The Shape of the Noncentral Chi-square Density
Statistics Theory
2011-06-28 v1 Classical Analysis and ODEs
Statistics Theory
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 lambda_nu>0 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.
Cite
@article{arxiv.1106.5241,
title = {The Shape of the Noncentral Chi-square Density},
author = {Yaming Yu},
journal= {arXiv preprint arXiv:1106.5241},
year = {2011}
}