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Likelihood Asymptotics

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  • Ib M. Skovgaard

Abstract

The paper gives an overview of modern likelihood asymptotics with emphasis on results and applicability. Only parametric inference in well‐behaved models is considered and the theory discussed leads to highly accurate asymptotic tests for general smooth hypotheses. The tests are refinements of the usual asymptotic likelihood ratio tests, and for one‐dimensional hypotheses the test statistic is known as r*, introduced by Barndorff‐Nielsen. Examples illustrate the applicability and accuracy as well as the complexity of the required computations. Modern likelihood asymptotics has developed by merging two lines of research: asymptotic ancillarity is the basis of the statistical development, and saddlepoint approximations or Laplace‐type approximations have simultaneously developed as the technical foundation. The main results and techniques of these two lines will be reviewed, and a generalization to multi‐dimensional tests is developed. In the final part of the paper further problems and ideas are presented. Among these are linear models with non‐normal error, non‐parametric linear models obtained by estimation of the residual density in combination with the present results, and the generalization of the results to restricted maximum likelihood and similar structured models.

Suggested Citation

  • Ib M. Skovgaard, 2001. "Likelihood Asymptotics," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(1), pages 3-32, March.
    • Handle: RePEc:bla:scjsta:v:28:y:2001:i:1:p:3-32
    DOI: 10.1111/1467-9469.00223
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    Cited by:

    1. Donald Alan Pierce & Ruggero Bellio, 2017. "Modern Likelihood-Frequentist Inference," International Statistical Review, International Statistical Institute, vol. 85(3), pages 519-541, December.
    2. Erlis Ruli & Laura Ventura, 2021. "Can Bayesian, confidence distribution and frequentist inference agree?," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(1), pages 359-373, March.
    3. Mohammad Reza Kazemi & Ali Akbar Jafari, 2019. "Inference about the shape parameters of several inverse Gaussian distributions: testing equality and confidence interval for a common value," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(5), pages 529-545, July.
    4. Christopher Withers & Saralees Nadarajah, 2010. "Tilted Edgeworth expansions for asymptotically normal vectors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(6), pages 1113-1142, December.
    5. A. C. Davison & D. A. S. Fraser & N. Reid & N. Sartori, 2014. "Accurate Directional Inference for Vector Parameters in Linear Exponential Families," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 302-314, March.
    6. S. Lin, 2013. "The higher order likelihood method for the common mean of several log-normal distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(3), pages 381-392, April.
    7. Ole E. Barndorff-Nielsen & Bent Nielsen & Neil Shephard & Carla Ysusi, 2002. "Measuring and forecasting financial variability using realised variance with and without a model," Economics Papers 2002-W21, Economics Group, Nuffield College, University of Oxford.
    8. V. Filimonov & G. Demos & D. Sornette, 2017. "Modified profile likelihood inference and interval forecast of the burst of financial bubbles," Quantitative Finance, Taylor & Francis Journals, vol. 17(8), pages 1167-1186, August.
    9. Cristine Rauber & Francisco Cribari-Neto & Fábio M. Bayer, 2020. "Improved testing inferences for beta regressions with parametric mean link function," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(4), pages 687-717, December.
    10. Melo, Tatiane F.N. & Vasconcellos, Klaus L.P. & Lemonte, Artur J., 2009. "Some restriction tests in a new class of regression models for proportions," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 3972-3979, October.
    11. Ferrari, Silvia L.P. & Cysneiros, Audrey H.M.A., 2008. "Skovgaard's adjustment to likelihood ratio tests in exponential family nonlinear models," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 3047-3055, December.
    12. Paramjit S. Gill, 2004. "Small-Sample Inference for the Comparison of Means of Log-Normal Distributions," Biometrics, The International Biometric Society, vol. 60(2), pages 525-527, June.
    13. John Robinson, 2004. "Multivariate tests based on empirical saddlepoint approximations," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(1), pages 1-14.
    14. Wong ACM & Zhang S, 2017. "A Directional Approach for Testing Homogeneity of Inverse Gaussian Scale-Like Parameters," Biostatistics and Biometrics Open Access Journal, Juniper Publishers Inc., vol. 3(2), pages 34-39, September.
    15. Ventura, Laura & Ruli, Erlis & Racugno, Walter, 2013. "A note on approximate Bayesian credible sets based on modified loglikelihood ratios," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2467-2472.
    16. Rukhin, Andrew L., 2016. "Confidence regions for comparison of two normal samples," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 273-280.
    17. Almudevar, Anthony, 2016. "Higher order density approximations for solutions to estimating equations," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 424-439.
    18. Gaurav Sharma & Thomas Mathew & Ionut Bebu, 2014. "Combining Multivariate Bioassays: Accurate Inference Using Small Sample Asymptotics," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 152-166, March.

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