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Sharp bounds for cumulative distribution functions

Abstract: Ratios of integrals can be bounded in terms of ratios of integrands under certain mono- tonicity conditions. This result, related with L?H?opital?s monotone rule, can be used to obtain sharp bounds for cumulative distribution functions. We consider the case of non- central cumulative gamma and beta distributions. Three different types of sharp bounds for the noncentral gamma distributions (also called Marcum functions) are obtained in terms of modified Bessel functions and one additional type of function: a second modified Bessel function, two error functions or one incomplete gamma function. For the noncen- tral beta case the bounds are expressed in terms of Kummer functions and one additional Kummer function or an incomplete beta function. These bounds improve previous results with respect to their range of application and/or its sharpness.

 Fuente: Journal of Mathematical Analysis and Applications, Volume 436, Issue 2, 15 April 2016, Pages 748-763

 Editorial: Academic Press Inc.

 Fecha de publicación: 15/04/2016

 Nº de páginas: 15

 Tipo de publicación: Artículo de Revista

 DOI: 10.1016/j.jmaa.2015.12.024

 ISSN: 0022-247X,1096-0813

 Proyecto español: MTM2012-34787

 Url de la publicación: https://doi.org/10.1016/j.jmaa.2015.12.024