{'fraction_non_alphanumeric': 0.043627450980392155, 'fraction_numerical': 0.016666666666666666, 'mean_word_length': 5.458860759493671, '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}}
Asymptotic distributions of two–sample rank statistics for continuous outcomes

We assume that there are two potentially infinite sampling sequences of units, sampled in a mutually independent way from Subpopulations 0 and 1, respectively, of an overall population. We define outcome random variables {Y0i} and {Y1i} for the ith unit sampled from Subpopulations 0 and 1, respectively. We will assume that Subpopulation i has a continuous common probability distribution for the Yij , with cumulative distribution function Fi(·) and probability density function fi(·). For h ∈ {0, 1} and i a positive integer, define Xhi = h, so that the ordinal X–variable indicates membership of the Subpopuation 1, rather than the Subpopulation 0. From these sequences, we may take finite samples (Subsample 0 and Subsample 1), containing units corresponding to the first N0 and N1 positive integers, respectively, and calculate sample statistics to estimate population parameters, to compare the two subpopulations. The population parameters that we aim to estimate here are Somers’ D, Harrell’s c, the Hodges–Lehmann percentile pairwise differences, and the differences between the subpopulation percentiles. All of these parameters can be estimated, with sample point estimates and confidence limits, using rank (or “nonparametric”) methods, and are discussed in Newson (2002)[7], or in Bonett and Price (2002)[1]. The subsequent sections will define each of these parameters, and their sample estimates, and discuss the asymptotic distributions of these sample estimates. All of these estimates are governed by versions of the the Central Limit Theorem, and are asymptotically Normally distributed as the smaller of the two subsample sizes becomes large, with asymptotic variances depending on the Fi(·) and the fi(·). Note that we will not necessarily assume that the two subpopulations are equally variable, although this was often done in the past, to avoid needing computers. (See, for example, Hodges and Lehmann (1963)[5] and Lehmann (1963)[6].)