Generalized Distribution Functions: The Linearly Ordered Case with Applications to Nonparametric Statistics

Abstract

We develop a theory for distribution functions which are defined on an abstract linearly ordered space. This extends and gives additional insight into the work of J. H. B. Kemperman who was concerned with a special class of linearly ordered spaces. Attention is given to nonparametric applications. The utility of Kemperman's work on tolerance regions and of related applications to goodness of fit tests, appearing in this paper, is enhanced. In particular, it is shown that a number of standard nonparametric procedures can be extended to cover almost any kind of random sample (e.g., multivariate and most time series data) occurring in practice.