Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to exploit the entire information encoded in them properly. We address this problem by considering an order based on (sets of) expectations of random variables mapping into such non-standard spaces. This order contains stochastic dominance and expectation order as extreme cases when no, or respectively perfect, cardinal structure is given. We derive a (regularized) statistical test for our proposed generalized stochastic dominance (GSD) order, operationalize it by linear optimization, and robustify it by imprecise probability models. Our findings are illustrated with data from multidimensional poverty measurement, finance, and medicine.

翻译：在统计学和机器学习中，具有局部尺度变化的空间（例如维度尺度不同的多维结构）相当常见。尽管如此，如何恰当利用其中编码的全部信息仍被视为一个开放性难题。我们通过考虑映射到此类非标准空间的随机变量的（集合）期望序来应对这一问题。当不存在或存在完整的基数结构时，该序分别以随机占优和期望序作为极端情形。我们推导出针对所提出的广义随机占优（GSD）序的（正则化）统计检验，通过线性优化使其可操作化，并利用不精确概率模型增强其稳健性。我们的研究结果通过多维贫困度量、金融和医学领域的数据进行了展示。