Representations of solutions to Stein equations and their derivatives

Abstract

We propose pointwise representation of the Stein solution to a first-order Stein equation associated with an univariate absolutely continuous distribution. The representation extends to all the derivatives of the solution and with respect to any derivative of the test function $h$. The mechanism behind these representations is an array of kernels that observe two key identities, yielding a specific calculus for the Stein framework considered here. The first two derivatives of the Stein solution are expressed without any assumption between the target density $p$ and the non-vanishing weight $w$, under classical conditions. The correction terms appearing in the representations explains how choosing the Stein kernel as weight have a special role in Taylor and pairing arguments. The calculus of the kernel extends these representations to any order and exhibit natural and explicit conditions to obtain a one term representation of the $n$-th derivative of the solution $f$ with respect to exactly one of the three neighbouring test-function orders of derivatives, $n-1,n,n+1$. These conditions are simple to ensure by choosing the weight accordingly. The representations yield pointwise envelopes, uniform and weighted Stein factors and ensure sharpness of the constants. We recover the known sharp results for the Gaussian and the Gamma distribution, improve known factors on the rest of the integrated Pearson distributions, which is the family under which every correction term cancels. We also study the Subbotin and symmetrized Maxwell families, for which the formulas retain additional pointwise terms. Applications sharpen constants in known distributional approximations and use the higher-order calculus to obtain Edgeworth corrections for the Beta approximation of the Pólya--Eggenberger urn and the quartic Subbotin approximation in the critical Curie--Weiss model.