Kumaraswamy Weibull Distribution Research Articles

Secant Kumaraswamy Family of Distributions: Properties, Regression Model, and Applications

In this study, Secant Kumaraswamy family of distributions is proposed and studied. This is motivated by the fact that no one distribution can model all types of data from different fields. Therefore, there is the need to develop distributions with desirable properties and flexible enough for modelling data exhibiting different characteristics. Some properties of the new family of distributions, including the quantile function, moments, moment generating function, and mean residual life function, are derived. Five special cases of the family of distributions are presented, and their flexibility is shown by the varying degrees of skewness and kurtosis and nonmonotonic hazard rates. The maximum likelihood estimation method is used to obtain estimators of the family of distributions. Two location-scale regression models are developed for the Secant Kumaraswamy Weibull distribution, which is a special case of the family of distributions. Six different real datasets are used to demonstrate the usefulness of the family of distributions and the regression models. The results show that the family of distributions can be used to model real datasets.

Comparison of estimation methods for the Kumaraswamy Weibull distribution

In this study, the performances of the different parameter estimation methods are compared for the Kumaraswamy Weibull distribution via Monte Carlo simulation study. Maximum Likelihood (ML), Least Squares (LS), Weighted Least Squares (WLS), Cramer-von Mises (CM) and Anderson Darling (AD) methods are used in the comparisons. The results of the Monte Carlo simulation study demonstrate that ML estimators for the parameters of the Kumaraswamy Weibull distribution are more efficient than the other estimators. It is followed by AD estimator. At the end of the study, a real data set taken from the literature is used to illustrate the applicability of the Kumaraswamy Weibull distribution.

A New Generalized Logarithmic–X Family of Distributions with Biomedical Data Analysis

In this article, an attempt is made to propose a novel method of lifetime distributions with maximum flexibility using a popular T–X approach together with an exponential distribution, which is known as the New Generalized Logarithmic-X Family (NGLog–X for short) of distributions. Additionally, the generalized form of the Weibull distribution was derived by using the NGLog–X family, known as the New Generalized Logarithmic Weibull (NGLog–Weib) distribution. For the proposed method, some statistical properties, including the moments, moment generating function (MGF), residual and reverse residual life, identifiability, order statistics, and quantile functions, were derived. The estimation of the model parameters was derived by using the well-known method of maximum likelihood estimation (MLE). A comprehensive Monte Carlo simulation study (MCSS) was carried out to evaluate the performance of these estimators by computing the biases and mean square errors. Finally, the NGLog–Weib distribution was implemented on four real biomedical datasets and compared with some other distributions, such as the Alpha Power Transformed Weibull distribution, Marshal Olkin Weibull distribution, New Exponent Power Weibull distribution, Flexible Reduced Logarithmic Weibull distribution, and Kumaraswamy Weibull distribution. The analysis results demonstrate that the new proposed model performs as a better fit than the other competitive distributions.

Lindley procedure and MCMC technique in Bayesian estimation for Kumaraswamy Weibull distribution

In this study, a comparison between three methods for estimating unknown parameters of the Kumaraswamy Weibull distribution for different sample sizes of type II censoring data is presented. Specifically, we compare the behaviors of maximum likelihood estimates, Lindley and Markov chain Monte Carlo (MCMC) estimates as Bayesian estimates. We have not found any work on this topic after reviews of the literature except one with little information about the inference of this important distribution. The simplest form for Lindley approximation of the posterior mean is proposed and approximate closed forms of acceptable Bayes estimates for the models of multi-parameters such as Kumaraswamy Weibull distribution is derived. A Monte Carlo simulation is conducted to investigate the performances of the proposed estimators. Finally, three real data examples are analyzed to illustrate the application possibility of the different proposed estimation methods. The results reveal that, although, good performance of the approximate forms of Lindley estimators, the estimators resulting in the MCMC technique are better in the sense of the mean squared errors.

Statistical Inference on a Finite Mixture of Exponentiated Kumaraswamy-G Distributions with Progressive Type II Censoring Using Bladder Cancer Data

A new family of distributions called the mixture of the exponentiated Kumaraswamy-G (henceforth, in short, ExpKum-G) class is developed. We consider Weibull distribution as the baseline (G) distribution to propose and study this special sub-model, which we call the exponentiated Kumaraswamy Weibull distribution. Several useful statistical properties of the proposed ExpKum-G distribution are derived. Under the classical paradigm, we consider the maximum likelihood estimation under progressive type II censoring to estimate the model parameters. Under the Bayesian paradigm, independent gamma priors are proposed to estimate the model parameters under progressive type II censored samples, assuming several loss functions. A simulation study is carried out to illustrate the efficiency of the proposed estimation strategies under both classical and Bayesian paradigms, based on progressively type II censoring models. For illustrative purposes, a real data set is considered that exhibits that the proposed model in the new class provides a better fit than other types of finite mixtures of exponentiated Kumaraswamy-type models.

Survival analysis of cancer patients using a new extended Weibull distribution.

One of the most important applications of statistical analysis is in health research and applications. Cancer studies are mostly required special statistical considerations in order to find the appropriate model for fitting the survival data. Existing classical distributions rarely fit such data well and an increasing interest has been shown recently in developing more flexible distributions by introducing some additional parameters to the basic model. In this paper, a new five-parameters distribution referred as alpha power Kumaraswamy Weibull distribution is introduced and studied. Particularly, this distribution extends the Weibull distribution based on a novel technique that combines two well known generalisation methods, namely, alpha power and T-X transformations. Different characteristics of the proposed distribution, including moments, quantiles, Rényi entropy and order statistics are obtained. The method of maximum likelihood is applied in order to estimate the model parameters based on complete and censored data. The performance of these estimators are examined via conducting some simulation studies. The potential importance and applicability of the proposed distribution is illustrated empirically by means of six datasets that describe the survival of some cancer patients. The results of the analysis indicated to the promising performance of the alpha power Kumaraswamy Weibull distribution in practice comparing to some other competing distributions.

Sec-G Class of Distributions: Properties and Applications

Although there are many continuous distributions in the literature, only a handful take advantage of the modeling power provided by trigonometric functions. To our knowledge, none of them are based on the so-called secant function, defined as the reciprocal of the cosine function. The secant function can go to large values whenever the cosine function goes to small values. The idea is to profit from this trigonometric property to modify well-known distribution tails and overall skewness features. With this in mind, in this paper, a new class of trigonometric distributions based on the secant function is introduced. It is called the Sec-G class. We discuss the main mathematical characteristics of this class, including series expansions of the corresponding cumulative distribution and probability density functions, as well as several probabilistic measures and functions. In particular, we present the moments, skewness, kurtosis, Lorenz, and Bonferroni curves, reliability coefficient, entropy measure, and order statistics. Throughout the study, emphasis is placed on the unique four-parameter continuous distribution of this class, defined with the Kumaraswamy-Weibull distribution as the baseline. The estimation of the model parameters is performed using the maximum likelihood method. We also carried out a numerical simulation study and present the results in graphic form. Three referenced datasets were analyzed, and it is proved that the proposed secant Kumaraswamy-Weibull model outperforms important competitors, including the Kumaraswamy-Weibull, Kumaraswamy-Weibull geometric, Kumaraswamy-Weibull Poisson, Kumaraswamy Burr XII, and Weibull models.

A New Bivariate Extended Generalized Inverted Kumaraswamy Weibull Distribution

This article presents a new bivariate extended generalized inverted Kumaraswamy Weibull (BIEGIKw-Weibull) distribution with nine parameters. Statistical properties of the new distribution are discussed. Forms of copulas, moments, conditional moments, bivariate reliability function, and bivariate hazard rate function are derived. Maximum likelihood estimators are formulated. Simulation is conducted for three different sets of parameters to verify the theoretical results and to discuss the new distribution properties. The performance of the maximum likelihood method is investigated via Monte Carlo simulation depending on the bias and the standard error. Simulated lifetime data is used as an application of the new model.

Statistical Inference For A Simple Step–Stress Model With Type–II Hybrid Censored Data From The Kumaraswamy Weibull Distribution

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors on the lifetimes of experimental units. In this paper, a step-stress model is considered in which the life-testing experiment gets terminated either at a pre-fixed time (say, Tm+1) or at a random time ensuring at least a specified number of failures (Say, y out of n). Under this model in which the data obtained are Type-II hybrid censored, the Kumaraswamy Weibull distribution is used for the underlying lifetimes. The maximum Likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model are derived. The confidence intervals of the parameters are also obtained. The hazard rate and reliability functions are estimated at usual conditions of stress. Monte Carlo simulation is carried out to investigate the precision of the maximum likelihood estimates. An application using real data is used to indicate the properties of the maximum likelihood estimators.

The extended generalized inverted Kumaraswamy Weibull distribution: Properties and applications

<abstract> In this paper we introduce a new six-parameters extension of the Weibull distribution. It will be called the extended generalized inverted Kumaraswamy Weibull (EGIKw-Weibull), that is commonly used to model lifetime data. Various useful properties of the new distribution are derived. A Monte Carlo simulation is employed to investigate the maximum likelihood estimator (MLE) for the parameters. Two real applications are presented. </abstract>

Transmuted Kumaraswamy Weibull Distribution with Covariates Regression Modelling to Analyze Reliability Data

This paper investigates the potential usefulness of the transmuted Kumaraswamy Weibull distribution by using quadratic rank transmutation map technique for modelling reliability data. Some structural properties of the transmuted Kumaraswamy Weibull distribution are discussed. We propose a location-scale regression model based on the transmuted log-Kumaraswamy Weibull distribution for modelling survival data. We discuss estimation of the model parameters by the method of maximum likelihood and provide two applications to illustrate the potentiality of the proposed family of lifetime distributions.

The Modified Kumaraswamy Weibull Distribution: Properties and Applications in Reliability and Engineering Sciences

We introduce the Kumaraswamy alpha power-G (KAP-G) family which extends the alpha power family (Mahdavi and Kundu, 2017) and some other families. We consider the Weibull as baseline for the KAP family and generate Kumaraswamy alpha power Weibull distribution which has right-skewed, left-skewed, symmetrical, and reversed-J shaped densities, and decreasing, increasing, bathtub, upside-down bathtub, increasing-decreasing–increasing, J shaped and reversed-J shaped hazard rates. The proposed distribution can model non-monotone and monotone failure rates which are quite common in engineering and reliability studies. Some basic mathematical properties of the new model are derived. The maximum likelihood estimation method is used to evaluate the parameters and the observed information matrix is determined. The performance and flexibility of the proposed family is illustrated via two real data applications.

Modelling Insurance Losses with a New Family of Heavy-Tailed Distributions

The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues. In this article, we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences. A specific sub-model form of our suggested family, named as a new extended heavy-tailed Weibull distribution is examined in detail. Some basic characterizations, including quantile function and raw moments have been derived. The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method. To judge the performance of the maximum likelihood estimators, a simulation analysis is performed in detail. Furthermore, some important actuarial measures such as value at risk and tail value at risk are also computed. A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed. The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set. The practical application shows that the proposed model is more flexible and efficient than the other six competing models including (i) the two-parameter models Weibull, Lomax and Burr-XII distributions (ii) the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions, and (iii) a well-known four-parameter Kumaraswamy Weibull distribution.

The Kumaraswamy Weibull Geometric Distribution with Applications

‎In this work‎, ‎we study the kumaraswamy weibull geometric ($Kw-WG$) distribution which includes as special cases‎, ‎several models such as the kumaraswamy weibull distribution‎, ‎kumaraswamy exponential distribution‎, ‎weibull geometric distribution‎, ‎exponential geometric distribution‎, ‎to name a few‎. ‎This distribution was monotone and non-monotone hazard rate functions‎, ‎which are useful in lifetime data analysis and reliability‎. ‎We derive some basic properties of the $Kw-WG$ distribution including noncentral $r$th‎-moments, ‎skewness‎, ‎kurtosis‎, ‎generating functions‎, ‎mean deviations‎, ‎mean residual life‎, ‎entropy‎, ‎order statistics and certain characterizations of our distribution‎. ‎The method of maximum likelihood is used for estimating the model parameters and a simulation study to investigate the behavior of this estimation is presented‎. ‎Finally‎, ‎an application of the new distribution and its comparison with recent flexible generalization of weibull distribution is illustrated via two real data sets‎.

Bayesian Approach for Constant-Stress Accelerated Life Testing for Kumaraswamy Weibull Distribution with Censoring

The accelerated life tests provide quick information on the life time distributions by testing materials or products at higher than basic conditional levels of stress such as pressure, high temperature, vibration, voltage or load to induce failures. In this paper, the acceleration model assumed is log linear model. Constant stress tests are discussed based on Type I and Type II censoring. The Kumaraswmay Weibull distribution is used. The estimators of the parameters, reliability, hazard rate functions and p-th percentile at normal condition, low stress, and high stress are obtained. In addition, credible intervals for parameters of the models are constructed. Optimum test plan are designed. Some numerical studies are used to solve the complicated integrals such as Laplace and Markov Chain Monte Carlo methods.

Estimation in the Exponentiated Kumaraswamy Dagum distribution with censored samples

In a recent note, Huang and Oluyede (2014) proposed a new model called the exponentiated Kumaraswamy Dagum (EKD) distribution with applications to income and lifetime data. In this note, this distribution is shown to be a very competitive model for describing censored observations in lifetime reliability problems. This work shows that in certain cases, the EKD distribution performs better than other parametric model such as the exponentiated Kumaraswamy Weibull distribution and its sub-models, which include some of the commonly used models in survival analysis and reliability analysis, such as the exponentiated Weibull, Weibull and exponential distributions.

Bayesian Approach for Constant-Stress Accelerated Life Testing for Kumaraswamy Weibull Distribution with Censoring

The accelerated life tests provide quick information on the life time distributions by testing materials or products at higher than basic conditional levels of stress such as pressure, high temperature, vibration, voltage or load to induce failures. In this paper, the acceleration model assumed is log linear model. Constant stress tests are discussed based on Type I and Type II censoring. The Kumaraswmay Weibull distribution is used. The estimators of the parameters, reliability, hazard rate functions and p-th percentile at normal condition, low stress, and high stress are obtained. In addition, credible intervals for parameters of the models are constructed. Optimum test plan are designed. Some numerical studies are used to solve the complicated integrals such as Laplace and Markov Chain Monte Carlo methods.

The New Kumaraswamy Kumaraswamy Weibull Distribution with Application

The Weibull distribution has been used over the past decades for modeling data in many fields so finding generalization of the Weibull distribution becomes very useful to fit more cases or get better fits than before. In this paper, the Kumaraswamy Kumaraswamy Weibull (Kw Kw W) distribution is presented for the first time and we show that it is generalizes many important distribution. The probability density function (pdf), the cumulative distribution function (cdf), moments, quantiles, the median, the mode, the mean deviation, the entropy, order statistics, L-moments, extreme value and parameters estimation based on maximum likelihood are obtained for the Kumaraswamy Kumaraswamy Weibull distribution. A simulation study and a real data set is used to illustrate the potentiality and application of the new Kumaraswamy Kumaraswamy Weibull distribution.

Bayesian inference from the Kumaraswamy-Weibull distribution with applications to real data

Bayesian inference from the Kumaraswamy-Weibull distribution with applications to real data

Statistical Inference for a Simple Constant Stress Model Based on Censored Sampling Data From the Kumaraswamy Weibull Distribution

In this paper, constant stress accelerated life tests are discussed based on Type I and Type II censored sampling data from Kumaraswmay Weibull distribution. The maximum likelihood estimators are derived for the unknown parameters. The log linear model is assumed as an accelerated model. In addition, confidence intervals for the model parameters are constructed. Optimum test plans, are developed to minimize the generalized asymptotic variance of the maximum likelihood estimators of the model parameters. Monte Carlo simulation is carried out to illustrate the theoretical results of the maximum likelihood estimates, confidence intervals and optimum test plans.