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General

Higher-Order Asymptotic Approximation: Laplace, Saddlepoint, and Related Methods

Pages 1358-1364 | Published online: 17 Feb 2012

References

  • Ablowitz , M. J. and Fokas , A. S. 1997 . Complex Variables: Introduction and Applications , Cambridge , , U.K. : Cambridge University Press .
  • Bak , J. and Newman , D. J. 1996 . Complex Analysis, , 2nd ed. , New York : Springer-Verlag .
  • Barndorff-Nielsen , O. E. 1980 . “Conditionality Resolutions” . Biometrika , 67 : 293 – 310 .
  • Barndorff-Nielsen , O. E. 1983 . “On a Formula for the Distribution of the Maximum Likelihood Estimator” . Biometrika , 70 : 343 – 365 .
  • Barndorff-Nielsen , O. E. 1986 . “Inference on Full and Partial Parameters Based on Standardized Signed Log-Likelihood Ratio” . Biometrika , 73 : 307 – 322 .
  • Barndorff-Nielsen , O. E. 1988 . Discussion of “Saddlepoint Methods and Statistical Inference” . Statistical Science , 3 : 288 – 289 . by N. Reid
  • Barndorff-Nielsen , O. E. 1990a . “p* and Laplace's Method” . REBRAPE (Brazilian Journal of Probability and Statistics) , 4 : 89 – 103 .
  • Barndorff-Nielsen , O. E. 1991 . “Modified Signed Log-Likelihood Ratio” . Biometrika , 78 : 557 – 563 .
  • Barndorff-Nielsen , O. E. 1994 . “Adjusted Versions of Profile Likelihood and Directed Likelihood, and Extended Likelihood” . Journal of the Royal Statistical Society , 56 : 125 – 140 . Ser. B
  • Barndorff-Nielsen , O. E. and Cox , D. R. 1979 . “Edgeworth and Saddlepoint Approximations With Statistical Applications” . Journal of the Royal Statistical Society , 41 : 279 – 312 . (with discussion), Ser. B
  • Barndorff-Nielsen , O. E. and Cox , D. R. 1994 . Inference and Asymptotics , London : Chapman and Hall .
  • Bleistein , N. and Handelsman , R. A. 1975 . Asymptotic Expansions of Integrals , New York : Dover .
  • Booth , J. , Hall , P. and Wood , A. T. A. 1994 . “On the Validity of Edgeworth and Saddlepoint Approximations” . Journal of Multivariate Analysis , 51 : 121 – 138 .
  • Casella , G. , DiCiccio , T. J. and Wells , M. T. 1995 . Discussion of “The Roles of Conditioning in Inference” . Statistical Science , 10 : 138 – 199 . by N. Reid
  • Daniels , H. 1954 . “Saddlepoint Approximations in Statistics” . Annals of Mathematical Statistics , 25 : 631 – 650 .
  • Daniels , H. 1958 . Discussion of “The Regression Analysis of Binary Sequences” . Journal of the Royal Statistical Society , 20 : 236 – 238 . by D. R. Cox, Ser. B
  • Daniels , H. 1983 . “Saddlepoint Approximations for Estimating Equations” . Biometrika , 70 : 89 – 96 .
  • Daniels , H. 1987 . “Tail Probability Approximations” . International Statistical Review , 55 : 37 – 48 .
  • Davis , P. and Rabinowitz , P. 1984 . Methods of Numerical Integration, , 2nd ed. , New York : Academic Press .
  • Davison , A. C. 1988 . “Approximate Conditional Inference in Generalized Linear Models” . Journal of the Royal Statistical Society , 50 : 445 – 461 . Ser. B
  • de Bruijn , N. G. 1961 . Asymptotic Methods in Analysis, , 2nd ed. , New York : Wiley .
  • Debye , P. 1909 . “Näherungsformeln für die Zylinderfunktionen für Grosse Werte des Arguments und Unbeschränkt Veränderliche Werte des Index” . Math. Ann. , 67 : 535 – 558 .
  • DiCiccio , T. J. , Kass , R. E. , Raftery , A. and Wasserman , L. 1997 . “Computing Bayes Factors by Combining Simulation and Asymptotic Approximations” . Journal of the American Statistical Association , 92 : 903 – 915 .
  • DiCiccio , T. J. and Martin , M. A. 1991 . “Approximations of Marginal Tail Probabilities for a Class of Smooth Functions With Applications to Bayesian and Conditional Inference” . Biometrika , 78 : 891 – 902 .
  • DiCiccio , T. J. and Martin , M. A. 1993 . “Analytical Approximations to Conditional Distribution Functions” . Biometrika , 80 : 781 – 790 .
  • Fraser , D. A. S. 1988 . “Normed Likelihood as Saddlepoint Approximation” . Journal of Multivariate Analysis , 27 : 181 – 193 .
  • Fraser , D. A. S. 1990 . “Tail Probabilities From Observed Likelihoods” . Biometrika , 77 : 65 – 76 .
  • Fulks , W. 1960 . “Asymptotics I: A Note on Laplace's Method” . American Mathematical Monthly , 67 : 880 – 882 .
  • Fulks , W. and Sather , J. O. 1961 . “Asymptotics II: Laplace's Method for Multiple Integrals” . Pacific Journal of Mathematics , 11 : 185 – 192 .
  • Gelman , A. and Meng , X-L. 1998 . “Simulating Normalizing Constants: From Importance Sampling to Bridge Sampling to Path Sampling” . Statistical Science , 13 : 163 – 185 . (with discussion)
  • Genz , A. and Monahan , J. 1998 . “Stochastic Integration Rules for Infinite Regions” . SIAM Journal on Scientific Computing , 19 : 426 – 439 .
  • Jensen , J. L. 1995 . Saddlepoint Approximations , Oxford , , U.K. : Clarendon Press .
  • Kass , R. E. , Tierney , L. and Kadane , J. B. 1988 . “Asymptotics in Bayesian Computation,” . In Bayesian Statistics 3 , Edited by: Bernardo , J. M. , DeGroot , M. H. , Lindley , D. V. and Smith , A. F. M. Oxford , , U.K. : Clarendon Press .
  • Kolassa , J. 1999 . “Convergence and Accuracy of Gibbs Sampling for Conditional Distributions in Generalized Linear Models” . The Annals of Statistics , 27 : 129 – 142 .
  • Kolassa , J. and Tanner , M. A. 1994 . “Approximate Conditional Inference in Exponential Families via the Gibbs Sampler” . Journal of the American Statistical Association , 89 : 697 – 702 .
  • Liu , Q. and Pierce , D. 1994 . “A Note on Gauss-Hermite Quadrature” . Biometrika , 81 : 624 – 629 .
  • Lugannani , R. and Rice , S. O. 1980 . “Saddlepoint Approximation for the Distribution of the Sum of Independent Random Variables” . Advances in Applied Probability , 12 : 475 – 490 .
  • Olver , F. W. J. 1974 . Asymptotics and Special Functions , New York : Academic Press .
  • Pierce , D. and Peters , D. 1992 . “Practical Use of Higher-Order Asymptotics for Multiparameter Exponential Families” . Journal of the Royal Statistical Society , 54 : 701 – 725 . (with discussion), Ser. B
  • Reid , N. 1995 . “The Roles of Conditioning in Inference” . Statistical Science , 10 : 138 – 157 . (with discussion)
  • Reid , N. 1996 . “Likelihood and Higher-Order Approximations to Tail Areas: A Review and Annotated Bibliography” . Canadian Journal of Statistics , 24 : 263 – 278 .
  • Reid , N. 2000 . “Likelihood” . Journal of the American Statistical Association , 95 : 0000 – 0000 .
  • Riemann , B. 1863 . “Sullo Svolgimento del Quoziente di due Serie Ipogeometriche in Frazione Continua Infinita,” , New York : Dover . reprinted in Complete Works, 1953
  • Robinson , J. 1982 . “Saddlepoint Approximations for Permutation Tests and Confidence Intervals” . Journal of the Royal Statistical Society , 44 : 91 – 101 . Ser. B
  • Severini , T. 1999 . “An Empirical Adjustment to the Likelihood Ratio Statistic” . Biometrika , 86 : 235 – 248 .
  • Skates , S. 1993 . “On Secant Approximations to Cumulative Distribution Functions” . Biometrika , 80 : 223 – 235 .
  • Skovgaard , I. 1987 . “Saddlepoint Expansions for Conditional Distributions” . Journal of Applied Probability , 24 : 875 – 887 .
  • Stigler , S. 1986 . The History of Statistics: The Measurement of Uncertainty Before 1900 , Cambridge , MA : Harvard University Press .
  • Strawderman , R. L. and Wells , M. T. 1998 . “Approximately Exact Inference for the Common Odds Ratio in Several 2 × 2 Tables” . Journal of the American Statistical Association , 93 : 1294 – 1306 . (with discussion)
  • Tierney , L. and Kadane , J. B. 1986 . “Accurate Approximations for Posterior Moments and Marginal Densities” . Journal of the American Statistical Association , 81 : 1032 – 1038 .
  • Tierney , L. , Kass , R. E. and Kadane , J. B. 1989 . “Fully Exponential Laplace Approximations to Expectations and Variances of Nonpositive Functions” . Journal of the American Statistical Association , 84 : 710 – 716 .
  • Watson , G. N. 1944 . A Treatise on the Theory of Bessel Functions, , 2nd ed. , London : Cambridge University Press .
  • Wood , A. T. A. , Booth , J. and Butler , R. 1993 . “Saddlepoint Approximations to the CDF of Some Statistics With Nonnormal Limit Distributions” . Journal of the American Statistical Association , 88 : 680 – 686 .
  • Zheng , X. 1998 . “Better Saddlepoint Confidence Intervals via Bootstrap Calibration” . Proceedings of the American Mathematical Society , 126 : 3669 – 3679 .

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