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Two new variables (var1, var2) are added to the current dataset.

Acknowledgments

This work forms part of the scientific research program of the Institute for Social and Economic Research, and was supported
by core funding from the University of Essex and the UK Economic and Social and Economic Research Council. The program
was created for use in joint work with Frank Cowell (London School of Economics) developing asymptotic standard errors for
inequality indices in the weighted data case.

sg106    Fitting Singh-Maddala and Dagum distributions by maximum likelihood

Stephen P. Jenkins, University of Essex, UK, [email protected]

Introduction

Economists and statisticians sometimes find it useful to fit parametric functional forms to data on a variable. smfit fits
the three-parameter Singh-Maddala (1976) distribution and dagumfit fits the Dagum (1977, 1980) distribution, in each case by
maximum likelihood (ML) methods, to a distribution of a random variable incvar, where unit record observations on incvar
are available. The Singh-Maddala distribution is also known as the Burr Type 12 distribution and the Dagum distribution as the
Burr Type 3 distribution. These three-parameter distributions have been shown to provide a good fit to empirical income data
relative to other parametric functional forms; see McDonald (1984), for example. For derivation of Lorenz orderings of pairs
of income distributions in terms of their Singh-Maddala and Dagum parameters, see Wifling and Kraemer (1993) and Kleiber
(1996). Of course the Singh-Maddala and Dagum distributions might be suitable for describing any skewed variable, not just
income.

Programmers may find smfit and dagumf it of interest because they are examples of the application of ml in a case which
is unlike a regression model (there are no covariates or dependent variable in the conventional sense).

The Singh-Maddala distribution

The Singh-Maddala distribution has distribution function
where a > 0, b > 0, q > 1∕α are parameters, for random variable X ≥ 0 (income). The parameters a and q are the key
distributional shape parameters; b is a scale parameter.

Letting z = 1 + (it∕δ)^α, then F(x) = 1 — z^-9, and the probability density function is

Fχ} = {aq∕b)z-^⁺¹∖x∕bγ^a-^

The likelihood function for a sample of incomes is specified as the product of the densities for each person (weighted where
relevant), and is maximized by smfit using Stata’s derivθ (numerical derivatives) method. In fact, transformations of the three
parameters are estimated (to impose the necessary restrictions) and the parameters derived from these.

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