AI 中文总结
研究非中心伽马随机变量和与差的密度函数的对数凸性与对数凹性，给出相关完整分类及准则，还推导出特殊情况下正态、方差-伽马及麦凯I型分布密度的对数凸凹性分类。
AI 中文摘要
我们研究了由两个独立的非中心伽马随机变量的和与差得到的密度函数的对数凸性和对数凹性。给出了非中心伽马差的单边对数凸性的完整分类，两个独立中心伽马随机变量和的完整对数凸性分类，以及中心差和共同尺度和的尖锐对数凹性准则。作为特殊情况，推导出了具有任意均值和方差的两个相关正态随机变量乘积密度的对数凸性分类，以及方差-伽马分布和麦凯I型分布密度的对数凸性和对数凹性分类。
英文摘要
We study log-convexity and log-concavity of densities obtained from sums and differences of two independent noncentral gamma random variables. We give a complete classification of one-sided log-convexity for noncentral gamma differences, a complete log-convexity classification for sums of two independent central gamma random variables, and sharp log-concavity criteria for central differences and for common-scale sums. As special cases, we deduce a log-convexity classification for the density of the product of two correlated normal random variables with arbitrary means and variances, and log-convexity and log-concavity classifications for the densities of the variance-gamma and McKay Type I distributions.
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