Annals of Statistics

Higher-order asymptotic normality of approximations to the modified signed likelihood ratio statistic for regular models

Heping He and Thomas A. Severini

Full-text: Open access


Approximations to the modified signed likelihood ratio statistic are asymptotically standard normal with error of order n−1, where n is the sample size. Proofs of this fact generally require that the sufficient statistic of the model be written as (θ̂, a), where θ̂ is the maximum likelihood estimator of the parameter θ of the model and a is an ancillary statistic. This condition is very difficult or impossible to verify for many models. However, calculation of the statistics themselves does not require this condition. The goal of this paper is to provide conditions under which these statistics are asymptotically normally distributed to order n−1 without making any assumption about the sufficient statistic of the model.

Article information

Ann. Statist., Volume 35, Number 5 (2007), 2054-2074.

First available in Project Euclid: 7 November 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F05: Asymptotic properties of tests
Secondary: 62F03: Hypothesis testing

Edgeworth expansion theory modified signed likelihood ratio statistic higher-order normality sufficient statistic Cramér-Edgeworth polynomial


He, Heping; Severini, Thomas A. Higher-order asymptotic normality of approximations to the modified signed likelihood ratio statistic for regular models. Ann. Statist. 35 (2007), no. 5, 2054--2074. doi:10.1214/009053607000000307.

Export citation


  • Bahadur, R. R. (1971). Some Limit Theorems in Statistics. SIAM, Philadelphia.
  • Barndorff-Nielsen, O. E. (1986). Inference on full or partial parameters based on the standardized signed log likelihood ratio. Biometrika 73 307–322.
  • Barndorff-Nielsen, O. E. (1991). Modified signed log likelihood ratio. Biometrika 78 557–563.
  • Barndorff-Nielsen, O. E. and Cox, D. R. (1994). Inference and Asymptotics. Chapman and Hall, London.
  • Barndorff-Nielsen, O. E. and Wood, A. T. A. (1998). On large deviations and choice of ancillary for $p^*$ and $r^*$. Bernoulli 4 35–63.
  • Bhattacharya, R. N. and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York.
  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
  • Casella, G. and Berger, R. L. (1990). Statistical Inference. Duxbury, Belmont, CA.
  • Cox, D. R. and Reid, N. (1987). Parameter orthogonality and approximate conditional inference (with discussion). J. Roy. Statist. Soc. Ser. B 49 1–39.
  • DiCiccio, T. J. and Martin, M. A. (1993). Simple modifications for signed roots of likelihood ratio statistics. J. Roy. Statist. Soc. Ser. B 55 305–316.
  • Huzurbazar, V. S. (1950). Probability distributions and orthogonal parameters. Proc. Cambridge Philos. Soc. 46 281–284.
  • Huzurbazar, V. S. (1956). Sufficient statistics and orthogonal parameters. Sankhyā Ser. A 17 217–220.
  • Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Univ. Press.
  • Kolassa, J. E. (1997). Series Approximation Methods in Statistics, 2nd ed. Lecture Notes in Statist. 88. Springer, New York.
  • Lee, E. T. and Wang, J. W. (2003). Statistical Methods for Survival Data Analysis. Wiley, Hoboken, NJ.
  • Perlman, M. D. (1972). On the strong consistency of approximate maximum likelihood estimators. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 1 263–281. Univ. California Press. Berkeley.
  • Reid, N. (1996). Likelihood and higher-order approximations to tail areas: A review and annotated bibliography. Canad. J. Statist. 24 141–166.
  • Severini, T. A. (1999). An empirical adjustment to the likelihood ratio statistic. Biometrika 86 235–247.
  • Skovgaard, I. M. (1981). Transformation of an Edgeworth expansion by a sequence of smooth functions. Scand. J. Statist. 8 207–217.
  • Skovgaard, I. M. (1981). Edgeworth expansions of the distributions of maximum likelihood estimators in the general (non-i.i.d.) case. Scand. J. Statist. 8 227–236.
  • Skovgaard, I. M. (1986). On multivariate Edgeworth expansions. Internat. Statist. Rev. 54 169–186.
  • Skovgaard, I. M. (1996). An explicit large-deviation approximation to one-parameter tests. Bernoulli 2 145–165.
  • Skovgaard, I. M. (2001). Likelihood asymptotics. Scand. J. Statist. 28 3–32.
  • Wald, A. (1949). Note on the consistency of the maximum likelihood estimator. Ann. Math. Statist. 20 595–601.