Indices of Inequality, Diversity, and Concentration

Abstract

In connection with total time on test processes the cumulative total time on test transform and its empirical version has received considerable interest in probabilistic and statistical papers in reliability theory. Also, the Gini index and some related measures of economic inequality and industrial concentration have been in use for a long time. For example, Good (1982) has recently pointed out, referring to Keynes (1921, pp. 398–399) that the Gini index was used before Gini (1912) by Lexis (1879) as a “quadratic index of homogenity”. The literature on the economic applications of the latter indices is so vast that we could not even attempt to mention all the relevant references. Below we list some commonly used indices, together with some new possibilities, and only list some relevant references we know of in connection with these indices without discussing the contents of these references. We give results on strong consistency and asymptotic normality. Some of these results have been known, or must have been known, in some special cases, or in general as well, perhaps under more stringent regularity conditions. Our versions are obtained as very simple applications of the so far achieved results, and we are not going to compare them to earlier ones.