Stein方程解及其导数的表示
Representations of solutions to Stein equations and their derivatives
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中文总结 AI 辅助
本文提出Stein方程解及其导数的点态表示，基于核微积分导出尖锐Stein因子，改进高斯、Gamma及Pearson分布族结果，并应用于分布逼近和Edgeworth修正。
中文摘要 AI 辅助
我们针对与一元绝对连续分布相关联的一阶Stein方程，提出了其Stein解的点态表示。该表示可推广至解的所有导数，以及测试函数$h$的任意导数。这些表示背后的机制是一组满足两个关键恒等式的核，从而为本文所考虑的Stein框架提供了一种特定的微积分。在经典条件下，Stein解的前两阶导数可在目标密度$p$与非零权重$w$之间不作任何假设的情况下表达。表示中出现的修正项解释了为何将Stein核选作权重在Taylor论证和配对论证中具有特殊作用。核的微积分将这些表示推广至任意阶，并给出了自然且显式的条件，以获得解$f$的第$n$阶导数关于三个相邻测试函数导数阶数$n-1,n,n+1$中恰好一个的单项表示。通过相应地选择权重，这些条件易于满足。这些表示产生了点态包络、一致和加权Stein因子，并确保了常数的尖锐性。我们恢复了高斯分布和Gamma分布的已知尖锐结果，改进了其余积分Pearson分布族（在该族中所有修正项均抵消）上的已知因子。我们还研究了Subbotin族和对称化Maxwell族，其公式保留了额外的点态项。应用方面，我们改进了已知分布逼近中的常数，并利用高阶微积分获得了Pólya–Eggenberger瓮模型的Beta逼近和临界Curie–Weiss模型中四次Subbotin逼近的Edgeworth修正。
英文摘要
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.
发表机构
- Université Libre de Bruxelles（布鲁塞尔自由大学）
- Vrije Universiteit Brussel（布鲁塞尔自由大学）
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