AI 中文总结
研究逻辑正态积分及其导数的数值评估，挑选最优级数展开，提出精确数值评估算法，可控制近似误差，还给出逻辑正态随机变量前两阶矩的明确结果。
AI 中文摘要
逻辑正态积分出现在具有高斯随机效应的逻辑模型和广义线性混合模型的统计估计问题中。我们研究该积分及其导数的数值评估，并给出特定点的封闭形式评估和级数展开。有连续的可能级数展开，我们挑选出一个对数值评估最优的。基于最优级数提出精确数值评估算法，尾部有良好近似误差控制。作为应用，给出逻辑正态随机变量前两阶矩的明确结果。
英文摘要
The logistic-normal integral appears in problems of statistical estimation for logistic models with Gaussian random effects, and generalized linear mixed models. We study the numerical evaluation of this integral and of its derivatives, and give closed form evaluations at certain points and series expansions. There is a continuum of possible series expansions, and we single out one series expansion which is optimal for numerical evaluation. We propose an algorithm for a precise numerical evaluation, based on the optimal series, with good approximation error control in the tails. As an application we give explicit results for the first four moments of a logistic-normal random variable.
Comments19 pages, 3 figures. v2: Fixed typos, and expanded the numerical tests. Expanded and simplified the statement of Prop.5.1
