Stata Technical Bulletin
21
Saved results
The global macros set by ml post, plus
S_a, S_b, S_q estimated parameters α. 6, ç. respectively
Access to estimated coefficients (transformations of the parameters) and their standard errors are available in the usual way:
see [U] 20.5 Accessing coefficients and standard errors, and [R] matrix get.
The Dagum distribution
The Dagum distribution has distribution function
where b > 0. h > 0. d > 1/b are parameters. for random variable X > 0 (income). Parameters b and d are the key distributional
shape parameters; h is a scale parameter.
1(x) 1 1 + hx~d
-b
The probability density function is
f(x) ι 1bdh')x(-d~^]/[l + hx<-dψ+V
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 dagumfit using Stata’s derivθ (numerical derivatives) method. Transformations of the
3 parameters are estimated (to impose the necessary restrictions) and the parameters derived from these.
The formulas used to derive the distributional summary statistics presented (optionally) are as follows. The rth moment
about the origin is given by
By substitution and using the result that G(1) = 1. implies that the moments can be written
bh(r^G(l -r∕d)G(b + r∕d)∕G(b+ V)
and hence
E(X) 1 [bhWd>]G(l - 1∕d)G(b + 1∕d)∕G(b + 1)
Var(X) = [bh^d'>]G(∖ - 2∣d)G(b + 2∣d)∣G(b + 1) - (B(X))2
from which the standard deviation and half the squared coefficient of variation can be derived. The percentiles are derived by
inverting the distribution function:
aip = ⅛(iM[√-ι∕b) -1](-ι∕4)
for each p = F(xp).
The Gini coefficient of inequality is given by
1 - Gini = [G(δ)G(2>+ 1∕d)]∕[G(2δ)G(δ+ 1Д]
The Lorenz curve ordinates L(p) at each p = F(xp) use the Beta cdf
L(p) 1 ibeta(δ+ 1Д 1 - 1∕d,√1/6))