The Annals of Statistics

Saddlepoint approximations and tests based on multivariate M-estimates

E. Ronchetti, G.A. Young, and J. Robinson

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Abstract

We consider multidimensional M-functional parameters defined by expectations of score functions associated with multivariate M-estimators and tests for hypotheses concerning multidimensional smooth functions of these parameters. We propose a test statistic suggested by the exponent in the saddlepoint approximation to the density of the function of the M-estimates. This statistic is analogous to the log likelihood ratio in the parametric case. We show that this statistic is approximately distributed as a chi-squared variate and obtain a Lugannani-Rice style adjustment giving a relative error of order $n^{-1}$. We propose an empirical exponential likelihood statistic and consider a test based on this statistic. Finally we present numerical results for three examples including one in robust regression.

Article information

Source
Ann. Statist., Volume 31, Number 4 (2003), 1154-1169.

Dates
First available in Project Euclid: 31 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1059655909

Digital Object Identifier
doi:10.1214/aos/1059655909

Mathematical Reviews number (MathSciNet)
MR2001646

Zentralblatt MATH identifier
1056.62023

Subjects
Primary: 62F11 62F05: Asymptotic properties of tests
Secondary: 62G09: Resampling methods

Keywords
Bootstrap tests composite hypothesis nonparametric likelihood relative error smooth functions of $M$-estimators

Citation

Robinson, J.; Ronchetti, E.; Young, G.A. Saddlepoint approximations and tests based on multivariate M -estimates. Ann. Statist. 31 (2003), no. 4, 1154--1169. doi:10.1214/aos/1059655909. https://projecteuclid.org/euclid.aos/1059655909


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