A First-Order Stein Characterization for Absolutely Continuous Bivariate Distributions

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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.