{'fraction_non_alphanumeric': 0.023148148148148147, 'fraction_numerical': 0.007716049382716049, 'mean_word_length': 4.743362831858407, '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 expansions of the distributions of some test statistics in generalized linear models

The asymptotic expansions of the distributions of three statistics, namely, the likelihood ratio, Wald and Rao's score statistics, are obtained for testing a subset of parameters in the generalized linear models. The asymptotic expansions are given in closed forms up to n −1 in the null case and up to n −1/2 in the nonnull case under a sequence of Pitman alternatives. It is found that the power of Wald and Rao's score tests are same up to n −1/2. Numerical examples are presented to illustrate the performance of the asymptotic null distributions.