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.