A FORMULA OF GINI'S CONCENTRATION RATIO

AND ITS APPLICATION TO LIFE TABLES

Kyo Hanada*

Gini's concentration ratio G is formulated in terms of a cumulative distribu-

tion function. The formula is quite simple and makes it easier to calculate

values, as well as to investigate the usefulness of the ratio. In addition, a kind

of Moment Problem is raised concerning Drewnowski's equality index D which

is defined as D=1-G. An application of the formula to life tables showed that

the index worked effectively as to measure equality of length of life. Numerical

quadrature is briefly discussed for an approximation of the calculation in the use

of life tables.

1. Introduction

The Lorentz curve is widely used in econometrics to express income distribu-

tion in terms of difference between the cumulative amount of income and that of

population. The Gini's concentration ratio which is derived from the Lorentz

curve is the index of inequality in a distribution. General explanation of the curve

and the ratio is written concisely by Kendall and Stuart [3].

Recently, equality in distributions of health related variables is becoming main

concern and there is the need for investigation of the applicability of various in-

dices. Drewnowski [1] proposed to apply the Gini's concentration ratio to measure

inequality existing in the distribution of health indicators, including health resources,

manpower and facilities. Preston et al. [4] applied the ratio to mortality differentials

among social classes. Uemura [6] proposed to apply the ratio to life tables and

expressed the need for further investigation of utilities of the ratio.

2. Formula of Gini's concentration ratio

The Lorentz curve Y=L(X) has the following equations with a parameter t,

where,  f(x)is  a density  function  defined  on  x〓0,  and  μ is the  expectation.

The  Gini's  concentration  ratio  G  is  a  ratio  of  the  area  surrounded by Y=

X and Y=L(X)to the area of the triangular Y=X, Y=O and X=1:

Substituting X and L(X) to the above equation, and using partial integration

concerning,

Received October 12, 1982. Revised April 22, 1983.

* The Medical Information System Development Center .

J.Japan Statist.Soc.

Vol.13 No.2 1983 95‑98