Estimating the exponential family using grouped data: An application to the New Zealand income distribution1

Abstract

The exponential family of distributions offers considerable scope for the analysis of income distributions because of its ability to ‘nest’ many densities and the possibility of deriving special cases explicitly from labour demand and supply models. This paper presents several estimation methods based on the use of grouped data. These methods are motivated by the fact that many income distribution data are available in grouped form. The methods are applied to New Zealand income distribution data for males and females in a number of age groups. The generalised gamma distribution is found to provide the best fit to the distributions of most age groups. Three of the four parameters of the generalized gamma distribution are expressed as functions of age and conditional generalised gamma distributions are estimated using maximum likelihood, modified for grouped data. The estimated model captured the major empirical features of the changing distribution of income with age.