Appended is a summary of replies to my query about the Gini coefficient.
The original posting is first, followed by the responses.
Thanks to those who replied.
Regards,
Stuart
--
Dr Stuart G. Young, Analyst, Royal Bank of Scotland
Tel/Fax: 0131 523 4903/6518, E-mail: [log in to unmask]
-----Original Message-----
From: Young, Stuart
Sent: Tuesday, October 05, 1999 8:54 AM
To: [log in to unmask]
Subject: QUERY: Gini coefficient
I've been asked to forward this to the list. Replies to me please,
and
I'll post a summary of responses in due course.
-----Original Message-----
From: Sykes, Chris
Does anybody know a generalised equation of the Gini curve (aka
Lorenz
curve) for categorical association? Assuming the curve is symmetric
about the line y = 1 - x, then is there an equation of the curve
that can
be expressed in terms of only one parameter, namely the Gini?
The curve will have the following characteristics:
0 <= x <= 1
0 <= y <= 1
x >= y for all x,y on the curve
slope is monotonic increasing
for any point (x,y) through which the curve passes, it
will also pass through the point (1 - y, 1 - x)
{symmetry}
The Gini parameter is to be constrained between 0 (random
association)
and 1 (perfect association).
Thanks,
Chris Sykes
Risk Modelling Team
Retail Lending Systems
Royal Bank of Scotland
Replies:
------
By rotating the axes clockwise 45 deg about the point x=y=0.5, the curve is
seen to be a simple quadratic, say Y = CX^2-A in the rotated coordinates.
They map to the original coordinates as Y = y-x and X = x+y-1, and from the
boundary conditions A = C. So this gives
(y-x) = C(x+y-1)^2 - C
or
(x+y-1)^2 - (y-x)/C - 1 = 0
It's messy to expand further.
Tim Cole
------
I wondered if you had an answer to your query given that I don't recall
seeing a posting. If I have read your question correctly, you are trying
to relate the Gini Coefficient to the shape of the Lorenz Curve.
Assuming the Gini Coefficient = { (2 / y bar * N squared) * (Sum of i *
Yi) -
([N + 1]/N)
where i is the weight taken from any given units rank, i.e. that bracket
contains a weighted sum of all unit data points times their rank. (The
reference for this is given in Cowell (1977) Measuring Inequality, London,
Philip Allan).
In principle, the Gini Coefficient is the ratio of the area between the
Lorenz Curve and the line of total equality (y = x) and the area between the
Lorenz Curve and the line (the axis) of total inequality. Symmetrical or
not, there are very (infinitely?) many ways of putting a curve in between
these two lines.
The Gini Coefficient is quite simplistic. It doesn't tell you anything
about where the inequalities lie (i.e. whether nearer y = 0 or y = 1). In
studying income inequality this is a big deal, because most people are more
concerned about inequality at the bottom of the income scale than anywhere
else (if you had a tax policy that reduced inequality by decreasing it from
1/3rd of the way up only you would see an improvement in the Gini
Coefficient). I don't see how you could write a full equation for the
curve given this one parameter, as the shape may vary considerably. I also
don't see why the Gini coefficient has to be constrained between 0 and 1,
because by the above defintion it is between 0 and 1.
Very interested to know if you get a full answer. My brain got stuck on
Ginis and Lorenzes, and I reckon the final answer to your underlying problem
will be quite different.
Paul Hewson
The Royal Bank of Scotland plc is registered in Scotland No 90312. Registered Office: 36 St Andrew Square, Edinburgh EH2 2YB.
The Royal Bank of Scotland plc is regulated by IMRO, SFA and Personal Investment Authority.
This e-mail message is confidential and for use by the addressee only. If the message is received by anyone other than the addressee, please return the message to the sender by replying to it and then delete the message from your computer.
'Internet e-mails are not necessarily secure. The Royal Bank of Scotland plc does not accept responsibility for changes made to this message after it was sent.'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|