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，具有局部变化度量尺度的空间（例如维度缩放方式不同）的多维结构相当普遍。然而，如何恰当利用其中编码的全部信息仍被视为一个开放性问题。我们通过考虑基于映射到这类非标准空间的随机变量期望（的集合）的序关系来解决这一问题。当不存在或存在完全基数结构时，该序关系分别以随机占优和期望序作为极端情形。我们针对提出的广义随机占优（GSD）序推导了一种（正则化）统计检验，通过线性优化使其可操作化，并借助不精确概率模型增强其稳健性。我们的发现通过多维贫困测量、金融和医学领域的数据进行了验证。