seminario

[Pietro Muliere <pietro.muliere@uni-bocconi.it>]

Istituto Metodi Quantitativi - Università L. Bocconi
Viale Isonzo, 25 - 20135 Milano
Tel. 02-58365629  - Fax 02-58365630

SEMINARIO



Gini Association ad Pseudo Lorenz
  Curve


Somesh Das Gupta

Indian Statistical Institute
Calcutta, India

giovedì, 10 maggio 2001 - ore 16.00
aula IMQ - stanza 137

_______________________________________________________________________________________________

Abstract:
It is shown that the Lorenz curve  Ly  of a nonnegative random variable Y 
lies below the pseudo Lorenz curve   Ly/  of Y relative to another 
nonnegative random variable X and  Lg()    lies above  L   for any 
nonnegative function g(X), where  m(x)  is the conditional expectation of Y 
given  X=x.  Blitz and Brittain  (Metron, 1964) considered the 
ratio   C(Y;X)  of the Lorenz area for L     to the Lorenz area for L  as a 
measure of association between Y and X. It is proved that this ratio 
C(Y;X), termed here as the Gini association index, lies between -1 and +1, 
and it equals +1 (or, -1)  if and only if Y is a monotonic increasing (or, 
decreasing) function of X almost surely. Two other measures of association, 
termed as Gini correlation ratio and Gini. regression index, are proposed, 
and their properties are studied. Lastly, the Lorenz area corresponding to 
the distribution of Y is split into  “ within” and  “between” X-arrays. It 
is shown that a weighted average of the Lorenz curve for the conditional 
distribution of Y given X lies above the Lorenz curve for the unconditional 
distribution of Y.