Saddlepoint approximations and tests based on multivariate M-estimates

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 . 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.