A first-order Stein characterization for absolutely continuous bivariate distributions

Abstract

A random variable X has a standard normal distribution if and only if $E[f^{\prime }(X)]=E[Xf(X)]$ 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.

KEYWORDS:

• Bivariate normal
• bivariate distributions
• Stein characterization

MSC2020 Subject Classification:

• 60E05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

LCA Umali is grateful for the support of the Philippines’ Commission on Higher Education (CHED) K-12 Transition Program Scholarship in the conduct of this research.