First-Order Stein Equations for Absolutely Continuous Bivariate Distributions and Some Applications

Final Defense
First-Order Stein Equations for Absolutely Continuous Bivariate Distributions and Some Applications

by Lester Charles A. Umali
PhD Mathematics Candidate

Date: Saturday, 09 December 2023
Time: 10 am
Venue: Online

Advisers:
Richard B. Eden, PhD
Timothy Robin Y. Teng, PhD
Ateneo de Manila University

Panelists:
Emmanuel A. Cabral, PhD (Reader)
Ateneo de Manila University
Job A. Nable, PhD (Reader)
Ateneo de Manila University
Christian Paul O. Chan Shio, PhD
Ateneo de Manila University
Crisanto A. Dorado, PhD
University of the Philippines Los Baños
Michael B. Frondoza, PhD
Mindanao State University-Iligan Institute of Technology

A random variable X has a standard normal distribution if and only if E[f ′ (X)] = E[X f (X)] for any contin- uous 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 characterizations have already been developed, all of them second-order equations. In this dissertation, we first developed a new Stein characterization for the bivariate normal distribution. We then constructed a generalized Stein characterization for other absolutely continuous bivariate distributions and illustrated how this Stein characterization looks like for some known bivariate distributions. Unlike many existing multivariate versions in the literature, our generalized Stein characterization has the univariate Stein identities as a special case. We also extended these first-order Stein identities, albeit only as a necessary condition, for a general multivariate distribution with dimension d ≥ 3.

In the second part of this dissertation, we developed and discussed some applications of these Stein identi- ties. Specifically, we used the first-order Stein identities to extend Siegel’s covariance formula for the multi- variate normal distribution to a class of absolutely continuous bivariate distributions. We also constructed recursive formulas for the higher moments and mixed moments for various bivariate distributions. Most of these recursions were then used to derive closed forms of higher and mixed moments for their correspond- ing distributions, particularly Isserlis’ theorem on higher moments for the bivariate normal distribution. Fi- nally, we used our Stein identities to compute for some exotic moments that involve the normal, log-normal, gamma and log-gamma random variables.

Key Words: Stein identities, bivariate distributions, Siegel’s formula, Isserlis’ theorem, higher and mixed moments