Bayesian Estimation of Beta-type Distribution Parameters Based on Grouped Data

Abstract

This study considers the estimation method of generalized beta (GB) distribution parameters based on grouped data from a Bayesian point of view. Because the GB distribution, which was proposed by McDonald and Xu (1995), includes several kinds of familiar distributions as special or limiting cases, it performs at least as well as those special or limiting distributions. Therefore, it is reasonable to estimate the parameters of the GB distribution. However, when the number of groups is small or when the number of parameters increases, it may become difficult to estimate the distribution parameters for grouped data using the existing estimation methods. This study uses a Tailored randomized block Metropolis-Hastings (TaRBMH) algorithm proposed by Chib and Ramamurthy (2010) to estimate the GB distribution parameters, and this method is applied to one simulated and two real datasets. Moreover, the Gini coefficients from the estimated parameters for the GB distribution are examined.

Approximate Bayesian computation for Lorenz curves from grouped data

“…This distribution was by proposed by McDonald and Xu (1995) and is the most flexible distribution of the family of beta-type distributions. The probability density function of the GB distribution is given by The hypothetical GB distribution can also be directly estimated from the grouped level income data by using the MCMC or maximum likelihood method (Kakamu and Nishino, 2016). An explicit formula of the Lorenz curve for the GB distribution is not available (McDonald and Ransom, 2008).…”

“…where Z ∼ Be(p, q) (Kakamu and Nishino, 2016). Since the Lorenz curve is location-free, we let θ = (a, c, p, q) and b is fixed to 1.…”

Classical and Bayesian Inference for Income Distributions using Grouped Data

“…Hajargasht and Griffiths (2020) shift the focus from income distributions to parametric Lorenz curves and provide a GMM framework covering two DGPs of empirical relevance, and Chen (2018) generalizes the GMM framework to incorporate varying data information. Bayesian approaches to the estimation of parametric income distributions are provided by Chotikapanich and Griffiths (2000), Kakamu (2016) and Kakamu and Nishino (2019). All Bayesian methods employ Monte Carlo Markov Chain (MCMC) techniques based on the Metropolis‐Hastings (MH) algorithm in order to obtain samples from the parameters’ joint posterior distribution.…”

“…All Bayesian methods employ Monte Carlo Markov Chain (MCMC) techniques based on the Metropolis‐Hastings (MH) algorithm in order to obtain samples from the parameters’ joint posterior distribution. While Chotikapanich and Griffiths (2000) employ the standard multinomial likelihood of McDonald (1984), the recent contributions of Kakamu (2016) and Kakamu and Nishino (2019) employ the joint likelihood of a set of order statistics as proposed by Nishino and Kakamu (2011), which is – however – appropriate for quantile‐data only. Moreover, both Bayesian settings do not account for unknown group boundaries and ignore the information of observed group mean incomes.…”

Classical and Bayesian Inference for Income Distributions using Grouped Data

“…Hajargasht and Griffiths (2020) shift the focus from income distributions to parametric Lorenz curves and provide a GMM framework covering two DGPs of empirical relevance, and Chen (2018) generalizes the GMM framework to incorporate varying data information. Bayesian approaches to the estimation of parametric income distributions are provided by , Kakamu (2016) and Kakamu and Nishino (2019). All Bayesian methods employ Monte Carlo Markov Chain (MCMC) techniques based on the Metropolis-Hastings (MH) algorithm in order to obtain samples from the parameters' joint posterior distribution.…”