Saddlepoint approximations and tests based on multivariate M-estimates

Robinson, J.; Ronchetti, Elvezio; Young, G. A.

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Journal article

Saddlepoint approximations and tests based on multivariate M-estimates

Robinson, J. University of Sydney
Ronchetti, Elvezio Istituto di finanza (IFin), Facoltà di scienze economiche, Università della Svizzera italiana, Svizzera
Young, G. A. University of Cambridge

2003

Published in:

The Annals of Statistics. - Institute of Mathematical Statistics. - 2003, vol. 31, no. 4, p. 1154-1169

English 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.

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