Apr 24

3:30 pm

## Gini Association and the Pseudo-Lorenz Curve

### Somesh Das Gupta

Seminar

Indian Statistical Institute

We were motivated by the problem of assessing the influence on the inequality in income by the corresponding inequality in some other related variable (say, the number of years of formal education completed). More generally, consider the pseudo-Lorenz curve of a nonnegative r.v. Y relative to (i.e., with respect to the ordering of) another related nonnegative r.v. X. It is shown that this pseudo-Lorenz curve L(Y/X) always lies above the Lorenz curve L(Y) of Y. A measure of association between Y and X, called the Gini Association, is proposed as the ratio of the Lorenz area of L(Y/X) to the Lorenz area of L(X). It is shown that this ratio lies between -1 and +1, and it measures the "degree" of monotonic relationship between Y and X. A measure, termed as the Gini Correlation Ratio, is introduced to measure the degree of monotonicity of E(Y/X = x) with respect to x. Lastly, the Gini index of Y is split into "between" and "within" X-arrays, and these components are studied for relevant interpretations.

Note: The Lorenz curve of a nonnegative random variable Y with positive mean mu is defined as

E[(Y/mu)I(Y < F^{-1}(p))], 0 < p < 1,

where I denotes the indicator function and F is the cdf of Y. The corresponding Lorenz area is

1/2 - area enclose by the Lorenz curve and the [0,1] interval on the p-axis.