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Re: st: Negative incomes and income components using -sgini-


From   Roger Newson <r.newson@imperial.ac.uk>
To   "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Subject   Re: st: Negative incomes and income components using -sgini-
Date   Thu, 28 Apr 2011 15:09:31 +0100

I too (as a simple biostatistician) am extremely puzzled by the idea of inputting negative incomes into a Gini coefficient.

After all, a Gini coefficient is defined by fantasizing that the population to which the Gini coefficient belongs participate in 2 lotteries. In the first, every individual buys one ticket. In the second, each individual buys a number of tickets proportional to that individual's income. The Gini coefficient is then the difference between 2 probabilities, namely the probability that the winner of the second lottery has a higher income than the winner of the first lottery and the probability that the winner of the first lottery has a higher income than the winner of the second lottery.

The first probability will always be at least at large as the second probability (at least if all incomes are non-negative and some incomes are positive), with both probabilities equal (to zero) if every individual in the population has the same income. And, if all incomes are zero, then there will be no winner of the second lottery (as no tickets will then be sold), and possibly no winner of the first lottery (unless the tickets for the first lottery are given away). However, I cannot see how anybody can have a negative probability of winning either lottery, even if some individuals have "negative incomes" because they are sinking deeper into debt and buying a negative number of tickets (because they are desparately selling stolen lottery tickets to make ends meet).

Perhaps the econometricians on the list might like to explain how sampling probabilities in a lottery can be negative. I look forward to seeing an explanation.

Best wishes

Roger


Roger B Newson BSc MSc DPhil
Lecturer in Medical Statistics
Respiratory Epidemiology and Public Health Group
National Heart and Lung Institute
Imperial College London
Royal Brompton Campus
Room 33, Emmanuel Kaye Building
1B Manresa Road
London SW3 6LR
UNITED KINGDOM
Tel: +44 (0)20 7352 8121 ext 3381
Fax: +44 (0)20 7351 8322
Email: r.newson@imperial.ac.uk
Web page: http://www.imperial.ac.uk/nhli/r.newson/
Departmental Web page:
http://www1.imperial.ac.uk/medicine/about/divisions/nhli/respiration/popgenetics/reph/

Opinions expressed are those of the author, not of the institution.

On 28/04/2011 13:55, David Coyne wrote:
Hello,

I've been attempting to decompose the Gini coefficient of a measure C,
where C=A-B. I have two questions related to this. First, if C has some
values that are negative, how does this affect the interpretation of the
Gini coefficient? Second, if B is nonnegative, (i.e. using -sgini- would
have me use C=A + (-B), so if (-B)<  0 for all observations) thus leaving
me with a negative contribution (s*g*r) to the Gini coefficient of C, does
that mean that B has an equalizing effect on C? Not sure of the
interpretation of this. Any insight and/or advice would be appreciated!

Thanks in advance,

--
David Coyne
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