The Shape of the Noncentral Chi-square Density
Abstract
A noncentral chi-square density is log-concave if the degree of freedom is nu>=2. We complement this known result by showing that, for each 0<nu<2, there exists lambdanu>0 such that the chi-square with nu degrees of freedom and noncentrality parameter lambda has a decreasing density if lambda <= lambdanu, and is bi-modal otherwise. The critical lambdanu is characterized by an equation involving a ratio of modified Bessel functions. When an interior mode exists we derive precise bounds on its location.
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