Abstract
The generalized Marcum functions Qμ(x,y) and Pμ(x,y) have as particular cases the non-central χ2 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.
Highlights
Introduction and definitionsGeneralized Marcum functions are defined as Qμ(x, y) = x (1−μ) +∞ t (μ−1) e−t−xIμ−1 √ 2 xt dt, (1)y where μ > 0 and Iμ(z) is the modified Bessel function [10, 10.25.2]
The generalized Marcum Q-function is an important function used in radar detection and communications. They occur in statistics and probability theory, where they are called non-central chi-square or non central gamma cumulative distributions
The central chi-square or gamma cumulative distributions P (a, y) and Q(a, y) are a particular case of the non-central distributions with non-centrality parameter x equal to zero: P (a, y) = Pa(0, y), Q(a, y) = Pa(0, y). These are functions related to the incomplete gamma function ratios
Summary
The generalized Marcum Q-function is an important function used in radar detection and communications They occur in statistics and probability theory, where they are called non-central chi-square or non central gamma cumulative distributions (see [5] and references cited therein). The central chi-square or gamma cumulative distributions P (a, y) and Q(a, y) are a particular case of the non-central distributions with non-centrality parameter x equal to zero: P (a, y) = Pa(0, y), Q(a, y) = Pa(0, y) These are functions related to the incomplete gamma function ratios. Particularizing for x = 0 we obtain bounds for the central case; additional bounds for the central case are obtained from new monotonicity properties for the incomplete gamma function ratios (section 4).
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