Descriptive concentration between two multivariate distributions and extension to pairs of normalized measures (Q1099542)

Let \(X_ A\) and \(X_ B\) be two random variables and A(x) and B(x) be their distribution functions, respectively, \(x\in {\mathbb{R}}^ n\). For each Borel subset J of \({\mathbb{R}}^ n\) (J\(\in {\mathcal B}^ n)\), define \(A=A[J]=\int_{J}dA(x)\) and \(B=B[J]=\int_{J}dB(x)\). The set of pairs \((A,B)\in [0,1]^ 2\) defined by \[ C^*=\{(A,B)=(A(J),B(J)): J\in {\mathcal B}^ n\}\quad \subset \quad [0,1]^ 2 \] describes the local association (on the same J) of the measures A[\(\cdot]\) and B[\(\cdot]\). The set \(C^*\) is closed but, in general, not convex. Properties of \(C^*\), its closed convex hull C, their boundaries, etc., are studied. The theory above allows the study in a unified way of many particular cases already considered in the literature (e.g., the Neyman-Pearson lemma.