{'fraction_non_alphanumeric': 0.023121387283236993, 'fraction_numerical': 0.008670520231213872, 'mean_word_length': 5.221556886227545, '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}}
Log-concavity and Inequalities for Chi-square, F and Beta Distributions with Applications in Multiple Comparisons

In several recent papers log-concavity results and related inequalities for a variety of distributions were obtained. This work is supposed to derive a nearly complete list of corresponding properties concerning the cdf's and some related functions for Beta as well as for central and non-central Chi-square and F distributions, where hitherto only partial results were available. To this end we introduce a generalized reproductive property, thereby extending the relationships between total positivity of order 2, log-concavity and reproductivity developed in Das Gupta and Sarkar (1984). The key to our results are log-concavity properties of the non-central Chi-square distribution with zero degrees of freedom introduced by Siegel (1979). Finally one of the results for the central F distribution is used to solve a monotonicity problem for a stepwise multiple F-test procedure for all pairwise comparisons of k means.