A general parametric Stein characterization

Abstract

We present a general characterization theorem for parametric probability distributions in terms of a differential operator akin to the socalled Stein operators from the literature on Stein's method.

Stein’s method for comparison of univariate distributions

“…Cancelling the first term in (59) and ensuring that all resulting assumptions are satisfied immediately leads to the following result.…”

“…Using • = 1, replacing x by X 2 and taking expectations gives (58). For (59), first fix f 1 ∈ F 1 and choose g = g h the corresponding solution of (61) with • = 1. By construction we can then take expectations and write…”

“…The power of Theorem 44 and of Stein's method in general lies in the freedom of choice on the r.h.s. of the identities : all functions f • , g • (where now • needs to be replaced by h, 1 or 2 according to which of (59) or (60) is used) can be chosen so as to optimise resulting bounds. We can even optimise the bounds over all suitable pairs (f, g).…”

“…In the sequel we will not use the full freedom of choice provided by Theorem 44, but rather focus on applications of identity (59) only. Indeed in this case much is known about g h and Dg h in case f 1 = 1 and X 1 is Gaussian (see [22]), Binomial (see [29]), Poisson (see [8]), Gamma (see [17,66]), etc.…”

“…Now let h be such that E|h(X)| < ∞ and E|h(W )| < ∞, and let g h = T −1 X (h − Eh[X])/Id; then g ∈ dom(∆ + , W, Id) ∩ dom(∆ + , X, Id). From (59) we obtain that Eh(W ) − Eh(X) = E g h (W + 1) T Bin(n,p) (Id)(W ) − T L(W ) (Id)(W ) .…”