×

Integral probability metrics and their generating classes of functions. (English) Zbl 0890.60011

Summary: We consider probability metrics of the following type: for a class \({\mathfrak F}\) of functions and probability measures \(P\), \(Q\) we define \(d_{\mathfrak F}(P, Q):= \sup_{f\in{\mathfrak F}}|\int fdP- \int fdQ|\). A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.

MSC:

60D05 Geometric probability and stochastic geometry
60E05 Probability distributions: general theory
60B10 Convergence of probability measures
PDFBibTeX XMLCite
Full Text: DOI