论文标题
广义Gini指数的鲁棒性
The robustness of the generalized Gini index
论文作者
论文摘要
在本文中，我们介绍了一个地图$φ$，我们将其称为\ textit {Zonoid Map}，从所有非阴性，有限的Borel措施的空间$ \ Mathbb {r}^n $带有有限的第一瞬间到$ \ m i \ thebb {r}^n $的Zonoids的空间。 This map, connecting Borel measure theory with zonoids theory, allows us to slightly generalize the Gini volume introduced, in the contest of Industrial Economics, by Dosi, Grazzi, Marengo and second author in 2016. This volume, based on the geometric notion of zonoid, is introduced as a measure of heterogeneity among firms in an industry and turned out to be quite interesting index as it is a multi-dimensional generalization of the well known并广泛使用的Gini指数。通过利用我们定义提供的数学竞赛，我们证明了地图$φ$的连续性，该$ $φ$又允许证明Glivenko-Cantelli定理的有效性，用于我们的广义Gini索引，因此对于Gini卷而言。在处理大量多维数据时，这两种结果均为$φ$和Glivenko-Cantelli定理，尤其有用。
In this paper we introduce a map $Φ$, which we call \textit{zonoid map}, from the space of all non-negative, finite Borel measures on $\mathbb{R}^n$ with finite first moment to the space of zonoids of $\mathbb{R}^n$. This map, connecting Borel measure theory with zonoids theory, allows us to slightly generalize the Gini volume introduced, in the contest of Industrial Economics, by Dosi, Grazzi, Marengo and second author in 2016. This volume, based on the geometric notion of zonoid, is introduced as a measure of heterogeneity among firms in an industry and turned out to be quite interesting index as it is a multi-dimensional generalization of the well known and broadly used Gini index. By exploiting the mathematical contest offered by our definition, we prove the continuity of the map $Φ$ which, in turns, allows to prove the validity of a Glivenko-Cantelli theorem for our generalized Gini index and, hence, for the Gini volume. Both results, continuity of $Φ$ and Glivenko-Cantelli theorem, are particularly useful when dealing with a huge amount of multi-dimensional data.
