A new class of composite indicators: The penalized power mean
提出一种基于惩罚幂均值的复合指标聚合方法，通过惩罚指标间异质性来减少排名波动，并用2019年人类发展指数数据验证其效果。
In this paper we propose a new aggregation method for constructing composite indicators based on a penalization of the power mean. The idea underlying this approach consists in multiplying the power mean by a factor that accounts for the horizontal heterogeneity among indicators while penalizing units with a larger heterogeneity. In line with the minimum loss of information principle, the penalization factor proposed is proven to be linked to the loss of information generated when the indicators are substituted with their power means. As a consequence, the aggregation approach gives rise to the class of penalized power means and the penalized Benefit of the Doubt aggregative approach. Including heterogeneity makes the aggregation approach more suitable for refined rankings. Interestingly, the penalized power mean of order one coincides with the Mazziotta Pareto Index. Some theoretical properties of the penalized power means are proven, thus supporting the Mazziotta Pareto index. An empirical analysis of the Human Development Index in 2019 is presented. Comparisons of the rankings induced by the penalized and non-penalized Benefit of the Doubt and power mean aggregation approaches are shown. There are three main findings: the penalized power means satisfy the properties characterizing weakly monotone aggregation functions; the penalization reduces ranking variations while differentiating units with close means; and the geometric mean provides composite indicators whose ranking is closest to those obtained with power means of different order.