发表机构
University of New Mexico（新墨西哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对三变量正态乘积分布的CDF计算，提出一种基于条件期望的降维算法，将三维积分化为二维求积，计算成本从$O(n^3)$降至$O(n^2)$，在六个参数设置下平均绝对误差为$3.1 \ imes 10^{-5}$，且计算时间不足一秒。
AI 中文摘要
在序贯中介模型中进行统计推断需要评估三个正态系数估计量乘积的累积分布函数（CDF），该操作涉及在边界为$xyz=v$的非凸区域上进行积分。虽然Delta方法提供了一阶近似的即时计算，但当两个或多个路径系数趋近于零时，其渐近方差会崩溃，导致在参数边界附近出现严重的覆盖不足。非参数自助法避免了梯度崩溃，但会产生$O(B \ imes N)$的计算成本，在大规模模拟研究或迭代功效分析中变得繁重。我们提出了一种基于模型的降维算法，在三变量高斯分布下，针对任意均值向量和正定协方差矩阵，解析地积分掉第三个变量。将$xy$平面划分为象限可隔离坐标轴处的符号变化，从而将问题简化为自适应二维求积，其固定网格成本为$O(n^2)$而非$O(n^3)$。在跨越六个参数设置的模拟基准测试中，该算法相对于$10^{8}$样本的蒙特卡洛参考实现了$3.1 \ imes 10^{-5}$的平均绝对误差，在一秒内完成分布评估，并产生经验覆盖率接近或高于名义水平的插入式分位数置信区间。
英文摘要
Statistical inference in sequential mediation models requires evaluating the cumulative distribution function (CDF) of the product of three normal coefficient estimators, an operation that entails integrating over a non-convex region with boundary $xyz=v$. While the Delta method provides an instantaneous first-order approximation, its asymptotic variance collapses whenever two or more path coefficients approach zero, producing severe undercoverage near parameter boundaries. Nonparametric bootstrapping avoids gradient collapse but incurs an $O(B \times N)$ computational cost that becomes burdensome in large-scale simulation studies or iterative power analyses. We propose a model-based dimension-reduction algorithm that integrates out the third variable analytically under the trivariate Gaussian distribution for arbitrary mean vectors and positive-definite covariance matrices. Partitioning the $xy$-plane into quadrants isolates the sign change at the coordinate axes, reducing the problem to an adaptive two-dimensional quadrature whose fixed-grid cost is $O(n^2)$ in place of $O(n^3)$. In simulation benchmarks across six parameter regimes, the algorithm achieves a mean absolute error of $3.1 \times 10^{-5}$ relative to a $10^{8}$-sample Monte Carlo reference, evaluating the distribution in under one second and yielding plug-in quantile confidence intervals whose empirical coverage was near or above nominal.
Comments30 pages, 1 figure, 3 tables. Submitted to Computational Statistics & Data Analysis. Code and simulation data: Mendeley Data, doi:10.17632/vdc44jyj5r.1
