Explicit Gamma–Stein Bounds for Satterthwaite’s Approximation

Gabriel Bailly, Ferdinand Rapin, Yvik Swan and Rainer von Sachs
Additional contact information
Gabriel Bailly: Université catholique de Louvain, LIDAM/ISBA, Belgium
Rainer von Sachs: Université catholique de Louvain, LIDAM/ISBA, Belgium

No 2026033, LIDAM Discussion Papers ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

Abstract: We derive explicit Gamma–Stein bounds for the Satterthwaite approximation of weighted sums of independent chi-square random variables and, more generally, of independent Gamma random variables. The approximating Gamma distribution is chosen by matching the first two moments. Our approach relies on new kernel representations of the solution to the Gamma–Stein equation and of its first derivatives on the positive real line. These representations yield weighted Stein-factor bounds adapted to the moment-matching structure and show that the quality of the approximation is controlled directly by the discrepancies between the individual scale parameters and the matched scale parameter. Kolmogorov bounds are obtained by smoothing. The results provide quantitative guarantees for the widely used Satterthwaite approximation and are applied to pooled variances, sample variances of stationary Gaussian time series, Wishart traces and sums of Wishart traces, and high-dimensional repeated-measures statistics.

Keywords: Satterthwaite approximation; Gamma approximation; Stein’s method; weighted chi-square distribution; Wishart distribution (search for similar items in EconPapers)
Pages: 27
Date: 2026-10-06
References: Add references at CitEc
Citations:

Downloads: (external link)
https://research.dial.uclouvain.be/

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:aiz:louvad:2026033

Access Statistics for this paper

More papers in LIDAM Discussion Papers ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA) Voie du Roman Pays 20, 1348 Louvain-la-Neuve (Belgium). Contact information at EDIRC.
Bibliographic data for series maintained by Alain Gillis ().