• Bernoulli
  • Volume 12, Number 3 (2006), 491-500.

Saddlepoint approximation in exponential models with boundary points

Joan Del Castillo and Anna López-Ratera

Full-text: Open access


A saddlepoint approximation is developed for the likelihood ratio test statistic in non-regular exponential models. For the one-parameter case, it is proved that the saddlepoint correction has a relative error of order O ( n - 1 ) , assuming that the canonical statistic has four finite moments. The results are applied in reliability theory and survival analysis, testing for exponentiality. Note that we are using the saddlepoint method for statistics without moment generating function.

Article information

Bernoulli, Volume 12, Number 3 (2006), 491-500.

First available in Project Euclid: 28 June 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

higher-order asymptotics likelihood ratio test reliability theory survival analysis models


Del Castillo, Joan; López-Ratera, Anna. Saddlepoint approximation in exponential models with boundary points. Bernoulli 12 (2006), no. 3, 491--500. doi:10.3150/bj/1151525132.

Export citation


  • [1] Barndorff-Nielsen, O. (1986) Inference on full or partial parameters based on the standardized signed log likelihood ratio, Biometrika, 73, 307-322.
  • [2] Barndorff-Nielsen, O. and Cox, D.R. (1994) Inference and Asymptotics. London: Chapman & Hall.
  • [3] Billingsley, P. (1995) Probability and Measure. New York: Wiley.
  • [4] del Castillo, J. and Puig, P. (1999a) The best test of exponential against singly truncated normal alternatives. J. Amer. Statist. Assoc., 94, 529-532.
  • [5] del Castillo, J. and Puig, P. (1999b) Invariant exponential models applied to reliability theory and survival analysis. J. Amer. Statist. Assoc., 94, 522-528.
  • [6] Daniels, H.E. (1954) Saddlepoint approximations in statistics. Ann. Math. Statist., 25, 631-650.
  • [7] Geyer, C.J. (1994) On the asymptotics of constrained M-estimation. Ann. Statist., 22, 1993-2010.
  • [8] Goutis, C. and Casella, G. (1999) Explaining the saddlepoint approximation. Amer. Statist., 53, 216-224.
  • [9] Jensen, J.L. (1992) The modified signed likelihood statistic and saddlepoint approximations. Biometrika, 79, 693-703.
  • [10] Jensen, J.L. (1995) Saddlepoint Aproximations. Oxford: Oxford University Press.
  • [11] Kolassa, J.E. (1994) Series Approximation Methods in Statistics, Lecture Notes in Statist. 88. New York: Springer-Verlag.
  • [12] Lee, E.T. (1992) Statistical Methods for Survival Data Analysis. New York: Wiley.
  • [13] Reid, N. (1996) Likelihood and higher-order approximations to tail areas: A review and annotated bibliography. Canad. J. Statist., 24, 141-166.
  • [14] Self, S.G. and Liang, K.Y. (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J. Amer. Statist. Assoc., 82, 605-610.