Sharp bounds for cumulative distribution functions

TL;DR

This paper derives sharp bounds for noncentral gamma and beta distribution functions using integral ratio bounds related to L'Hôpital's rule, improving previous bounds in range and accuracy.

Contribution

It introduces new sharp bounds for noncentral gamma and beta distributions, expanding their applicability and precision compared to prior results.

Findings

Derived three types of bounds for noncentral gamma distributions using Bessel functions.

Established bounds for noncentral beta distributions with Kummer functions.

Improved the range and sharpness of existing distribution bounds.

Abstract

Ratios of integrals can be bounded in terms of ratios of integrands under certain monotonicity conditions. This result, related with L'H\^{o}pital's monotone rule, can be used to obtain sharp bounds for cumulative distribution functions. We consider the case of noncentral 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 noncentral 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.