A unified framework for estimation of truncated bivariate normal distribution with non-regular domains: applications in medicometrics

Truncation is a core issue in the multivariate statistics and distribution theory. Concurrently, the bivariate normal distribution (BND) holds critical significance in R2. Despite the pivotal importance of truncated BNDs in biomedical and environmental sciences, the challenge of parameter estimation for this distribution on non-regular truncated domains, including rectangle, remains inadequately tackled. This paper introduces a novel normalized expectation–maximization (N-EM) algorithm to address this issue, which can be achieved by innovatively partitioning R2 and providing closed-form expressions for both the first- and second-order central moments of one–sided truncated distributions of four kinds. Furthermore, we expand the rectangle truncated domain to encompass parallelograms and even non-regular truncated domains, presenting an embedding Monte Carlo N-EM (MCN-EM) algorithm for the estimation in non-regular truncated domains. Our N-EM algorithm surpasses existing methods, solving complex scenarios with proven stability in simulations. Finally, the application of paired medical indicator data for serum protein and albumin provides valuable information for regional health monitoring.

截断问题是多元统计与分布理论中的核心研究议题。与此同时，二维正态分布（bivariate normal distribution, BND）在二维欧氏空间R²中具有关键的理论与应用价值。尽管截断二维正态分布在生物医学与环境科学领域具有举足轻重的地位，但针对矩形等非规则截断域下该分布的参数估计难题，至今仍未得到充分解决。本文提出一种全新的归一化期望最大化（normalized expectation–maximization, N-EM）算法以解决该问题：通过创新性地划分二维欧氏空间R²，并推导得到四类单侧截断分布的一阶与二阶中心矩闭式表达式。进一步地，本文将矩形截断域拓展至平行四边形乃至一般非规则截断域，并提出嵌入蒙特卡洛归一化期望最大化（embedding Monte Carlo N-EM, MCN-EM）算法以实现非规则截断域下的参数估计。所提出的N-EM算法性能优于现有方法，可有效处理复杂估计场景，并在仿真实验中验证了其稳定性与有效性。最后，本文将所提方法应用于血清蛋白与白蛋白的配对医学指标数据集，为区域公共卫生监测提供了有价值的分析参考。