Volume 25, 2021
|Page(s)||325 - 345|
|Published online||27 July 2021|
New error bounds for Laplace approximation via Stein’s method
Department of Mathematics, The University of Manchester,
M13 9PL, UK.
* Corresponding author: firstname.lastname@example.org
Accepted: 28 June 2021
We use Stein’s method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 571–587] for Stein’s method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.
Mathematics Subject Classification: 60F05 / 62E17
Key words: Stein’s method / Laplace approximation / rate of convergence / random sums / Rayleigh distribution
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.