*! version 1.1.0 10january2012

help for gb2distMichal Brzezinski (January 2012)

Title

gb2dist -- Compute poverty and inequality indices (including the Gini index) for the Generalized Beta distribution of the second kind (GB2)

Syntax

gb2dist, pline(#)

Description

gb2dist computes poverty and inequality indices and their standard errors for the 4 parameter Generalized Beta distribution of the second kind (GB2). gb2dist is usually used just after
GB2 has been fitted to individual data using gb2fit
or after GB2 has been fitted to group data using gbgfit. Inequality indices computed include the Gini index and the Generalized Entropy (GE)
indices with parameter alpha = -1, 0, 1 and 2. For more information on the GE indices, see svygei. Poverty indices computed belong to the Foster-Greer-Thorbecke (FGT) family with
poverty aversion parameters alphaa = 0, 1 and 2. See povdeco, for mor information on the FGT measures. Standard errors are computed using delta method. They are reliable only
for estimates from gb2fit.

Options

pline(#) specifies the value of the poverty line.

Remarks

The Gini coefficient for the GB2 distribution was derived in MacDonald (1984) and takes the form

Gini = 2*B(2*p+1/a,2*q-1/a)/(p*B(p,q)*B(p+1/a,q-1/a))*{1/p*3F2[1,p+q,2*p+1/a;p+1,2*(p+q);1]
-1/(p+1/a)*3F2[1,p+q,2*p+1/a;p+1/a+1,2*(p+q);1]},

where B(u,v) = G(u)*G(v)/G(u+v) is the beta function, G(u)=exp(lngamma(u)) is the gamma function, and 3F2(.;1) is a generalized hypergeometric function defined as

3F2[a1,a2,a3;b1,b2;1] = SUM(((a1)i*(a2)i*(a3)i*1^i)/(((b1)i*(b2)i)*1/i!)),

where (a)i is defined as exp(lngamma(a+i))/exp(lngamma(a)).

gb2dist computes function 3F2(a1,a2,a3;b1,b2;1) using 3-term recursive algorithm provided in Wimp (1981).

GE indices are computed according to methods developed by Jenkins (2009).

FGT poverty measures are computed using formulas developed by Hajargsht et al. (2011).

Examples

. sysuse nlsw88,clear

. gb2fit wage

. gb2dist, pline(4)

References

Hajargsht, G., Griffiths, W.E., Brice, J., Rao, D.S.P., Chotikapanich, D. (2011).
GMM Estimation of Income Distributions from Grouped Data, University of Melbourne, Department of Economics,
Research Paper Number 1129.

Jenkins, S.P. (2009). Distributionally-sensitive inequality indices
and the GB2 income distribution. Review of Income and Wealth, 55 (2),
392-398, DOI: 10.1111/j.1475-4991.2009.00318.x.

McDonald, J.B. (1984). Some generalized functions for the size
distribution of income. Econometrica, 52 (3), 647-663,
http://www.jstor.org/stable/1913469.

Wimp, J. (1981). The computation of 3F2(1).
International Journal of Computer Mathematics, 10 (1), 55-62,
DOI: 10.1080/00207168108803266.

Also see

Online: help for gb2fit, gbgfit, gb2pred, qgb2, pgb2, ineqdeco, svygei if installed.

Author

Michal Brzezinski <mbrzezinski@wne.uw.edu.pl>, Faculty of Economic Sciences, University of Warsaw, Poland.