Comparing cross-country estimates of Lorenz curves using a Dirichlet distribution across estimators and datasets

We verify and extend the results of Chotikapanich and Griffiths (2002), who fit a family of five Lorenz curves to one year of Swedish and Brazilian income distribution
data using unconstrained non-linear least squares and maximum likelihood based on a
parametric Dirichlet distribution. We proceed in three steps.

First, we fit the Chotikapanich and Griffiths Lorenz curves using unconstrained nonlinear
least squares and maximum likelihood in Matlab R2013a (functions lsqcurvefit
and fminunc, respectively) and Stata 13 MP (function nl) and the Chotikapanich and
Griffiths data. We find that, for less heavily parameterized Lorenz curves, our point
estimates and standard errors match Chotikapanich and Griffiths. However, for more
heavily paramaterized Lorenz curves our standard error estimates are unstable, in that
nearly identical point estimates yield quite different standard error estimates, depending
on the set of initial conditions. This instability was also experienced by Chotikapanich
and Griffiths.

Second, we impose the necessary and sufficient conditions from Sarabia et al. (1999)
to ensure that the five Lorenz curves are invariant to increasing convex exponential and
power transformations, which turns our unconstrained estimation procedures into constrained
non-linear least squares and constrained maximum likelihood. Although imposing
the constraints changes some parameter estimates, the implied Gini coefficients
are similar between the unconstrained and constrained estimators for both non-linear
least squares and maximum likelihood across Lorenz curve specifications.

Third, we fit the Lorenz curves on five additional years of Swedish and Brazilian income
distribution data from the World Bank, using both constrained and unconstrained non-linear least squares and maximum likelihood. We find that the within-country and within-time
implied Gini coefficients are stable and are insensitive to our choice of estimator or Lorenz
curve specification. We conclude that the Chotikapanich and Griffiths results are robust.

References
Chotikapanich, D., Griffiths, W. E., 2002. Estimating lorenz curves using a dirichlet distribution. Journal of Business & Economic Statistics 20 (2), 290-295.

Sarabia, J.-M., Castillo, E., Slottje, D. J., 1999. An ordered family of lorenz curves. Journal of
Econometrics 91 (1), 43–60.