A new family of generalized distributions
01 Jul 2011-Journal of Statistical Computation and Simulation (Taylor & Francis)-Vol. 81, Iss: 7, pp 883-898
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...
Citations
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10 Apr 2013
TL;DR: In this article, a new method is proposed for generating families of continuous distributions, where a random variable is used to transform another random variable and the resulting family, the $$T$$¯¯ -=-=-=-=-=-=-=-=-=-=-=-=- family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function.
Abstract: In this paper, a new method is proposed for generating families of continuous distributions. A random variable $$X$$
, “the transformer”, is used to transform another random variable $$T$$
, “the transformed”. The resulting family, the $$T$$
-
$$X$$
family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function of the random variable $$X$$
. Many new distributions, which are members of the family, are presented. Several known continuous distributions are found to be special cases of the new distributions.
694 citations
Cites background from "A new family of generalized distrib..."
...(1.11) Several generalized distributions from (1.11) have been studied in the literature including the Kumaraswamy Weibull distribution by Cordeiro et al. [7], the Kumaraswamy generalized gamma distribution by de Castro et al. [9], and the Kumaraswamy generalized half-normal distribution by Cordeiro et al. [8]....
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...Recently, Jones [22] and Cordeiro and de Castro [6] extended the beta-generated family of distributions by replacing the beta distribution in (1....
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...Recently, Jones [22] and Cordeiro and de Castro [6] extended the beta-generated family of distributions by replacing the beta distribution in (1.9) with the Kumaraswamy distribution, b(t) = αβxα−1(1 − xα)β−1, x ∈ (0, 1), Kumaraswamy [25]....
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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.
348 citations
TL;DR: The generalized beta-generated (GBG) distributions as discussed by the authors are the most tractable class of distributions and have tractable properties: moments, generating function, quantiles, deviations and reliability.
Abstract: This article introduces generalized beta-generated (GBG) distributions. Sub-models include all classical beta-generated, Kumaraswamy-generated and exponentiated distributions. They are maximum entropy distributions under three intuitive conditions, which show that the classical beta generator skewness parameters only control tail entropy and an additional shape parameter is needed to add entropy to the centre of the parent distribution. This parameter controls skewness without necessarily differentiating tail weights. The GBG class also have tractable properties: we present various expansions for moments, generating function, quantiles, deviations and reliability. The model parameters are estimated by maximum likelihood and the usefulness of the new class is illustrated by means of some real data sets.
290 citations
Cites background from "A new family of generalized distrib..."
...For instance, the Beta-Birnbaum Saunders distribution introduced by Cordeiro and Lemonte (2011) is a special case of the generalized beta Birnbaum-Saunders distribution, which has the classical Birnbaum and Saunders (1969) distribution as parent....
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TL;DR: The generalized beta-generated (GBG) distributions as mentioned in this paper are a subclass of GBG distributions, and they have tractable properties: they can be expanded for moments, generating function and quantiles.
242 citations
TL;DR: In this article, a new five-parameter continuous distribution, the so-called McDonald Lomax distribution, was proposed and studied, which has as special sub-models new four-and threeparameter distributions, including expansions for the density function, explicit expressions for the moments, generating and quantile functions, mean deviations and Renyi entropy.
Abstract: A new five-parameter continuous distribution, the so-called McDonald Lomax distribution, that extends the Lomax distribution and some other distributions is proposed and studied. The model has as special sub-models new four- and three-parameter distributions. Various structural properties of the new distribution are derived, including expansions for the density function, explicit expressions for the moments, generating and quantile functions, mean deviations and Renyi entropy. The score function is derived and the estimation is performed by maximum likelihood. We also obtain the observed information matrix. An application illustrates the usefulness of the proposed model.
159 citations
Cites background or methods from "A new family of generalized distrib..."
...We adopt a different approach to much of the literature so far: rather than considering the classical beta generator [3] or the Kumaraswamy generator [4] applied to a baseline distribution, we propose the use of a more flexible McDonald generator applied to the Lomax distribution....
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...The class of distributions (2) includes two important special sub-classes: the beta generalized distributions [3] for c = 1 and the Kumaraswamy generalized distributions [4] for a = c....
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References
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4,258 citations
Book•
01 Jun 1981
TL;DR: A number of new classes of life distributions arising naturally in reliability models are treated systematically and each provides a realistic probabilistic description of a physical property occurring in the reliability context, thus permitting more realistic modeling of commonly occurring reliability situations.
Abstract: : This is the first of two books on the statistical theory of reliability and life testing. The present book concentrates on probabilistic aspects of reliability theory, while the forthcoming book will focus on inferential aspects of reliability and life testing, applying the probabilistic tools developed in this volume. This book emphasizes the newer, research aspects of reliability theory. The concept of a coherent system serves as a unifying theme for much of the book. A number of new classes of life distributions arising naturally in reliability models are treated systematically: the increasing failure rate average, new better than used, decreasing mean residual life, and other classes of distributions. As the names would seem to indicate, each such class of life distributions provides a realistic probabilistic description of a physical property occurring in the reliability context. Also various types of positive dependence among random variables are considered, thus permitting more realistic modeling of commonly occurring reliability situations.
3,876 citations
"A new family of generalized distrib..." refers background in this paper
...In reliability and life testing experiments, many times the data are modelled by finite-range distributions, see, for example, [10]....
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IBM1
TL;DR: The authors define L-moments as the expectations of certain linear combinations of order statistics, which can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions.
Abstract: L-moments are expectations of certain linear combinations of order statistics. They can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. The theory involves such established procedures as the use of order statistics and Gini's mean difference statistic, and gives rise to some promising innovations such as the measures of skewness and kurtosis and new methods of parameter estimation
2,668 citations
Journal Article•
TL;DR: In this paper, a nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard.
Abstract: On introduit une nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard
2,470 citations
"A new family of generalized distrib..." refers background in this paper
...Further, the KwN distribution with a = 2 and b = 1 coincides with the skew normal distribution with the shape parameter equal to one [13]....
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TL;DR: The generalized additive model for location, scale and shape (GAMLSS) as mentioned in this paper is a general class of statistical models for a univariate response variable, which assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects.
Abstract: Summary. A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y, as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton–Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.
2,386 citations
"A new family of generalized distrib..." refers methods in this paper
...Numerical maximization of the log-likelihood above is accomplished by using the RS method 18 available in the gamlss package 14 in R Development Core Team 19 ....
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