Stein Solution Factors via Mills Ratios: Affine Birth Rates and Symmetric Potential Distributions
Abstract
We study Stein solution factors for indicator test functions by a common Mills-ratio method in discrete and continuous settings. In the discrete case, a general birth--death formulation gives exact solution envelopes and recovers the first-increment theory of Brown and Xia. For binomial and Poisson targets, and for negative binomial targets with shape parameter r≥1, the additional algebraic structure of affine birth rates yields improved uniform bounds for the Stein solution. In the continuous case, we consider symmetric densities proportional to e−V. Under natural structural assumptions on the potential V, Mills ratios yield explicit finite bounds for the indicator Stein solution and show that the uniform derivative and drift Stein factors have the sharp value $1$. Under an additional one-crossing condition, the solution-factor optimization can be carried out exactly and gives the optimal value $1/(4p(0))$. The even-power targets arising in statistical mechanics, including the quartic critical Curie--Weiss law, provide the original examples, and the calculation extends to the wider Subbotin family. For 1<β<2 in the Subbotin family, the exact envelope has two off-center maximizers characterized by a unique incomplete-gamma equation, while the two first-order factors remain equal to $1$. The results exhibit a common discrete--continuous mechanism behind improved and, in the continuous sharp regime, optimal Stein solution factors.
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BibTeX
@misc{eichelsbacher2026stein,
title = {{Stein Solution Factors via Mills Ratios: Affine Birth Rates and Symmetric Potential Distributions}},
author = {Peter Eichelsbacher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25380},
eprint = {2609.25380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25380},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stein Solution Factors via Mills Ratios: Affine Birth Rates and Symmetric Potential Distributions
AU - Peter Eichelsbacher
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25380
UR - https://arxiv.org/abs/2609.25380
AB - We study Stein solution factors for indicator test functions by a common Mills-ratio method in discrete and continuous settings. In the discrete case, a general birth--death formulation gives exact solution envelopes and recovers the first-increment theory of Brown and Xia. For binomial and Poisson targets, and for negative binomial targets with shape parameter r≥1, the additional algebraic structure of affine birth rates yields improved uniform bounds for the Stein solution. In the continuous case, we consider symmetric densities proportional to e^−V. Under natural structural assumptions on the potential V, Mills ratios yield explicit finite bounds for the indicator Stein solution and show that the uniform derivative and drift Stein factors have the sharp value $1$. Under an additional one-crossing condition, the solution-factor optimization can be carried out exactly and gives the optimal value $1/(4p(0))$. The even-power targets arising in statistical mechanics, including the quartic critical Curie--Weiss law, provide the original examples, and the calculation extends to the wider Subbotin family. For 1<β<2 in the Subbotin family, the exact envelope has two off-center maximizers characterized by a unique incomplete-gamma equation, while the two first-order factors remain equal to $1$. The results exhibit a common discrete--continuous mechanism behind improved and, in the continuous sharp regime, optimal Stein solution factors.
ER -