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 <i>bivariate normal distribution</i> (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 <i>normalized expectation–maximization</i> (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 <i>Monte Carlo N-EM</i> (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.