Some Generalized Functions for the Size Distribution of Income

McDonald, James B.

doi:10.2307/1913469

Cited by 653 publications

(366 citation statements)

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“…The McGBB distribution with parameters n, α, β, γ is a mixture of binomial Bin(n, π) and the McDonald's generalized beta distribution (McDonald, 1984(McDonald, , 1995 of the first kind GB1(α, β, γ) where the latter distribution is defined as:…”

Section: The Mcdonald Generalized Beta-binomial Model (Mcgbb)mentioning

confidence: 99%

On Some Mixture Models for Over-dispersed Binary Data

Lawal¹

2017

IJSP

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In this paper, we consider several binomial mixture models for fitting over-dispersed binary data. The models range from the binomial itself, to the beta-binomial (BB), the Kumaraswamy distributions I and II (KPI & KPII) as well as the McDonald generalized beta-binomial mixed model (McGBB). The models are applied to five data sets that have received attention in various literature. Because of convergence issues, several optimization methods ranging from the Newton-Raphson to the quasi-Newton optimization algorithms were employed with SAS PROC NLMIXED using the Adaptive Gaussian Quadrature as the integral approximation method within PROC NLMIXED. Our results differ from those presented in Li, Huang and Zhao (2011) for the example data sets in that paper but agree with those presented in Manoj, Wijekoon and Yapa (2013). We also applied these models to the case where we have a k vector of covariates (x 1 , x 2 , . . . , x k ) ′ . Our results here suggest that the McGBB performs better than the other models in the GLM framework.All computations in this paper employed PROC NLMIXED in SAS. We present in the appendix a sample of the SAS program employed for implementing the McGBB model for one of the examples.

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Section: The Mcdonald Generalized Beta-binomial Model (Mcgbb)mentioning

confidence: 99%

On Some Mixture Models for Over-dispersed Binary Data

Lawal¹

2017

IJSP

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“…The McDonald's Generalized Beta distribution of the first kind (McDonald, 1984(McDonald, , 1995 is considered in our work. It has been shown that this distribution has more flexibility than the Beta and Kumaraswamy distributions in literature (see for example : Alexander, Cordeiro, Ortega, & Sarabia, 2012).…”

Section: Univariate Continuousmentioning

confidence: 99%

“…Let P be a random variable following a McDonald's Generalized Beta distribution of the first kind (McDonald, 1984(McDonald, , 1995 with three shape parameters α, β and γ, denoted GB1 (α, β, γ). The probability density function of P is then given by…”

Section: Mcdonald's Generalized Beta Distribution Of the First Kindmentioning

confidence: 99%

The McDonald Generalized Beta-Binomial Distribution: A New Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling Overdispersion

Chandrabose¹,

Wijekoon²,

Yapa³

2013

IJSP

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The binomial outcome data are widely encountered in many real world applications. The Binomial distribution often fails to model the binomial outcomes since the variance of the observed binomial outcome data exceeds the nominal Binomial distribution variance, a phenomenon known as overdispersion. One way of handling overdispersion is modeling the success probability of the Binomial distribution using a continuous distribution defined on the standard unit interval. The resultant general class of univariate discrete distributions is known as the class of Binomial mixture distributions. The Beta-Binomial (BB) distribution is a prominent member of this class of distributions. The Kumaraswamy-Binomial (KB) distribution is another recent member of this class. In this paper we focus the emphasis on the McDonald's Generalized Beta distribution of the first kind as the mixing distribution and introduce a new Binomial mixture distribution called the McDonald Generalized Beta-Binomial distribution(McGBB). Some theoretical properties of McGBB are discussed. The parameters of the McGBB distribution are estimated via maximum likelihood estimation technique. A real world dataset is modeled by using the new McGBB mixture distribution, and it is shown that this model gives better fit than its nested models. Finally, an extended simulation study is presented to compare the McGBB distribution with its nested distributions in handling overdispersed binomial outcome data.

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“…We propose an extension of the Kw distribution by taking G 1 (x; ω) as the cdf (3) and g 2 (x; τ ) as the generalized beta density of the first kind (GB1) Gupta and Nadarajah (2004a);McDonald (1984) defined by…”

Section: Introductionmentioning

confidence: 99%

An Extension of the Kumaraswamy Distribution

Carrasco¹,

Cordeiro²

2017

IJSP

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We propose and study a new five-parameter continuous distribution in the unit interval through a specific probability integral transform. The new distribution, under some parameter constraints, is an identified parametric model that includes as special cases six important models such as the Kumaraswamy and beta distributions. We obtain ordinary and incomplete moments, quantile and generating functions, mean deviations, Rényi entropy and moments of order statistics. The estimation of the model parameters is performed by maximum likelihood, and hypothesis tests are discussed. Additionally, through a simulation study we investigate the behavior of the maximum likelihood estimator, also we investigate the impact of ignoring identifiability problems. The usefulness of the proposed distribution is illustrated by means of a real data set.

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Some Generalized Functions for the Size Distribution of Income

Cited by 653 publications

References 0 publications

On Some Mixture Models for Over-dispersed Binary Data

On Some Mixture Models for Over-dispersed Binary Data

The McDonald Generalized Beta-Binomial Distribution: A New Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling Overdispersion

An Extension of the Kumaraswamy Distribution

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