{'fraction_non_alphanumeric': 0.03341584158415842, 'fraction_numerical': 0.0, 'mean_word_length': 5.082706766917293, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
G-Ordered Functions, with Applications in Statistics. I. Theory.

Abstract : This paper develops the theory of functions isotonic with respect to the more general ordering, and presents applications of this theory in statistics. Using the theory of reflection groups, reflection ordering (a generalization of functions decreasing in transposition) is defined. Reflection ordering is closely related to G-majorization (a point x G-majorizes a point y if y is an element of the convex hull of the G-orbit of x) and G-ordered functions contain G-monotone functions as special cases (G-monotone increasing functions preserve the G-majorization ordering). Many preservation properties are developed for G-ordered functions and a preservation theorem is proved for G-monotone functions under an integral transform.