The Annals of Statistics

Saddlepoint approximations and tests based on multivariate M-estimates

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

Full-text: Open access


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

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

First available in Project Euclid: 31 July 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • ALMUDEVAR, A., FIELD, C. and ROBINSON, J. (2000). The density of multivariate M-estimates. Ann. Statist. 28 275-297.
  • BARNDORFF-NIELSEN, O. E. and COX, D. R. (1984). Bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator. J. Roy. Statist. Soc. Ser. B. 46 483-495.
  • DANIELS, H. E. and YOUNG, G. A. (1991). Saddlepoint approximation for the studentized mean, with an application to the bootstrap. Biometrika 78 169-179.
  • DAVISON, A. C. and HINKLEY, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Univ. Press.
  • DAVISON, A. C., HINKLEY, D. V. and WORTON, B. J. (1995). Accurate and efficient construction of bootstrap likelihoods. Statistics and Computing 5 257-264.
  • DICICCIO, T. J. and ROMANO, J. P. (1990). Nonparametric confidence limits by resampling methods and least favorable families. Internat. Statist. Rev. 58 59-76.
  • FAN, R. Y. K. and FIELD, C. A. (1995). Approximations for marginal densities of M-estimators. Canad. J. Statist. 23 185-197.
  • FIELD, C. A. (1982). Small sample asy mptotic expansions for multivariate M-estimates. Ann. Statist. 10 672-689.
  • FIELD, C. A. and RONCHETTI, E. (1990). Small Sample Asy mptotics. IMS, Hay ward, CA.
  • GATTO, R. (2000). Multivariate saddlepoint test for the wrapped normal model. J. Statist. Comput. Simulation 65 271-285.
  • GATTO, R. and RONCHETTI, E. (1996). General saddlepoint approximations of marginal densities and tail probabilities. J. Amer. Statist. Assoc. 91 666-673.
  • HAMPEL, F. R., RONCHETTI, E. M., ROUSSEEUW, P. J. and STAHEL, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley, New York.
  • HERITIER, S. and RONCHETTI, E. (1994). Robust bounded-influence tests in general parametric models. J. Amer. Statist. Assoc. 89 897-904.
  • JENSEN, J. L. and WOOD, A. T. A. (1998). Large deviation and other results for minimum contrast estimators. Ann. Inst. Statist. Math. 50 673-695.
  • JING, B. and ROBINSON, J. (1994). Saddlepoint approximations for marginal and conditional probabilities of transformed variables. Ann. Statist. 22 1115-1132.
  • MONTI, A. C. and RONCHETTI, E. (1993). On the relationship between empirical likelihood and empirical saddlepoint approximation for multivariate M-estimators. Biometrika 80 329-338.
  • My KLAND, P. A. (1995). Dual likelihood. Ann. Statist. 23 396-421.
  • OWEN, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237-249.
  • SKOVGAARD, I. M. (1990). On the density of minimum contrast estimators. Ann. Statist. 18 779-789.
  • TINGLEY, M. A. and FIELD, C. A. (1990). Small-sample confidence intervals. J. Amer. Statist. Assoc. 85 427-434.
  • WELSH, A. H. (1996). Aspects of Statistical Inference. Wiley, New York.