A First-Order Stein Characterization for Absolutely Continuous Bivariate Distributions

• Main Authors: Umali, Lester Charles A., Eden, Richard B., Teng, Timothy Robin Y.
• Format: text
• Published: Archīum Ateneo 2023
• Subjects: bivariate distributions; Bivariate normal; Stein characterization; Physical Sciences and Mathematics; Statistics and Probability
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/244; https://doi.org/10.1080/03610926.2023.2250485
• Institution: Ateneo De Manila University

A random variable X has a standard normal distribution if and only if (Formula presented.) for any continuous and piecewise continuously differentiable function f such that the expectations exist. This first-order characterizing equation, called the Stein identity, has been extended to other univariate distributions. For the multivariate normal distribution, a number of Stein identities have already been developed, all of them second order equations. In this study, we developed a new Stein characterization for the bivariate normal distribution. Unlike many existing multivariate versions in the literature, ours is a system of first-order equations which has the univariate Stein identity as a special case. We also constructed a generalized Stein characterization for other absolutely continuous bivariate distributions. Finally, we illustrated how this Stein characterization looks like for some known absolutely continuous bivariate distributions.