# "AUC" for a vector

This is almost definitely a case for "not knowing what to search for", but am interested in how to refer to the following:

For a set of $$n$$ reals (say non-negative), I'm interested in characterizing how "concentrated" the values are in the top few (a la Pareto $$80/20$$). This could be for instance for thinking about percent of total variance of top few principal components.

Is there a way to refer to this function? I'd want a single value, similar to AUROC in statistics.

Context: I'm interested in optimizing over this function as a tradeoff, similar to penalties on norms like $$L^1$$ or $$L^0$$.

• This may be relevant: en.wikipedia.org/wiki/Gini_coefficient#Definition Jul 2 at 0:14
• What is the percent of total variance? And what are principal components of a set? There are like eight other phrases I don't understand but maybe they are economics terms. Jul 2 at 0:17
• Thanks @TravisWillse , I think Gini coefficient works! Jul 2 at 0:19
• Cheers. In that case I'll write a short answer. Jul 2 at 0:58

One such metric is the Gini coefficient $$G$$, which quantifies how much a distribution of values deviates from perfect equality: Concisely, it is half of the relative mean absolute difference of a list of values, i.e., if the values are $$x_1, \ldots, x_n$$, we define $$\boxed{G = \frac{1}{n^2 \bar x}\sum_{1 \leq i < j \leq n} |x_i - x_j|} ,$$ where $$\bar x$$ is the mean of $$x_1, \ldots, x_n$$. Provided that all $$x_i$$ are nonnegative, the Gini quotient takes values between $$0$$ (perfect equality) and $$1$$ (perfect inequality). It is often used to quantify inequality in income or wealth.