The Shape of the Noncentral Chi-square Density

Details The Shape of the Noncentral Chi-square Density The Shape of the Noncentral Chi-square Density

2011-06-26

arxiv.org/abs/1106.5241v1

Mathematics

Statistics Theory

Scientific paper

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 00 such that the chi-square with nu degrees of freedom and noncentrality parameter lambda has a decreasing density if lambda <= lambda_nu, and is bi-modal otherwise. The critical lambda_nu 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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