The paper considers the distribution of a general linear combination of central and non-central chi-square random variables by exploring the branch cut regions that appear in the standard Laplace inversion process. Due to the original interest from the directional statistics, the focus of this paper is on the density function of such distributions and not on their cumulative distribution function. In fact, our results confirm that the latter is a special case of the former. Our approach provides new insight by generating alternative characterizations of the probability density function in terms of a finite number of feasible univariate integrals. In particular, the central cases seem to allow an interesting representation in terms of the branch cuts, while general degrees of freedom and non-centrality can be easily adopted using recursive differentiation. Numerical results confirm that the proposed approach works well while more transparency and therefore easier control in the accuracy is ensured.

翻译：本文通过探索标准拉普拉斯反演过程中出现的支割线区域，研究了中心与非中心卡方随机变量一般线性组合的分布问题。由于最初源于方向统计学的兴趣，本文重点聚焦于此类分布的密度函数而非累积分布函数。事实上，我们的结果证实后者是前者的特例。该方法通过有限个可行单变量积分生成概率密度函数的替代表征，提供了新的理论视角。特别地，中心情形在支割线框架下呈现出富有启发性的表示形式，而一般自由度和非中心参数则可通过递归微分轻松纳入。数值结果证实了所提方法的有效性，同时确保了更高透明度与更易把控的计算精度。