Monotonicity properties and bounds for the chi-square and gamma distributions

TL;DR

This paper investigates the monotonicity and convexity of generalized Marcum functions related to non-central chi-square and gamma distributions, deriving sharper bounds and convergent sequences to improve existing inequalities.

Contribution

It provides new monotonicity properties and tighter bounds for Marcum functions and related distributions, enhancing previous estimation methods.

Findings

Derived sharper upper and lower bounds for Marcum functions.

Established monotonicity and convexity properties for generalized Marcum functions.

Developed convergent sequences of bounds that improve existing inequalities.

Abstract

The generalized Marcum functions $Q_{\mu}(x,y)$ and $P_{\mu}(x,y)$ have as particular cases the non-central $\chi^2$ 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 exiting inequalities.