Title: Stein method, algebra and statistics
Authors: Ehsan Azmoodeh - University of Liverpool (United Kingdom) [presenting]
Abstract: Let $d \ge 1$. Consider target probability distributions of the form $Y=h(N_1,...,N_d)$ where here $(N_1,...,N_d)$ is a $d$-dimensional standard Gaussian vector and, $h$ is a polynomial in $d$ variables. We introduce the novel notion of an algebraic polynomial Stein operator and show that in dimension $d=1$, any polynomial Stein operator is, in fact, algebraic. Furthermore, we discuss -- from a non-commutative algebra viewpoint -- in details, the class of polynomial Stein operator associated with the standard Gaussian distribution, denoted by $PSO(N)$. $N$ stands for the one-dimensional standard Gaussian random variable. We show, among many other findings, that $PSO(N)$ is a principal right-ideal of the first Weyl algebra generated by the so-called divergence operator. We will discuss applications in mathematical statistics.