Sharp bounds for cumulative distribution functions

Abstract

Ratios of integrals can be bounded in terms of ratios of integrands under certain monotonicity conditions. This result, related with L'Hôpital'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.

References in corpus (3)

• Connections between the Generalized Marcum Q-Function and a class of Hypergeometric Functions
• GammaCHI: a package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)
• Efficient algorithms for the inversion of the cumulative central beta distribution

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• A new type of sharp bounds for ratios of modified Bessel functions