{smcl}
{* *! version 1.1.0 10january2012}{...}
{hline}
help for {hi:gb2dist}{right:Michal Brzezinski (January 2012)}

{hline}

{title:Title}

{p2colset 5 16 20 2}{...}
{p2col: {hi:gb2dist} {hline 2}}Compute poverty and inequality indices (including the Gini index) for the Generalized Beta distribution of the second kind (GB2){p_end}
{p2colreset}{...}


{title:Syntax}

{p 4 4 2}
{cmd:gb2dist}, {cmd:pline(}{it:#}{cmd:)}

{title:Description}

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

{title:Options}

{synoptset}
{synopthdr}
{synoptline}
{synopt:{cmd:pline(}{it:#}{cmd:)}}specifies the value of the poverty line.

{title:Remarks}

{p 4 4 2}
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*3{it:F}2[1,p+q,2*p+1/a;p+1,2*(p+q);1]
	       -1/(p+1/a)*3{it:F}2[1,p+q,2*p+1/a;p+1/a+1,2*(p+q);1]},

{p 4 4 2}
where B(u,v) = G(u)*G(v)/G(u+v) is the beta function, G(u)=exp({help lngamma}(u)) is the gamma function, and 3{it:F}2(.;1) is a generalized hypergeometric function defined as
	
	3{it:F}2[a1,a2,a3;b1,b2;1] = SUM(((a1)i*(a2)i*(a3)i*1^i)/(((b1)i*(b2)i)*1/i!)), 

{p 4 4 2}
where (a)i is defined as exp({cmd:lngamma}(a+i))/exp({cmd:lngamma}(a)).

{p 4 4 2}
{cmd:gb2dist} computes function 3{it:F}2(a1,a2,a3;b1,b2;1) using 3-term recursive algorithm provided in Wimp (1981).

{p 4 4 2}
GE indices are computed according to methods developed by Jenkins (2009).

{p 4 4 2}
FGT poverty measures are computed using formulas developed by Hajargsht et al. (2011).

{title:Examples}

{phang2}{cmd:. sysuse nlsw88,clear}{p_end}
{phang2}{cmd:. gb2fit wage}{p_end}
{phang2}{cmd:. gb2dist, pline(4)}{p_end}


{title:References}

{phang}
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. 

{phang}
Jenkins, S.P. (2009). Distributionally-sensitive inequality indices 
and the GB2 income distribution. {it:Review of Income and Wealth}, {cmd: 55} (2), 
392-398, {browse "http://dx.doi.org/10.1111/j.1475-4991.2009.00318.x":DOI: 10.1111/j.1475-4991.2009.00318.x}.

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

{phang}
Wimp, J. (1981). The computation of 3{it:F}2(1). 
{it:International Journal of Computer Mathematics}, {cmd: 10} (1), 55-62, 
{browse "http://dx.doi.org/10.1080/00207168108803266":DOI: 10.1080/00207168108803266}.


{title:Also see}

{p 4 13 2}
Online: help for {help gb2fit}, {help gbgfit}, {help gb2pred}, {help qgb2}, {help pgb2}, {help ineqdeco}, {help svygei} if installed. 


{title:Author}

{p 4 4 2}
Michal Brzezinski <mbrzezinski@wne.uw.edu.pl>, Faculty of Economic Sciences, University of Warsaw, Poland.