Müller, Alfred Integral probability metrics and their generating classes of functions. (English) Zbl 0890.60011 Adv. Appl. Probab. 29, No. 2, 429-443 (1997). 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. Cited in 1 ReviewCited in 53 Documents MSC: 60D05 Geometric probability and stochastic geometry 60E05 Probability distributions: general theory 60B10 Convergence of probability measures Keywords:integral probability metrics; maximal generator; uniformity in weak convergence; stop-loss metric PDFBibTeX XMLCite \textit{A. Müller}, Adv. Appl. Probab. 29, No. 2, 429--443 (1997; Zbl 0890.60011) Full Text: DOI