The mathematical properties of a family of generalized beta distribution, including beta-normal, skewed-t, log-F, beta-exponential, beta-Weibull distributions have recently been studied in several publications. This paper applies these distributions to the modeling of the size distribution of income and computes the maximum likelihood estimation estimates of parameters. Their performances are compared to the widely used generalized beta distributions of the first and second types in terms of measures of goodness of fit.
for 0 < y < 1, where α > 0, β > 0 and the beta function B(α, β) = Γ(α + β)/ [Γ(α)Γ(β)]. Note the domain of G(•) is (0, 1). Use the fact that the range of a cumulative distribution function (cdf) is (0, 1), replacing the upper limit y of the integration in (1.1) with a cdf F (•) has been studied in several papers reviewed below. The resulting probability density function is
where f (•) is the derivative of F (•)and therefore is the corresponding probability density function if F (•) is a distribution function. For simplicity, this distribution will be called the generalized beta-F distribution hereafter. Jones (2004) introduced this as the probability density function of the transformed random variable X = F -1 (Y ) where Y is a Beta random variable with parameters of α and β. The density function form in (1.2) was also alternatively described as a simple generalization of the use of the collection of order statistics distributions associated with F. Jones (2004) and Ferreira, etc. (2004) explored general properties of this family of distributions and examined the special cases of skewed-t and log-F distributions. Since F (•) can be any distribution function, the family of this generalized beta-F distribution is a very rich one and can be further explored. This family of distribution was first introduced by Singh et al. (1988) and has since been studied for several distribution functions. In this paper, it is applied to the analyses of placecountry-regionU.S. family income data.
Numerous distributions (see McDonald, 1984, and references therein), including gamma, beta, Singh-Maddala (or Burr), Pareto, Weibull, and generalized beta of first and second kinds, have been used to model the size distribution of income. McDonald (1984) fit the above models to the income data of 1970, 1975, and 1980 and concluded that the generalized beta of the second type provided the best relative fit and that the Singh-Maddala (SM) distribution provided a better fit than the generalized beta of the first kind. McDonald also discussed the relationships between several widely used models for the income distribution, those relationships can be expanded to the family of the generalized beta-F distribution in (1.2) that includes some of the distributions as its special cases.
In this paper, examples of the family of the generalized beta-F distribution described in (1.2) in existing literature are summarized in section 2. The distributions tabulated in Table 1 are fit to the U.S. family income data presented in a grouped format on the website of the Census Bureau. Outlines of the maximum likelihood estimation of unknown population parameters involved in the generalized beta-F distribution function and in the F (•)function are derived for the grouped income data in section 3. The equations to be maximized and the gradients do not have closed forms and depend on the function F (•) of interest. Besides the parameter estimates and associated estimated value for the mean, goodness-of-fit values including sum of the squared errors, sum of the absolute deviations and chi-squares are reported for comparisons in section 4. The performance comparisons of the distributions are also presented.
The probability functions of interest and their means and moments are summarized in Table 1 in this section. Technical details on the characteristics such as the shapes, moments, skewness, and limiting distribution as some parameters tend to extreme values of each distribution can also be found in the provided reference.
The generalized beta of the first (GB1) and second (GB2) kind (McDonald, 1984) are respectively defined by
They are special cases of the generalized beta-F distribution with
2), respectively. The underlying distribution of income in Thurow (1970) with a=1 in (2.1) is therefore also a special case with a distribution function F of a uniform distribution over the interval (0, b). The Singh-Maddala distribution with a density function of aβy ( a-1)/[1+(y/b) a ] β+1 is a special case of generalized beta of the second kind with the beta parameter α = 1. Note that these distributions are unimodal. Eugene (2002)studies the properties of a beta-normal (BN) distribution, i.e., F is a normal distribution function. Gupta and Nadarajah (2004) further derived a different form of the moments of the beta-normal distribution. The betanormal can be both bimodal and unimodal. Eugene (2002) showed that it is skewed to the right when α > β and skewed-to the left when α < β. When α = β, it is symmetric aboutµ. It has heavy symmetric tails when α < 1and β < 1 in which bimodality eventually occurs as α(=β) decreases. Also when α > 1 andβ > 1, it has long symmetric tails with a higher peak associated with a larger value of α(=β).
The two particular distributions that Jones (2004) believed to provide the most tractable instances of families with power and exponential tails are the skew-t distribution (Beta-T ) and the log-F (Beta-Logistic) distribution. The
