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 to L'Hopital'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.

[1]  Javier Segura.  Monotonicity properties and bounds for the chi-square and gamma distributions , 2014, Appl. Math. Comput..

[2]  Árpád Baricz,et al.  Tight bounds for the generalized Marcum Q-function , 2009 .

[3]  George K. Karagiannidis,et al.  On the Monotonicity of the Generalized Marcum and Nuttall ${Q}$ -Functions , 2007, IEEE Transactions on Information Theory.

[4]  T. H. Gronwall.  An Inequality for the Bessel Functions of the First Kind with Imaginary Argument , 1932 .

[5]  Nico M. Temme,et al.  Algorithm 939 , 2014, ACM Trans. Math. Softw..

[6]  N. M. Temme,et al.  The Asymptotic and Numerical Inversion of the Marcum Q‐Function , 2014, 1404.0302.

[7]  Iosif Pinelis,et al.  L'Hospital Rules for Monotonicity and the Wilker-Anglesio Inequality , 2004, Am. Math. Mon..

[8]  Javier Segura,et al.  Bounds for ratios of modified Bessel functions and associated Turán-type inequalities , 2011 .

[9]  G. Shimura,et al.  Confluent hypergeometric functions on tube domains , 1982 .

[10]  Shidong Zhou,et al.  On the Monotonicity, Log-Concavity, and Tight Bounds of the Generalized Marcum and Nuttall $Q$-Functions , 2010, IEEE Transactions on Information Theory.

[11]  Jeff Cheeger,et al.  Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds , 1982 .

[12]  Javier Segura,et al.  The Schwarzian-Newton method for solving nonlinear equations, with applications , 2015, Math. Comput..

[13]  Matti Vuorinen,et al.  Monotonicity Rules in Calculus , 2006, Am. Math. Mon..

[14]  Árpád Baricz,et al.  Geometrically concave univariate distributions , 2010 .

[15]  Matti Vuorinen,et al.  Asymptotic expansions and in-equalities for hypergeometric functions , 1997 .

[16]  Suvrit Sra,et al.  The multivariate Watson distribution: Maximum-likelihood estimation and other aspects , 2011, J. Multivar. Anal..

[17]  José F. Paris,et al.  Connections Between the Generalized Marcum $Q$ -Function and a Class of Hypergeometric Functions , 2013, IEEE Transactions on Information Theory.

[18]  P. Cerone.  Special functions: approximations and bounds , 2007 .

[19]  Matti Vuorinen,et al.  Inequalities for quasiconformal mappings in space , 1993 .

[20]  D. Karp,et al.  Log-concavity for series in reciprocal gamma functions and applications , 2012, 1206.4814.

[21]  Nico M. Temme,et al.  GammaCHI: A package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral) , 2015, Comput. Phys. Commun..

[22]  Lokenath Debnath,et al.  A generalization of L'Hôspital-type rules for monotonicity and its application , 2009, Appl. Math. Lett..

[23]  Richard B. Paris,et al.  Incomplete gamma and related functions , 2010, NIST Handbook of Mathematical Functions.

[24]  A. Gray.  Bessel Functions , 1899, Nature.

[25]  Nico M. Temme,et al.  Efficient algorithms for the inversion of the cumulative central beta distribution , 2016, Numerical Algorithms.

[26]  R. Chattamvelli.  A Note on the Noncentral Beta Distribution Function , 1995 .