Monotonicity properties and bounds for the chi-squareand gamma distributions.

Abstract

The generalized Marcum functions Qlðx; yÞ and Plðx; yÞ have as particular cases the noncentral v2 and gamma cumulative distributions, which become central distributions (incomplete gamma function ratios) when the non-centrality parameter x is set to zero. We analyze monotonicity and convexity properties for the generalized Marcum functions and for ratios of Marcum functions of consecutive parameters (differing in one unity) and we obtain upper and lower bounds for the Marcum functions. These bounds are proven to be sharper than previous estimations for a wide range of the parameters. Additionally we show how to build convergent sequences of upper and lower bounds. The particularization to incomplete gamma functions, together with some additional bounds obtained for this particular case, lead to combined bounds which improve previously existing inequalities