturing survival rates across varying gender, age groups, and treatment categories. Simulated data was used to fit the models, with model identification criteria (AIC, BIC, and R2) applied for evaluation. Results indicate that the AFT model is particularly sensitive to interaction terms, showing significant effects for older age groups (50 - 60 years) and treatment interaction, while the Cox model provides a more stable fit across all age groups. The Survival model displayed variability, with its performance diminishing when interaction terms were introduced, particularly in older age groups. Overall, while the AFT model captures the complexities of interactions in the data, the Cox model’s stability suggests it may be better suited for general analyses without strong interaction effects. The findings highlight the importance of model selection in survival analysis, especially in complex disease progression scenarios like HIV/AIDS.

Quantitative Study on Characteristics of the Weibull Distribution

Aim To quantitatively study three characteristics of the Weibull distribution. Methods Theoritical analysis of the three characteristics of parameters of the Weibull distribution was done and mathematics software was used to make some chart analysis. Results 17 equations and 7 figures were made. Conclusion Under the standard form, the class of the Weibull probable density founction(pdf) curves appear double peak shape. Under the standard form, the maximum value point curve of the Weibull pdf takes line t =0 and t=1 as asymptotes. When β = 3 30-3 40, the Weibull distribution is the most similar to the normal distribution.

三参数Weibulldistribution场合下定数截尾矩估计

本文讨论了在定数截尾样本下三参数威布尔分布的矩估计问题,并改进了文[4]的结果。

Performance of Weibull function as a diameter distribution model for Pinus thunbergii stands in the eastern coast of South Korea

This study was carried out to determine the performance of percentile-based Weibull diameter distribution model for Pinus thunbergii stands thriving along the eastern coast of South Korea. The parameter recovery technique was used to estimate the three parameters of the Weibull model. The analysis demonstrated satisfactory results based on the following test statistics for the principal percentile models： fit index （FI） range from 0.501（minimum diameter） to 0.932 （50th diameter percentiles） and root mean square error （RMSE） range from 0.112 （quadratic mean diameter） to 3.572 （minimum diameter）. The developed model was further evaluated by determining the mean bias （E） in trees per ha （TPH） for each diameter class, and the results showed highest over-prediction in the 20 cm, and under-prediction in the x6 cm and 24 cm diameter classes. The goodness of fit tested by Kolmogorov- Smirnov （KS） test showed no significant differences （P〉0.05） between the observed and predicted diameter distributions for almost all plots. Using site index and aboveground biomass （AGB） models developed for P. thunbergii in South Korea, a model to predict the AGB per ha for each diameter class and subsequently the total AGB of the stand was created. An application guide was also created, which will serve as a decision-support tool for forest managers in quantifying the future total AGB in P. thunbergii stands located in the eastern coast of South Korea and, subsequently, the quantification of potential carbon stocks aside from being a vital input in designing efficient management and protection strategies for these stands.

Parameter Estimations for Some Modifications of the Weibull Distribution

Proposed by the Swedish engineer and mathematician Ernst Hjalmar Waloddi Weibull (1887-1979), the Weibull distribution is a probability distribution that is widely used to model lifetime data. Because of its flexibility, some modifications of the Weibull distribution have been made from several researches in order to best adjust the non-monotonic shapes. This paper gives a study on the performance of two specific modifications of the Weibull distribution which are the exponentiated Weibull distribution and the additive Weibull distribution.

A New Study on Combustion Behavior of Pine Sawdust Characterized by the Weibull Distribution

An isothermal operation is implemented by employing a thermogravimetric analyzer (TGA) for simulating the thermal decomposition behavior of 58μm pine sawdust in air atmosphere.An independent parallel reaction model is adopted in this study to describe the thermal decomposition mechanism.The Weibull distribution function is used to record and analyze the weight loss during isothermal decomposition at different temperatures(500,600,700,and 800°C).The total weight loss of the pine sawdust is assumed as a linear combination of individual weight loss from three components,including the char and two volatile matters.The plot of the thermal decomposition rate curve leads to kinetic parameters such as the reaction rate constants and the reaction order.The results show that the Weibull distribution function successfully represents decomposition curves of three components,and fits the experimental data very well.Therefore,this study provides a simple way to evaluate the decomposition rate of biomass combustion in a real combustor.

Comparison of the Bayesian Methods on Interval-Censored Data for Weibull Distribution

This study considers the estimation of Maximum Likelihood Estimator and the Bayesian Estimator of the Weibull distribution with interval-censored data. The Bayesian estimation can’t be used to solve the parameters analytically and therefore Markov Chain Monte Carlo is used, where the full conditional distribution for the scale and shape parameters are obtained via Metropolis-Hastings algorithm. Also Lindley’s approximation is used. The two methods are compared to maximum likelihood counterparts and the comparisons are made with respect to the mean square error (MSE) to determine the best for estimating of the scale and shape parameters.

Interval Estimation in a Two Parameter Weibull Distribution Based on Type-2 Censored Data

In this paper, we consider the construction of the approximate profile- likelihood confidence intervals for parameters of the 2-parameter Weibull distribution based on small type-2 censored samples. In previous research works, the traditional Wald method has been used to construct approximate confidence intervals for the 2-parameter Weibull distribution under type-2 censoring scheme. However, the Wald technique is based on normality assumption and thus may not produce accurate interval estimates for small samples. The profile-likelihood and Wald confidence intervals are constructed for the shape and scale parameters of the 2-parameter Weibull distribution based on simulated and real type-2 censored data, and are hence compared using confidence length and coverage probability.

Application of the Weibull Distribution to Estimate the Volume of Water Pumping by a Windmill

Given the steady increase in use of alternative energy sources to supply part of the energy demand of the world, and considering that Mexico has enough wind resources to produce all the electricity required by the country, in this work, it is carried out the development of a system of wind resource assessment to determine the volume of water that a windmill pumping system can provide in a specific area. To this end, it has developed a computer program for wind resource assessment by measurements of wind speed, direction, temperature, barometric pressure and relative humidity. Measuring equipments were mounted in a wind measurement tower of 10 m high, within the premises of the University of Quintana Roo which is located along the coast. Until this day, there are records over a period of five months of the variables of wind speed, wind direction, temperature, relative humidity and atmospheric pressure. These variables were used for statistical calculations using the Weibull distributions. To give a better use of the calculated data, they were applied to the evaluation of a Pumping System.

The Marshall-Olkin Right Truncated Fréchet-Inverted Weibull Distribution: Its Properties and Applications

In this paper, a new probability distribution is proposed by using Marshall and Olkin transformation. Some of its properties such as moments, moment generating function, order statistics and reliability functions are derived. The method of maximum likelihood is used to estimate the model parameters. The graphs of the reliability function and hazard rate function are plotted by taken some values of the parameters. Three real life applications are introduced to compare the behaviour of the new distribution with other distributions.

A New Approach for Parameter Estimation of Mixed Weibull Distribution:A Case Study in Spindle

In order to improve the accuracy and efficiency of graphical method and maximum likelihood estimation( MLE) in Mixed Weibull distribution parameters estimation,Graphical-GA combines the advantage of graphical method and genetic algorithm( GA) is proposed.Firstly,with the analysis of Weibull probability paper( WPP),mixed Weibull is identified to data fitting.Secondly,the observed value of shape and scale parameters are obtained by graphical method with least square,then optimizing the parameters of mixed Weibull with GA.Thirdly,with the comparative analysis on graphical method,piecewise Weibull and two-Weibull,it shows graphical-GA mixed Weibull is the best.Finally,the spindle MTBF point estimation and interval estimation are got based on mixed Weibull distribution.The results indicate that graphical-GA are improved effectively and the evaluation of spindle can provide the basis for design and reliability growth.

Fracture Statistics Dominated by Hot-Cutting Defect and Deviation from Weibull Distribution

This paper characterizes the rarely noticeable hot-cutting defect and statistically models the fracture governed by this new type defect. The morphology of the defect on fired ceramic is examined and quantitatively featured through comparing the fracture strength governed by intrinsic defect and hot-cutting defect. Weibull distribution is utilized to fit the observed strength data and chi-square goodness-of-fit test is conducted to analyze the deviation. Kernel density estimation is introduced to explore the underlying strength distribution dominated by hot-cutting defect. Based on the shape information from kernel density estimating,gamma and lognormal distribution are compared and the hot-cutting defect governing fracture statistics is finally confirmed by chisquare test. Results show that the newly-defined hot-cutting defect is more dangerous than the intrinsic defect and the priori Weibull distribution fails to describe the fracture statistic governed by the edge defect while the lognormal with a slightly right skewed shape fits it very well.

Evaluation of the location parameter in Weibull distribution of CTOD testing results of high strength steel weldments

In this paper, the location parameter δ_(a) in Weibull distribution i.e. the lower limiting CTOD toughness of high strength steel weldments is evaluated from Gleeble simulated coarse grained specimens. The Charpy-V transition temperature obtained from these specimens is transformed to a fracture toughness K_(IC) value at a given temperature using Sanz' relation. Then, the location parameter a is δ_(a) evaluated. The predicted location parameter arrives at agreement with the lowest CTOD value as obtained from thick multilayer weldments. The improvement of the fracture toughness of multilayer weldments needs the increase of the lowering limiting CTOD.toughness i.e. the toughness of the coarse grained zone.

Parameter Estimation of Mixed Weibull Distributions Using Cuckoo Search

The lifetime data of products with multiple failure modes which are collected from life testing are often fitted by the mixed Weibull distributions. Since the mixed Weibull distributions contain no less than five parameters,the parameter estimation is difficult and inaccurate. In order to enhance the accuracy,a new method of parameter estimation based on Cuckoo search( CS) is proposed. An optimization model for the mixed Weibull distribution is formulated by minimizing the residual sum of squares. The optimal parameters are searched via CS algorithm. In the case study,the lifetime data come from the life testing of diesel injectors and are fitted by the twocomponent Weibull mixture. Regarding the maximum absolute error and the accumulative absolute error between estimated and observed values as the accuracy index of parameter estimation,the results of four parameter estimation methods that the graphic estimation method,the nonlinear least square method,the optimization method based on particle swarm optimization( PSO) and the proposed method are compared. The result shows that the proposed method is more efficient and more accurate than the other three methods.

Investigations into Some Simple Expressions of the Gamma Function in Wind Power Theoretical Estimate by the Weibull Distribution

The Weibull distribution is a probability density function (PDF) which is widely used in the study of meteorological data. The statistical analysis of the wind speed v by using the Weibull distribution leads to the estimate of the mean wind speed , the variance of v around and the mean power density in the wind. The gamma function Γ is involved in those calculations, particularly Γ (1+1/k), Γ (1+2/k) and Γ (1+3/k). The paper reports the use of the Weibull PDF f(v) to estimate the gamma function. The study was performed by looking for the wind speeds related to the maximum values of f(v), v2 f(v) and v3 f(v). As a result, some approximate relationships were obtained for Γ (1+1/k), Γ (1+2/k) and Γ (1+3/k), that use some fitting polynomial functions. Very good agreements were found between the exact and the estimated values of Γ (1+n/k) that can be used for the estimation of the mean wind speed , the variance σ2 of the wind speed v;around the mean speed and the average wind power density.

A Novel Modified Alpha Power Transformed Weibull Distribution and Its Engineering Applications

This paper suggests a new modified version of the traditional Weibull distribution by adding a new shape parameter utilising the modified alpha power transformed technique.We refer to the new model as modified alpha power transformed Weibull distribution.The attractiveness and significance of the new distribution lie in its power to model monotone and non-monotone failure rate functions,which are quite familiar in environmental investigations.Its hazard rate function can be decreasing,increasing,bathtub and upside-down then bathtub shaped.Diverse structural properties of the proposed model are acquired including quantile function,moments,entropies,order statistics,residual life and reversed failure rate function.The parameters of the distribution were estimated using the maximum likelihood function.The maximum likelihood method is employed to estimate the model parameters and the approximate confidence intervals are also computed.Via a simulation study,the performance of the point and interval estimates are compared using different criteria.Employing real lifetime data sets,we verify that the offered model furnishes a better fit than some other lifetime models including Weibull,gamma and alpha powerWeibull models.

The Exponential Flexible Weibull Extension Distribution

This paper is devoted to study a new generalization of the flexible Weibull with three parameters. This model is referred to as the exponential flexible Weibull extension (EFWE) distribution which exhibits bathtub-shaped hazard rate function. Some statistical properties such as the mode, median, the moment, quantile function, the moment generating function and order statistics are discussed. Moreover, the maximum likelihood method for estimating the model parameters and the Fisher’s information matrix is given. Finally, the advantage of the EFWE distribution is concluded by an application using real data.

A New Three-Parameter Inverse Weibull Distribution with Medical and Engineering Applications

The objective of this article is to provide a novel extension of the conventional inverse Weibull distribution that adds an extra shape parameter to increase its flexibility.This addition is beneficial in a variety of fields,including reliability,economics,engineering,biomedical science,biological research,environmental studies,and finance.For modeling real data,several expanded classes of distributions have been established.The modified alpha power transformed approach is used to implement the new model.The datamatches the new inverseWeibull distribution better than the inverse Weibull distribution and several other competing models.It appears to be a distribution designed to support decreasing or unimodal shaped distributions based on its parameters.Precise expressions for quantiles,moments,incomplete moments,moment generating function,characteristic generating function,and entropy expression are among the determined attributes of the new distribution.The point and interval estimates are studied using the maximum likelihood method.Simulation research is conducted to illustrate the correctness of the theoretical results.Three applications to medical and engineering data are utilized to illustrate the model’s flexibility.

Bivariate Beta–Inverse Weibull Distribution:Theory and Applications

Probability distributions have been in use for modeling of random phenomenon in various areas of life.Generalization of probability distributions has been the area of interest of several authors in the recent years.Several situations arise where joint modeling of two random phenomenon is required.In such cases the bivariate distributions are needed.Development of the bivariate distributions necessitates certain conditions,in a field where few work has been performed.This paper deals with a bivariate beta-inverse Weibull distribution.The marginal and conditional distributions from the proposed distribution have been obtained.Expansions for the joint and conditional density functions for the proposed distribution have been obtained.The properties,including product,marginal and conditional moments,joint moment generating function and joint hazard rate function of the proposed bivariate distribution have been studied.Numerical study for the dependence function has been implemented to see the effect of various parameters on the dependence of variables.Estimation of the parameters of the proposed bivariate distribution has been done by using the maximum likelihood method of estimation.Simulation and real data application of the distribution are presented.

Efficient Process Monitoring Under General Weibull Distribution

Product testing is a key ingredient in maintaining the quality of a pro-duction process.The production process is considered an efficient process if it is capable of quick identification of faulty products.The items produced by any production process are usually packed and acceptance or rejection of the pack depends upon its conformity to some specified quality level.Generally,the specified quality level is based upon the number of defective items found in the inspected number of items.Such decisions are based upon some rules and usually acceptance of the pack is based upon a fewer number of defective items in the pack.Such questions can be answered by using acceptance sampling plans.The acceptance sampling plans assume the fact that the quality level of the item follows some probability distribution.The sampling plans based upon some classical probability distributions are available but often it happens that the quality behavior of the product does not follow a simple probability model and hence the available sampling plans fail.In this paper,we have developed acceptance sampling plans when the product life follows a general Weibull distribution.The sampling plans have been constructed by considering the crisp and fuzzy behavior of the acceptance probability.These sampling plans have been constructed by assuming an infinite lot size.It has been found that the number of items required for inspection decreases with an increase in some parameters.