A Power Log-logistic Modified Weibull Distribution: Model, Properties and a Simulation

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.

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

The Sustainable Development Goals (SDGs) of the United Nations consist of 17 general objectives, subdivided into 169 targets to be achieved by 2030. Several SDG indices and indicators require continuous analysis and evaluation, and most of these indices are supported in the unit interval ((0,1)). To incorporate the flexibility of the modified Weibull (MW) distribution in doubly constrained datasets, the first objective of this work is to propose a new unit probability distribution based on the MW distribution. For this, a transformation of the MW distribution is applied, through which the unit modified Weibull (UMW) distribution is obtained. The second objective of this work is to introduce a quantile regression model for random variables with UMW distribution, reparameterized in terms of the quantiles of the distribution. Maximum likelihood estimators (MLEs) are used to estimate the model parameters. Monte Carlo simulations are performed to evaluate the MLE properties of the model parameters in finite sample sizes. The introduced methods are used for modeling some sustainability indicators related to the SDGs, also considering the reading skills of dyslexic children, which are indirectly associated with SDG 4 (Quality Education) and SDG 3 (Health and Well-Being).

Models with the bathtub‐shaped hazard rate function are widely used in lifetime analysis and reliability engineering. In this paper, we adopted the reduced new modified Weibull (RNMW) distribution with a bathtub‐shaped hazard rate function. Under consideration that the population units are failing with two independent causes of failure and the failure time is distributed with RNMW distribution, we formulate the model which is known as competing risks model. The model parameters under the type‐II censoring scheme are estimated with the maximum likelihood method with the corresponding asymptotic confidence intervals. Also, the Bayes point and credible intervals with the help of MCMC methods are constructed. The real and simulated datasets are analyzed for illustrative purposes. Finally, the estimators are compared with the Monte Carlo simulation study.

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.

Inference and prediction for modified Weibull distribution based on doubly censored samples

A modified Weibull distribution is evaluated for characterizing the statistical strength of ceramic fibers and whiskers with varying diameters from filament to filament. Many commercial ceramic fibers and whiskers have a significant range of diameters. A single-modal Weibull distribution is found inadequate to describe the statistical strength of these fibers and whiskers because of the effect of fiber diameter variation on strength. Procedures for extracting distribution parameters for the modified Weibull distribution from experimental data are presented. Comparison of the modified Weibull distribution with the single-modal Weibull distribution is made for the strength data from Nicalon fibers, Nextel (Al2O3) fibers, hydridopolysilzazlane (HPZ) Si-N-C-O fibers, Al2O3 whiskers, Si3N4 whiskers, and SiC whiskers. Due to its ability to account for the diameter effect on strength, the modified Weibull distribution can yield a more accurate β value than the single-modal Weibull distribution. The Modified Weibull distribution is shown to fit experimental data well and is recommended for characterizing the strength of ceramic fibers and whiskers, the diameters of which vary from filament to filament.

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.

Wind speed (WS) is important information to determine the potential for wind energy in an area. Wind speed has been widely expressed by the probability density function (Pdf), one of which uses the Weibull Distribution (WD). Not all WS data can be analyzed by WD because some deficiencies need to be corrected. Modified Weibull Distribution (MWD) is proposed to improve the existing WD models. In addition, this paper also compares the performance of MWD against WD using WS data measured in Medan City. To validate the two models (WD and MWD), the coefficient of determination (R-squared) and the mean square root error (RMSE) were used. In addition, data validation tests were also carried out using Chi-square and Kolmogorov-Smirnov. The result obtained is that MWD has a more acceptable fit than WD for this case.Keywords: Wind speed, Distribution function, Weibull distribution, Modified Weibull distributionJEL Classifications: C13; C22; C36; C93; L94; Q42DOI: https://doi.org/10.32479/ijeep.11625

Dielectric spectroscopy of the human blood is a powerful and convenient non-invasive testing technique that can be used to diagnose diseases such as diabetes and leukemia. One needs to consider rigorous experimental procedures and mathematical models to make the results of this type of test comparable. The present paper will discuss previously published results to further investigate the statistical modeling of the dielectric properties of human blood. The analysis shows that previously published results were related to Modified Weibull (MW) distributions of relaxation times, not Gaussian distributions, as reported. This re-analysis prevents the ill definition of fitting parameters, making sure they present physically justifiable values. Besides, for fluids presenting a Modified Weibull distribution of relaxation times, novel exact and closed-form expressions for the real and imaginary parts of complex permittivities were obtained in terms of generalized hypergeometric functions. Also, a high accuracy approximation was built for the imaginary part of the complex permittivity, creating an easy-to-use alternative expression for practitioners. The new results are used to fit experimental results for human blood, showing that more robust estimators are built for the parameters involved, which can be used as thresholds to classify the dielectric behavior of blood as normal (healthy) or anomalous (sick).

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.

For lifetime data analysis, failure time analysis, or survival analysis, the Weibull distribution is commonly used due to its various hazard functions, which can be increasing, decreasing, or constant. We have extended the traditional twoparameter Weibull distribution to accommodate hazard functions that are increasing, decreasing, constant and bathtubshaped. Utilizing a competing risks approach, we applied this modified Weibull distribution to both simulated data and an observed mice dataset. Our findings indicate that the modified Weibull distribution provides a better fit to the mice dataset compared to the traditional Weibull distribution. We estimated the parameters of the modified Weibull distribution using Maximum likelihood estimation (MLE) and Bayesian methods. For MLE, we employed the Newton-Raphson numerical method, while for the Bayesian approach, we used the Metropolis-Hastings algorithm, an MCMC method. Additionally, we plotted hazard curves for both the simulated and mice datasets. The Kaplan-Meier survival curves were plotted along with the survival curve of the modified Weibull distribution.. KEYWORDS :Modified weibull distribution, Competing risks, MCMC, Information criterion, MLE.

The knowledge of the probability density function of wind speed is key information for determining wind energy potential of the specified region. In literature, the Weibull distribution is widely-used and accepted distribution to express the probability density function of wind speed data. However, the Weibull distribution does not exhibit good fitting for all wind speed data measured at different geographical locations throughout the world. Thus, in this study, it is proposed that a better fitting of wind speed data and a better estimating wind power density is possible with new modified Weibull distribution (MWD). Also, we compare the performance of the MWD relative to the Weibull distribution by using wind speed data measured in the different regions of Turkey. The results point out that the MWD shows good fitting for most of the considered wind speed data cases. Thus, the MWD can be alternative for assessment of wind energy potential.

Discretization of continuous lifetime distribution is an interesting and intuitively appealing approach to derive a discrete lifetime model. This study derived a discretized form of Reduced Modified Weibull distribution known as the Marshall-Olkin Discrete Reduced Modified Weibull (MDRMW) distribution. The mathematical and statistical properties of MDRMW distribution were derived and compared with existing distributions of Discrete Reduced Modified Weibull distribution (DRMW), Exponentiated Discrete Weibull distribution (EDW) and Two Parameters Discrete Lindley distribution (TDL). Maximum likelihood method was used to derive the statistics of MDRMW parameters. The Aarset Reliability dataset was fitted for the existing and derived distribution and AIC and Kolmogorov Smirrnoff (KS) were compared. The shape of MDRMW distribution was unimodal and monotonic decreasing. The plot of hazard rate function could be decreasing or bath-tub. The AIC and KS values of Aarset reliability data analysis were 483.9 and 0.17579; 507.8 and 0.24435; 485.2 and 0.17897 for MDRMW, DRMW and TDL respectively. The AIC and KS values of Leukemia survival data analysis were 668.2 and 0.11053; 751.9 and 0.39285 respectively. The Aarset reliability data analysis showed that MDRMW compared favorably with existing distributions. The MDRMW and DRMW handled Leukemia survival data set as against EDW and TDL. The values of AIC and KS for MDRMW were lower than DRMW, EDW and TDL. This showed that MDRMW was better than the existing distributions.

Novel modified distribution functions of fiber length in fiber reinforced thermoplastics