Electronic Journal of Statistics

Likelihood decision functions

Marco E. G. V. Cattaneo

Full-text: Open access

Abstract

In both classical and Bayesian approaches, statistical inference is unified and generalized by the corresponding decision theory. This is not the case for the likelihood approach to statistical inference, in spite of the manifest success of the likelihood methods in statistics. The goal of the present work is to fill this gap, by extending the likelihood approach in order to cover decision making as well. The resulting likelihood decision functions generalize the usual likelihood methods (such as ML estimators and LR tests), while maintaining some of their key properties, and thus providing a theoretical foundation for established and new likelihood methods.

Article information

Source
Electron. J. Statist., Volume 7 (2013), 2924-2946.

Dates
First available in Project Euclid: 2 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1385995295

Digital Object Identifier
doi:10.1214/13-EJS869

Mathematical Reviews number (MathSciNet)
MR3148372

Zentralblatt MATH identifier
1280.62016

Subjects
Primary: 62C05: General considerations 62A01: Foundations and philosophical topics

Keywords
Likelihood approach to statistics decision theory foundations of statistics conditional inference minimax invariances asymptotics

Citation

Cattaneo, Marco E. G. V. Likelihood decision functions. Electron. J. Statist. 7 (2013), 2924--2946. doi:10.1214/13-EJS869. https://projecteuclid.org/euclid.ejs/1385995295


Export citation

References

  • [1] Antonucci, A., Cattaneo, M. and Corani, G. (2012). Likelihood-based robust classification with Bayesian networks. In, Advances in Computational Intelligence, part 3 ( S. Greco, B. Bouchon-Meunier, G. Coletti, M. Fedrizzi, B. Matarazzo and R. R. Yager, eds.) 491–500. Springer.
  • [2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk., Math. Finance 9 203–228.
  • [3] Azzalini, A. (1996)., Statistical Inference: Based on the Likelihood. Chapman & Hall.
  • [4] Bahadur, R. R. (1967). Rates of convergence of estimates and test statistics., Ann. Math. Statist. 38 303–324.
  • [5] Barnard, G. A. (1949). Statistical inference., J. Roy. Statist. Soc. Ser. B 11 115–149.
  • [6] Barnard, G. A. (1967). The use of the likelihood function in statistical practice. In, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. I: Statistics ( L. M. Le Cam and J. Neyman, eds.) 27–40. University of California Press.
  • [7] Barnard, G. A. (1972). The logic of statistical inference., British J. Philos. Sci. 23 123–132.
  • [8] Barnard, G. A., Jenkins, G. M. and Winsten, C. B. (1962). Likelihood inference and time series., J. Roy. Statist. Soc. Ser. A 125 321–372.
  • [9] Barnard, G. A. and Sprott, D. A. (1983). Likelihood. In, Encyclopedia of Statistical Sciences, vol. 4 ( S. Kotz, N. L. Johnson and C. B. Read, eds.) 639–644. Wiley.
  • [10] Basu, D. (1975). Statistical information and likelihood., Sankhyā Ser. A 37 1–71.
  • [11] Berger, J. (1985a). The frequentist viewpoint and conditioning. In, Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, vol. I ( L. M. Le Cam and R. A. Olshen, eds.) 15–44. Wadsworth.
  • [12] Berger, J. O. (1985b)., Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer.
  • [13] Berger, J. O. and Wolpert, R. L. (1988)., The Likelihood Principle, 2nd ed. Institute of Mathematical Statistics.
  • [14] Birnbaum, A. (1962). On the foundations of statistical inference., J. Amer. Statist. Assoc. 57 269–326.
  • [15] Board, J. L. G. and Sutcliffe, C. M. S. (1994). Estimation methods in portfolio selection and the effectiveness of short sales restrictions: UK evidence., Management Sci. 40 516–534.
  • [16] Brenner, D., Fraser, D. A. S. and McDunnough, P. (1982). On asymptotic normality of likelihood and conditional analysis., Canad. J. Statist. 10 163–172.
  • [17] Brown, L. D. (1986)., Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Institute of Mathematical Statistics.
  • [18] Cattaneo, M. (2005). Likelihood-based statistical decisions. In, ISIPTA’05, Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications ( F. G. Cozman, R. Nau and T. Seidenfeld, eds.) 107–116. SIPTA.
  • [19] Cattaneo, M. (2007). Statistical Decisions Based Directly on the Likelihood Function. PhD thesis, ETH, Zurich.
  • [20] Cattaneo, M. (2010). Likelihood-based inference for probabilistic graphical models: Some preliminary results. In, PGM 2010, Proceedings of the Fifth European Workshop on Probabilistic Graphical Models ( P. Myllymäki, T. Roos and T. Jaakkola, eds.) 57–64. HIIT Publications.
  • [21] Cattaneo, M. (2013). On maxitive integration. Technical Report No. 147, Department of Statistics, LMU, Munich.
  • [22] Cattaneo, M. and Wiencierz, A. (2012). Likelihood-based imprecise regression., Internat. J. Approx. Reason. 53 1137–1154.
  • [23] Cattaneo, M. and Wiencierz, A. (2013). On the implementation of LIR: The case of simple linear regression with interval data., Comput. Statist., in press.
  • [24] Denneberg, D. (1994)., Non-Additive Measure and Integral. Kluwer.
  • [25] Diehl, H. and Sprott, D. A. (1965). Die Likelihoodfunktion und ihre Verwendung beim statistischen Schluß [The likelihood function and its use in statistical inference (in German with English summary)]., Statist. Hefte 6 112–134.
  • [26] Edwards, A. W. F. (1969). Statistical methods in scientific inference., Nature 222 1233–1237.
  • [27] Edwards, A. W. F. (1970). Likelihood., Nature 227 92.
  • [28] Edwards, A. W. F. (1992)., Likelihood, exp. ed. Johns Hopkins University Press.
  • [29] Evans, M. J., Fraser, D. A. S. and Monette, G. (1986). On principles and arguments to likelihood., Canad. J. Statist. 14 181–199.
  • [30] Fisher, R. A. (1973)., Statistical Methods and Scientific Inference, 3rd ed. Hafner Press.
  • [31] Föllmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints., Finance Stoch. 6 429–447.
  • [32] Fraser, D. A. S. and McDunnough, P. (1984). Further remarks on asymptotic normality of likelihood and conditional analyses., Canad. J. Statist. 12 183–190.
  • [33] Giang, P. H. and Shenoy, P. P. (2005). Decision making on the sole basis of statistical likelihood., Artificial Intelligence 165 137–163.
  • [34] Goutis, C. and Casella, G. (1995). Frequentist post-data inference., Internat. Statist. Rev. 63 325–344.
  • [35] Hacking, I. (1964). On the foundations of statistics., British J. Philos. Sci. 15 1–26.
  • [36] Hill, B. M. (1965). Inference about variance components in the one-way model., J. Amer. Statist. Assoc. 60 806–825.
  • [37] Hills, M. (2005). Likelihood. In, Encyclopedia of Biostatistics, vol. 4, 2nd ed. ( P. Armitage and T. Colton, eds.) 2775–2779. Wiley.
  • [38] Hudson, D. J. (1971). Interval estimation from the likelihood function., J. Roy. Statist. Soc. Ser. B 33 256–262.
  • [39] Joshi, V. M. (1983). Likelihood principle. In, Encyclopedia of Statistical Sciences, vol. 4 ( S. Kotz, N. L. Johnson and C. B. Read, eds.) 644–647. Wiley.
  • [40] Kalbfleisch, J. G. (1985)., Probability and Statistical Inference, vol. 2: Statistical Inference, 2nd ed. Springer.
  • [41] Kiefer, J. (1977). Conditional confidence statements and confidence estimators., J. Amer. Statist. Assoc. 72 789–827.
  • [42] Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters., Ann. Math. Statist. 27 887–906.
  • [43] Klotz, J. H., Milton, R. C. and Zacks, S. (1969). Mean square efficiency of estimators of variance components., J. Amer. Statist. Assoc. 64 1383–1402.
  • [44] Kullback, S. and Leibler, R. A. (1951). On information and sufficiency., Ann. Math. Statist. 22 79–86.
  • [45] Lehmann, E. L. and Romano, J. P. (2005)., Testing Statistical Hypotheses, 3rd ed. Springer.
  • [46] Leonard, T. (1978). Density estimation, stochastic processes and prior information., J. Roy. Statist. Soc. Ser. B 40 113–146.
  • [47] Levy, H. and Sarnat, M. (1970). International diversification of investment portfolios., Amer. Econ. Rev. 60 668–675.
  • [48] Lindsey, J. K. (1996)., Parametric Statistical Inference. Oxford University Press.
  • [49] Lindsey, J. K. (1999). Some statistical heresies., The Statistician 48 1–40.
  • [50] Lindsey, J. K. (2005). Likelihood principle. In, Encyclopedia of Biostatistics, vol. 4, 2nd ed. ( P. Armitage and T. Colton, eds.) 2779–2782. Wiley.
  • [51] Markowitz, H. (1952). Portfolio selection., J. Finance 7 77–91.
  • [52] Montoya, J. A., Díaz-Francés, E. andSprott, D. A. (2009). On a criticism of the profile likelihood function., Statist. Papers 50 195–202.
  • [53] Pawitan, Y. (2001)., In All Likelihood: Statistical Modelling and Inference Using Likelihood. Oxford University Press.
  • [54] Portnoy, S. (1971). Formal Bayes estimation with application to a random effects model., Ann. Math. Statist. 42 1379–1402.
  • [55] Reid, N. (2000). Likelihood., J. Amer. Statist. Assoc. 95 1335–1340.
  • [56] Robinson, G. K. (1979). Conditional properties of statistical procedures., Ann. Statist. 7 742–755.
  • [57] Royall, R. M. (1997)., Statistical Evidence: A Likelihood Paradigm. Chapman & Hall.
  • [58] Schervish, M. J. (1995)., Theory of Statistics. Springer.
  • [59] Searle, S. R., Casella, G. and McCulloch, C. E. (1992)., Variance Components. Wiley.
  • [60] Shilkret, N. (1971). Maxitive measure and integration., Indag. Math. 33 109–116.
  • [61] Sprott, D. A. (2000)., Statistical Inference in Science. Springer.
  • [62] Stone, M. and Springer, B. G. F. (1965). A paradox involving quasi prior distributions., Biometrika 52 623–627.
  • [63] Thompson, W. A. Jr. (1962). The problem of negative estimates of variance components., Ann. Math. Statist. 33 273–289.
  • [64] Tiao, G. C. and Tan, W. Y. (1965). Bayesian analysis of random-effect models in the analysis of variance. I. Posterior distribution of variance-components., Biometrika 52 37–53.
  • [65] van der Vaart, A. W. (1998)., Asymptotic Statistics. Cambridge University Press.
  • [66] Wald, A. (1949). Note on the consistency of the maximum likelihood estimate., Ann. Math. Statist. 20 595–601.
  • [67] Wald, A. (1950)., Statistical Decision Functions. Wiley.