Topic

# Kumaraswamy distribution

About: Kumaraswamy distribution is a(n) research topic. Over the lifetime, 213 publication(s) have been published within this topic receiving 3393 citation(s). The topic is also known as: Kumaraswamy's double bounded distribution.

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TL;DR: In this paper, a new family of generalized distributions for double-bounded random processes with hydrological applications is described, including Kw-normal, Kw-Weibull and Kw-Gamma distributions.

Abstract: Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with a...

604 citations

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Open University

^{1}TL;DR: In this article, a two-parameter family of distributions on (0, 1) is explored, which has many similarities to the beta distribution and a number of advantages in terms of tractability.

Abstract: A two-parameter family of distributions on (0,1) is explored which has many similarities to the beta distribution and a number of advantages in terms of tractability (it also, of course, has some disadvantages). Kumaraswamy’s distribution has its genesis in terms of uniform order statistics, and has particularly straightforward distribution and quantile functions which do not depend on special functions (and hence afford very easy random variate generation). The distribution might, therefore, have a particular role when a quantile-based approach to statistical modelling is taken, and its tractability has appeal for pedagogical uses. To date, the distribution has seen only limited use and development in the hydrological literature.

354 citations

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TL;DR: This work introduces and study some mathematical properties of the Kumaraswamy Weibull distribution that is a quite flexible model in analyzing positive data and provides explicit expressions for the moments and moment generating function.

Abstract: For the first time, we introduce and study some mathematical properties of the Kumaraswamy Weibull distribution that is a quite flexible model in analyzing positive data. It contains as special sub-models the exponentiated Weibull, exponentiated Rayleigh, exponentiated exponential, Weibull and also the new Kumaraswamy exponential distribution. We provide explicit expressions for the moments and moment generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and Renyi entropy. The moments of the order statistics are calculated. We also discuss the estimation of the parameters by maximum likelihood. We obtain the expected information matrix. We provide applications involving two real data sets on failure times. Finally, some multivariate generalizations of the Kumaraswamy Weibull distribution are discussed.

288 citations

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TL;DR: The methods of maximum likelihood and parametric bootstrap and a Bayesian procedure are proposed for estimating the model parameters and explicit expressions are derived for the moments of order statistics for the Gumbel distribution.

Abstract: The Gumbel distribution is perhaps the most widely applied statistical distribution for problems in engineering. We propose a generalization—referred to as the Kumaraswamy Gumbel distribution—and provide a comprehensive treatment of its structural properties. We obtain the analytical shapes of the density and hazard rate functions. We calculate explicit expressions for the moments and generating function. The variation of the skewness and kurtosis measures is examined and the asymptotic distribution of the extreme values is investigated. Explicit expressions are also derived for the moments of order statistics. The methods of maximum likelihood and parametric bootstrap and a Bayesian procedure are proposed for estimating the model parameters. We obtain the expected information matrix. An application of the new model to a real dataset illustrates the potentiality of the proposed model. Two bivariate generalizations of the model are proposed.

88 citations

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TL;DR: In this article, the authors proposed a generalization of the Kumaraswamy distribution, referred to as the exponentiated KG distribution, and derived the moments, moment generating function, mean deviations, Bonferroni and Lorentz curves, density of the order statistics and their moments.

Abstract: The paper by Kumaraswamy (Journal of Hydrology 46 (1980) 79–88) introduced a probability distribution for double bounded random processes which has considerable attention in hydrology and related areas. Based on this distribution, we propose a generalization of the Kumaraswamy distribution refereed to as the exponentiated Kumaraswamy distribution. We derive the moments, moment generating function, mean deviations, Bonferroni and Lorentz curves, density of the order statistics and their moments. We also present a related distribution, so-called the log-exponentiated Kumaraswamy distribution, which extends the generalized exponential (Aust. N. Z. J. Stat. 41 (1999) 173–188) and double generalized exponential (J. Stat. Comput. Simul. 80 (2010) 159–172) distributions. We discuss maximum likelihood estimation of the model parameters. In applications to real data sets, we show that the log-exponentiated Kumaraswamy model can be used quite effectively in analyzing lifetime data.

83 citations