The Annals of Applied Probability

Chi-square approximation by Stein’s method with application to Pearson’s statistic

Robert E. Gaunt, Alastair M. Pickett, and Gesine Reinert

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Abstract

This paper concerns the development of Stein’s method for chi-square approximation and its application to problems in statistics. New bounds for the derivatives of the solution of the gamma Stein equation are obtained. These bounds involve both the shape parameter and the order of the derivative. Subsequently, Stein’s method for chi-square approximation is applied to bound the distributional distance between Pearson’s statistic and its limiting chi-square distribution, measured using smooth test functions. In combination with the use of symmetry arguments, Stein’s method yields explicit bounds on this distributional distance of order $n^{-1}$.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 2 (2017), 720-756.

Dates
Received: July 2015
First available in Project Euclid: 26 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1495764365

Digital Object Identifier
doi:10.1214/16-AAP1213

Mathematical Reviews number (MathSciNet)
MR3655852

Zentralblatt MATH identifier
1370.60042

Subjects
Primary: 60F05: Central limit and other weak theorems 62G10: Hypothesis testing 62G20: Asymptotic properties

Keywords
Stein’s method chi-square approximation Pearson’s statistic rate of convergence

Citation

Gaunt, Robert E.; Pickett, Alastair M.; Reinert, Gesine. Chi-square approximation by Stein’s method with application to Pearson’s statistic. Ann. Appl. Probab. 27 (2017), no. 2, 720--756. doi:10.1214/16-AAP1213. https://projecteuclid.org/euclid.aoap/1495764365


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