Bernoulli

  • Bernoulli
  • Volume 10, Number 3 (2004), 421-446.

Likelihood functions based on parameter-dependent functions

Thomas A. Severini

Full-text: Open access

Abstract

Consider likelihood inference about a scalar function ψ of a parameter θ. Two methods of constructing a likelihood function for ψ are conditioning and marginalizing. If, in the model with ψ held fixed, T is ancillary, then a marginal likelihood may be based on the distribution of T, which depends only on ψ; alternatively, if a statistic S is sufficient when ψ is fixed, then a conditional likelihood function may be based on the conditional distribution of the data given S. The statistics T and S are generally required to be the same for each value of ψ. In this paper, we consider the case in which either T or S is allowed to depend on ψ. Hence, we might consider the marginal likelihood function based on a function Tψ or the conditional likelihood given a function Sψ. The properties and construction of marginal and conditional likelihood functions based on parameter-dependent functions are studied. In particular, the case in which Tψ and Sψ may be taken to be functions of the maximum likelihood estimators is considered and approximations to the resulting likelihood functions are presented. The results are illustrated on several examples.

Article information

Source
Bernoulli, Volume 10, Number 3 (2004), 421-446.

Dates
First available in Project Euclid: 7 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1089206405

Digital Object Identifier
doi:10.3150/bj/1089206405

Mathematical Reviews number (MathSciNet)
MR2061439

Zentralblatt MATH identifier
1053.62029

Keywords
ancillary statistics conditional likelihood likelihood inference marginal likelihood maximum likelihood estimators modified profile likelihood nuisance parameters

Citation

Severini, Thomas A. Likelihood functions based on parameter-dependent functions. Bernoulli 10 (2004), no. 3, 421--446. doi:10.3150/bj/1089206405. https://projecteuclid.org/euclid.bj/1089206405


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References

  • [1] Barndorff-Nielsen, O.E. (1980) Conditionality resolutions. Biometrika, 67, 293-310.
  • [2] Barndorff-Nielsen, O.E. (1983) On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343-365.
  • [3] Barndorff-Nielsen, O.E. (1994) Adjusted versions of profile likelihood and directed likelihood, and extended likelihood. J. Roy. Statist. Soc. Ser. B, 56, 125-140.
  • [4] Barndorff-Nielsen, O.E. and Cox, D.R. (1994) Inference and Asymptotics. London: Chapman and Hall.
  • [5] Basu, D. (1955) On statistics independent of a complete sufficient statistic. Sankhya, 15, 377-380.
  • [6] Basu, D. (1958) On statistics independent of a sufficient statistic. Sankhya, 20, 223-226.
  • [7] Chamberlin, S.R. and Sprott, D.A. (1989) Linear systems of pivotals and associated pivotal likelihoods with applications. Biometrika, 76, 685-691. Abstract can also be found in the ISI/STMA publication
  • [8] Cox, D.R. and Reid, N. (1987) Parameter orthogonality and approximate conditional inference. J. Roy. Statist. Soc. Ser. B, 49, 1-39.
  • [9] Ferguson, H., Reid, N. and Cox, D.R. (1991) Estimating functions from modified profile likelihood. In V.P. Godambe (ed.), Estimating Functions. Oxford: Oxford University Press.
  • [10] Fraser, D.A.S. (1967) Data transformations and the linear model. Ann. Math. Statist., 38, 1456-1465.
  • [11] Fraser, D.A.S. (1968) The Structure of Inference. New York: Wiley.
  • [12] Fraser, D.A.S. (1972) The determination of likelihood and the transformed regression model. Ann. Math. Statist., 43, 898-916.
  • [13] Fraser, D.A.S. (1979) Inference and Linear Models. Toronto: McGraw-Hill.
  • [14] Fraser, D.A.S. and Reid, N. (1989) Adjustments to profile likelihood. Biometrika, 76, 477-488. Abstract can also be found in the ISI/STMA publication
  • [15] Fraser, D.A.S. and Reid, N. (1995) Bayes posteriors for scalar interest parameters. In J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith (eds), Bayesian Statistics 5. Oxford: Oxford University Press.
  • [16] Hoffmann-Jørgensen, J. (1994) Probability with a View towards Statistics, Volume II. New York: Chapman and Hall.
  • [17] Kalbfleisch, J.D. and Sprott, D.A. (1970) Application of likelihood methods to models involving large numbers of parameters (with discussion). J. Roy. Statist. Soc. Ser. B, 32, 175-208.
  • [18] Kalbfleisch, J.D. and Sprott, D.A. (1973) Marginal and conditional likelihoods. Sankhya Ser. A, 35, 311-328.
  • [19] McCullagh, P. and Nelder, J. (1989) Generalized Linear Models, 2nd edition. London: Chapman and Hall.
  • [20] McCullagh, P. and Tibshirani, R. (1990) A simple method for the adjustment of profile likelihoods. J. Roy. Statist. Soc. Ser. B, 52, 325-344. Abstract can also be found in the ISI/STMA publication
  • [21] Severini, T.A. (1994) On the approximate elimination of nuisance parameters by conditioning. Biometrika, 81, 649-661. Abstract can also be found in the ISI/STMA publication
  • [22] Severini, T.A. (1998) An approximation to the modified profile likelihood function. Biometrika, 85, 403-411. Abstract can also be found in the ISI/STMA publication
  • [23] Severini, T.A. (2000a) Likelihood Methods in Statistics. Oxford: Oxford University Press.
  • [24] Severini, T.A. (2000b) The likelihood ratio approximation to the conditional distribution of the maximum likelihood estimate in the lattice case. Biometrika, 87, 939-945. Abstract can also be found in the ISI/STMA publication
  • [25] Skovgaard, I.M. (1996) An explicit large-deviation approximation to one-parameter tests. Bernoulli, 2, 145-165. Abstract can also be found in the ISI/STMA publication
  • [26] Tjur, T. (1980) Probability Based on Radon Measures. New York: Wiley.