Showing papers on "Kumaraswamy distribution published in 2019"

On estimation of reliability in a multicomponent stress-strength model for a Kumaraswamy distribution based on progressively censored sample

Abstract: Based on progressively Type-II censored samples, this paper deals with the estimation of multicomponent stress-strength reliability by assuming the Kumaraswamy distribution. Both stress and strength are assumed to have a Kumaraswamy distribution with different the first shape parameters, but having the same second shape parameter. Different methods are applied for estimating the reliability. The maximum likelihood estimate of reliability is derived. Also its asymptotic distribution is used to construct an asymptotic confidence interval. The Bayes estimates of reliability have been developed by using Lindley’s approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms. The uniformly minimum variance unbiased and Bayes estimates of reliability are obtained when the common second shape parameter is known. The highest posterior density credible intervals are constructed for reliability. Monte Carlo simulations are performed to compare the performances of the different methods, and one data set is analyzed for illustrative purposes.

Truncated Inverted Kumaraswamy Generated Family of Distributions with Applications

Abstract: In this article, we introduce a new general family of distributions derived to the truncated inverted Kumaraswamy distribution (on the unit interval), called the truncated inverted Kumaraswamy generated family. Among its qualities, it is characterized with tractable functions, has the ability to enhance the flexibility of a given distribution, and demonstrates nice statistical properties, including competitive fits for various kinds of data. A particular focus is given on a special member of the family defined with the exponential distribution as baseline, offering a new three-parameter lifetime distribution. This new distribution has the advantage of having a hazard rate function allowing monotonically increasing, decreasing, and upside-down bathtub shapes. In full generality, important properties of the new family are determined, with an emphasis on the entropy (Renyi and Shannon entropy). The estimation of the model parameters is established by the maximum likelihood method. A numerical simulation study illustrates the nice performance of the obtained estimates. Two practical data sets are then analyzed. We thus prove the potential of the new model in terms of fitting, with favorable results in comparison to other modern parametric models of the literature.

The Exponentiated Kumaraswamy-G Class: General Properties and Application

Abstract: We propose a new family of distributions called the exponentiated Kumaraswamy-G class with three extra positive parameters, which generalizes the Cordeiro and de Castro's family. Some special distributions in the new class are discussed. We derive some mathematical properties of the proposed class including explicit expressions for the quantile function, ordinary and incomplete moments, generating function, mean deviations, reliability, Renyi entropy and Shannon entropy. The method of maximum likelihood is used to fit the distributions in the proposed class. Simulations are performed in order to assess the asymptotic behavior of the maximum likelihood estimates. We illustrate its potentiality with applications to two real data sets which show that the extended Weibull model in the new class provides a better fit than other generalized Weibull distributions.

Generalized inverted Kumaraswamy generated family of distributions: theory and applications

Abstract: This paper proposes a new generator function based on the inverted Kumaraswamy distribution and introduces ‘generalized inverted Kumaraswamy-G’ family of distributions. We provide a compreh...

Inflated Kumaraswamy distributions

Abstract: The Kumaraswamy distribution is useful for modeling variables whose support is the standard unit interval, i.e., (0, 1). It is not uncommon, however, for the data to contain zeros and/or ones. When that happens, the interest shifts to modeling variables that assume values in [0, 1), (0, 1] or [0, 1]. Our goal in this paper is to introduce inflated Kumaraswamy distributions that can be used to that end. We consider inflation at one of the extremes of the standard unit interval and also the more challenging case in which inflation takes place at both interval endpoints. We introduce inflated Kumaraswamy distributions, discuss their main properties, show how to estimate their parameters (point and interval estimation) and explain how testing inferences can be performed. We also present Monte Carlo evidence on the finite sample performances of point estimation, confidence intervals and hypothesis tests. An empirical application is presented and discussed.

Stress–Strength Parameter Estimation Based on Type-II Hybrid Progressive Censored Samples for a Kumaraswamy Distribution

Abstract: In this paper, when the stress and strength are two independent Kumaraswamy random variables, we derive the point and interval estimate of the stress–strength parameter, from both frequentist and Bayesian viewpoints, under the Type-II hybrid progressive censoring scheme. In fact, the problem is solved in two cases. First, with the assumption that the stress and strength have different first shape parameters and the common second shape parameter, we attain maximum likelihood estimate (MLE), approximation MLE, and two Bayesian approximation estimates, Lindley's approximation and the Markov chain Monte Carlo (MCMC) method, due to lack of closed forms. Also, the asymptotic confidence interval of $R$ is constructed by asymptotic distribution of it. Moreover, by using the MCMC method, we achieve the highest posterior density credible intervals. Second, with the assumption that the common shape parameter is known, we attain the MLE, the exact Bayes estimate, and the uniformly minimum-variance unbiased estimate of $R$ . Also, we construct the asymptotic and Bayesian intervals for the stress–strength parameter. Furthermore, to compare the performance of various methods, we apply the Monte Carlo simulation. Finally, one real dataset is analyzed for demonstrative aims.

The Topp Leone Generalized Inverted Kumaraswamy Distribution: Properties and Applications

Abstract: We propose a new four parameters continuous model called the Topp Leone generalized inverted Kumaraswamy distribution which extends the generalized inverted Kumaraswamy distribution. Basic mathematical properties of the new distribution are invstigated such as; quantile function, raw and incomplete moments, generating functions, probability weighted moments, order statistics, Rényi entropy, stochastic ordering and stress strength model. The method of maximum likelihood is used to estimate the model parameters of the new distribution. A Monte Carlo simulation is conducted to examine the behavior of the parameter estimates. The applicability of the new model is demonstrated by means of two real applications.

Type II Topp–Leone Inverted Kumaraswamy Distribution with Statistical Inference and Applications

Abstract: In this paper, we present and study a new four-parameter lifetime distribution obtained by the combination of the so-called type II Topp–Leone-G and transmuted-G families and the inverted Kumaraswamy distribution. By construction, the new distribution enjoys nice flexible properties and covers some well-known distributions which have already proven themselves in statistical applications, including some extensions of the Bur XII distribution. We first present the main functions related to the new distribution, with discussions on their shapes. In particular, we show that the related probability density function is left, right skewed, near symmetrical and reverse J shaped, with a notable difference regarding the right tailed, illustrating the flexibility of the distribution. Then, the related model is displayed, with the estimation of the parameters by the maximum likelihood method and the consideration of two practical data sets. We show that the proposed model is the best one in terms of standard model selection criteria, including Akaike information and Bayesian information criteria, and goodness of fit tests against three well-established competitors. Then, for the new model, the theoretical background on the maximum likelihood method is given, with numerical guaranties of the efficiency of the estimates obtained via a simulation study. Finally, the main mathematical properties of the new distribution are discussed, including asymptotic results, quantile function, Bowley skewness and Moors kurtosis, mixture representations for the probability density and cumulative density functions, ordinary moments, incomplete moments, probability weighted moments, stress-strength reliability and order statistics.

Kumaraswamy Marshall-Olkin Exponential distribution

Abstract: In this paper, a new generalization of the Kumaraswamy distribution namely, the Kumaraswamy Marshall-Olkin Exponential distribution (KwMOE) is introduced and studied. Various properties are explore...

Reliability Models Using the Composite Generalizers of Weibull Distribution

Abstract: In this article, we study the composite generalizers of Weibull distribution using exponentiated, Kumaraswamy, transmuted and beta distributions. The composite generalizers are constructed using both forward and reverse order of each of these distributions. The usefulness and effectiveness of the composite generalizers and their order of composition is investigated by studying the reliability behavior of the resulting distributions. Two sets of real-world data are analyzed using the proposed generalized Weibull distributions.

Comparisons of six different estimation methods for log-Kumaraswamy distribution

Abstract: In this paper, it is considered the problem of estimation of unknown parameters of log-Kumaraswamy distribution via Monte-Carlo simulations. Firstly, it is described six different estimation methods such as maximum likelihood, approximate bayesian, least-squares, weighted least-squares, percentile, and Cramer-von-Mises. Then, it is performed a Monte-Carlo simulation study to evaluate the performances of these methods according to the biases and mean-squared errors of the estimators. Furthermore, two real data applications based on carbon fibers and the gauge lengths are presented to compare the fits of log-Kumaraswamy and other fitted statistical distributions.

Statistical Properties and Different Methods of Estimation for Type I Half Logistic Inverted Kumaraswamy Distribution

Abstract: In this paper, we introduce and study a new three-parameter lifetime distribution constructed from the so-called type I half-logistic-G family and the inverted Kumaraswamy distribution, naturally called the type I half-logistic inverted Kumaraswamy distribution. The main feature of this new distribution is to add a new tuning parameter to the inverted Kumaraswamy (according to the type I half-logistic structure), with the aim to increase the flexibility of the related inverted Kumaraswamy model and thus offering more precise diagnostics in data analyses. The new distribution is discussed in detail, exhibiting various mathematical and statistical properties, with related graphics and numerical results. An exhaustive simulation was conducted to investigate the estimation of the model parameters via several well-established methods, including the method of maximum likelihood estimation, methods of least squares and weighted least squares estimation, and method of Cramer-von Mises minimum distance estimation, showing their numerical efficiency. Finally, by considering the method of maximum likelihood estimation, we apply the new model to fit two practical data sets. In this regards, it is proved to be better than recent models, also derived to the inverted Kumaraswamy distribution.

On some properties of Generalized Transmuted Kumaraswamy distribution

Abstract: This articles introduces a new lifetime model called the generalized transmuted Kumaraswamy distribution which extends the Kumaraswamy distribution from the family proposed by Nofal et al., (2017). We provide hazard and survival functions of the proposed distribution. The statistical properties of the proposed model are provided and the method of Maximum Likelihood Estimation (MLE) was proposed in estimating its parameters.

Relations for Moments of Dual Generalized Order Statistics for a New Inverse Kumaraswamy Distribution

Abstract: The inverse Kumaraswamy distribution has been presented as a new distribution. The recurrence relation for moments of dual generalized order statistics has been presented for the new inverse Kumaraswamy distribution. These include the recurrence relations for single, inverse, product and ratio moments of dual generalized order statistics for the new inverse Kumaraswamy distribution. Special cases of the recurrence relations have also been obtained.

A New Distribution on the Simplex with Auto-Encoding Applications

Abstract: We construct a new distribution for the simplex using the Kumaraswamy distribution and an ordered stick-breaking process. We explore and develop the theoretical properties of this new distribution and prove that it exhibits symmetry (exchangeability) under the same conditions as the well-known Dirichlet. Like the Dirichlet, the new distribution is adept at capturing sparsity but, unlike the Dirichlet, has an exact and closed form reparameterization--making it well suited for deep variational Bayesian modeling. We demonstrate the distribution's utility in a variety of semi-supervised auto-encoding tasks. In all cases, the resulting models achieve competitive performance commensurate with their simplicity, use of explicit probability models, and abstinence from adversarial training.

Empirical Bayes estimator of parameter, reliability and hazard rate for Kumaraswamy distribution

Abstract: This paper proposes empirical Bayes estimators of parameter, reliability and hazard function for Kumaraswamy distribution under the linear exponential loss function for progressively type II censored samples with binomial removal and type II censored samples. The proposed estimators have been compared with the corresponding Bayes estimators for their simulated risks. The applicability of the proposed estimators have been illustrated through ulcer patient data.

A New Distribution on the Simplex with Auto-Encoding Applications

Abstract: We construct a new distribution for the simplex using the Kumaraswamy distribution and an ordered stick-breaking process. We explore and develop the theoretical properties of this new distribution and prove that it exhibits symmetry under the same conditions as the well-known Dirichlet. Like the Dirichlet, the new distribution is adept at capturing sparsity but, unlike the Dirichlet, has an exact and closed form reparameterization--making it well suited for deep variational Bayesian modeling. We demonstrate the distribution's utility in a variety of semi-supervised auto-encoding tasks. In all cases, the resulting models achieve competitive performance commensurate with their simplicity, use of explicit probability models, and abstinence from adversarial training.

The Stress-Strength Models for the Proportional Hazards Family and Proportional Reverse Hazards Family

Abstract: The stress-strength model has been widely used for reliability design of systems. The reliability of the model is defined as the probability that the strength is larger than the stress. This chapter considers the stress-strength model when both the stress and the strength variables follow the two-parameter proportional hazards family or the proportional reverse hazards family. These two distribution families include many commonly-used distributions, such as the Weibull distribution, the Gompertz distribution, the Kumaraswamy distribution and the generalized exponential distribution, etc. Based on complete samples and record values, we derive the maximum likelihood estimation for the these stress-strength reliability. We also present the generalized confidence intervals for these stress-strength reliability. The simulation results show that the proposed generalized confidence intervals work well.

Comparing Different Estimators for the shape Parameter and the Reliability function of Kumaraswamy Distribution

Abstract: Accepted: 17/2/2019 Abstract In this paper, we used maximum likelihood method and the Bayesian method to estimate the shape parameter (θ), and reliability function (R(t)) of the Kumaraswamy distribution with two parameters  θ (under assuming the exponential distribution, Chi-squared distribution and Erlang-2 type distribution as prior distributions), in addition to that we used method of moments for estimating the parameters of the prior distributions. Bayes estimators derived under the squared error loss function. We conduct simulation study, to compare the performance for each estimator, several values of the shape parameter (θ) from Kumaraswamy distribution for data-generating, for different samples sizes (small, medium, and large). Simulation results have shown that the Best method is the Bayes estimation according to the smallest values of mean square errors(MSE) for all samples sizes (n).

MMK for heterogeneous data: identifiability, estimation and application

Abstract: Heterogeneous real datasets need complex probabilistic structures for a correct modeling. On the other hand, several generalizations of the Kumaraswamy distribution have been developed in the past few decades in an attempt to obtain better data adjustments that are limited in the interval (0,1). In this paper, we propose a mixture model of Kumaraswamy distributions (MMK) as a probabilistic structure for heterogeneous datasets with support in (0,1) and as an important generalization of the Kumaraswamy distribution. We derive the moments, the moment-generating function and analyze the failure rate function. Also, we prove the identifiability of the class of all finite mixtures of Kumaraswamy distributions. Via the EM-algorithm, we find estimates of maximum likelihood for the parameters of the MMK. Finally, we test the performance of the estimates by Monte Carlo simulation and illustrate an application of the proposed model using a real dataset.

Evaluation of mean and variance approximations in three point estimation of task completion times using the beta and the Kumaraswamy distribution

Abstract: Estimation of task and project completion times within IT projects remains one of the most error-prone, but also most critical duties of an IT project manager. Various three-point methods of PERT have been evaluated by assuming that the true distribution is a beta-distribution. We evaluate PERT methods by comparing additionally with the Kumaraswamy distribution, which has an equal claim to be the true a-priori distribution for project completion times. We use skew and kurtosis in order to define test sets instead of simply choosing a range of shape parameters. We validate various approximations proposed in the literature and show that valid approximations are possible.