Abstract
We outline an approach to statistical inference based on belief functions. For estimation, a consonant belief functions is constructed from the likelihood function. For prediction, the method is based on an equation linking the unobserved random quantity to be predicted, to the parameter and some underlying auxiliary variable with known distribution. The approach allows us to compute a predictive belief function that reflects both estimation and random uncertainties. The method is invariant to one-to-one transformations of the parameter and compatible with Bayesian inference, in the sense that it yields the same results when provided with the same information. It does not, however, require the user to provide prior probability distributions. The method is applied to stochastic frontier analysis with cross-sectional data. We demonstrate how predictive belief functions on inefficiencies can be constructed for this problem and used to assess the plausibility of various assertions.
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References
Aickin, M.: Connecting Dempster-Shafer belief functions with likelihood-based inference. Synthese 123, 347–364 (2000)
Aigner, D.J., Lovell, C.A.K., Schmidt, P.: Formulation and estimation of stochastic frontier production function models. J. Econom. 6, 21–37 (1977)
Barnard, G.A., Jenkins, G.M., Winsten, C.B.: Likelihood inference and time series. J. R. Stat. Soc. 125(3), 321–372 (1962)
Bayarri, M.J., DeGroot, M.H.: Difficulties and ambiguities in the definition of a likelihood function. J. Ital. Stat. Soc. 1(1), 1–15 (1992)
Ben Abdallah, N., Mouhous Voyneau, N., Denœux, T.: Combining statistical and expert evidence using belief functions: application to centennial sea level estimation taking into account climate change. Int. J. Approx. Reason. 55, 341–354 (2014)
Berger, J.O., Wolpert, R.L.: The likelihood principle: A Review, Generalizations, and Statistical Implications, 2nd edn. Lecture Notes-Monograph Series, vol 6. Institute of Mathematical Statistics, Hayward (1988)
Birnbaum, A.: On the foundations of statistical inference. J. Am. Stat. Assoc. 57(298), 269–306 (1962)
Bjornstad, J.F.: Predictive likelihood: a review. Stat. Sci. 5(2), 242–254, 05 (1990)
Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38, 325–339 (1967)
Dempster, A.P.: A generalization of Bayesian inference (with discussion). J. R. Stat. Soc. B 30, 205–247 (1968)
Denœux, T.: Likelihood-based belief function: justification and some extensions to low-quality data. Int. J. Approx. Reason. 55(7), 1535–1547 (2014)
Dubois, D., Prade, H.: A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets. Int. J. Gen. Syst. 12(3), 193–226 (1986)
Edwards, A.W.F.: Likelihood (expanded edn.). The John Hopkins University Press, Baltimore (1992)
Greene, W.H.: Econometric Analysis, 6th edn. Prentice Hall, Upper Saddle River (2008)
Jondrow, J., Lovell, C.A.K., Materov, I.S., Schmidt, P.: On the estimation of technical efficiency in the stochastic production function model. J. Econom. 19, 233–238 (1982)
Kanjanatarakul, O., Sriboonchitta, S., Denœux, T.: Forecasting using belief functions: an application to marketing econometrics. Int. J. Approx. Reason. 55(5), 1113–1128 (2014)
Kanjanatarakul, O., Lertpongpiroon P., Singkharat S. and Sriboonchitta, S.: Econometric forecasting using linear regression and belief functions, In: Cuzzolin, F. (ed.), Belief Functions: Theory and Applications. Oxford, Springer, LNAI 8764, pp. 304–312 (2014)
Mathiasen, P.E.: Prediction functions. Scand. J. Stat. 6(1), 1–21 (1979)
Robbins, H.: An empirical Bayes approach to statistics. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Volume 1: Contributions to the Theory of Statistics, pp. 157–163 (1956)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Smets, Ph: Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. Int. J. Approx. Reason. 9, 1–35 (1993)
Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)
Wasserman, L.A.: Belief functions and statistical evidence. Can. J. Stat. 18(3), 183–196 (1990)
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Kanjanatarakul, O., Kaewsompong, N., Sriboonchitta, S., Denœux, T. (2015). Estimation and Prediction Using Belief Functions: Application to Stochastic Frontier Analysis. In: Huynh, VN., Kreinovich, V., Sriboonchitta, S., Suriya, K. (eds) Econometrics of Risk. Studies in Computational Intelligence, vol 583. Springer, Cham. https://doi.org/10.1007/978-3-319-13449-9_12
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