×

Minimum distance estimation in imprecise probability models. (English) Zbl 1177.62028

Summary: The present article considers estimating a parameter \(\theta \) in an imprecise probability model \((\overline P_\theta)_{\theta \in \varTheta}\). This model consists of coherent upper previsions \(\overline P_\theta\) which are given by finite numbers of constraints on expectations. A minimum distance estimator is defined in this case and its asymptotic properties are investigated. It is shown that the minimum distance can be approximately calculated by discretizing the sample space. Finally, the estimator is applied in a simulation study and on a real data set.

MSC:

62F12 Asymptotic properties of parametric estimators
62G05 Nonparametric estimation
62F35 Robustness and adaptive procedures (parametric inference)
62G20 Asymptotic properties of nonparametric inference
62G30 Order statistics; empirical distribution functions
62G35 Nonparametric robustness
65C60 Computational problems in statistics (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Augustin, T., Optimale Tests bei Intervallwahrscheinlichkeit (1998), Vandenhoeck & Ruprecht: Vandenhoeck & Ruprecht Göttingen · Zbl 0910.62005
[2] Augustin, T., Neyman-Pearson testing under interval probability by globally least favorable pairs reviewing Huber-Strassen theory and extending it to general interval probability, Journal of Statistical Planning and Inference, 105, 149-173 (2002) · Zbl 1010.62006
[3] Bhaskara Rao, K. P.S.; Bhaskara Rao, M., Theory of Charges (1983), Academic Press Inc.: Academic Press Inc. New York · Zbl 0516.28001
[4] Bickis, M., Bickis, U., 2007. Predicting the next pandemic: an exercise in imprecise hazards. In: de Cooman, G., Vejnarová, J., Zaffalon, M. (Eds.), Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications. SIPTA, Prague, pp. 41-46.; Bickis, M., Bickis, U., 2007. Predicting the next pandemic: an exercise in imprecise hazards. In: de Cooman, G., Vejnarová, J., Zaffalon, M. (Eds.), Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications. SIPTA, Prague, pp. 41-46.
[5] Buja, A., Simultaneously least favorable experiments. I. Upper standard functionals and sufficiency, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 65, 3, 367-384 (1984) · Zbl 0506.62002
[6] De Cooman, G.; Miranda, E., Weak and strong laws of large numbers for coherent lower previsions, Journal of Statistical Planning and Inference, 138, 8, 2409-2432 (2008) · Zbl 1176.60017
[7] Donoho, D.; Liu, R., The “automatic” robustness of minimum distance functionals, The Annals of Statistics, 16, 2, 552-586 (1988) · Zbl 0684.62030
[8] Dunford, N.; Schwartz, J., Linear Operators. I. General Theory (1958), Wiley-Interscience Publishers: Wiley-Interscience Publishers New York
[9] Fan, K., Minimax theorems, Proceedings of the National Academy of Sciences of the United States of America, 39, 42-47 (1953) · Zbl 0050.06501
[10] Hable, R., 2008. imprProbEst: minimum distance estimation in an imprecise probability model. Contributed R-Package on CRAN, Version 1.0, 2008-10-23; maintainer Hable, R.; Hable, R., 2008. imprProbEst: minimum distance estimation in an imprecise probability model. Contributed R-Package on CRAN, Version 1.0, 2008-10-23; maintainer Hable, R.
[11] Hable, R., 2009a. Data-based decisions under complex uncertainty. Ph.D. Thesis, Ludwig-Maximilians-Universität (LMU) Munich. URL \(\langle;\) http://edoc.ub.uni-muenchen.de/\(9874/ \rangle;\).; Hable, R., 2009a. Data-based decisions under complex uncertainty. Ph.D. Thesis, Ludwig-Maximilians-Universität (LMU) Munich. URL \(\langle;\) http://edoc.ub.uni-muenchen.de/\(9874/ \rangle;\). · Zbl 1277.62018
[12] Hable, R., Discretization of sample spaces in data-based decision theory under imprecise probabilities, International Journal of Approximate Reasoning, 50, 7, 1115-1128 (2009) · Zbl 1185.62022
[13] Hable, R., 2009c. A minimum distance estimator in an imprecise probability model—computational aspects and applications. In: 6th International Symposium on Imprecise Probability: Theories and Applications (ISIPTA’09). URL \(\langle;\) http://www.sipta.org/isipta09/proceedings/005.html \(\rangle;\).; Hable, R., 2009c. A minimum distance estimator in an imprecise probability model—computational aspects and applications. In: 6th International Symposium on Imprecise Probability: Theories and Applications (ISIPTA’09). URL \(\langle;\) http://www.sipta.org/isipta09/proceedings/005.html \(\rangle;\). · Zbl 1204.62025
[14] Hampel, F.; Ronchetti, E.; Rousseeuw, P.; Stahel, W., Robust Statistics (1986), Wiley: Wiley New York
[15] Hoffmann-Jørgensen, J., Probability with a View Toward Statistics, vol. I (1994), Chapman & Hall: Chapman & Hall New York · Zbl 0821.62003
[16] Hoffmann-Jørgensen, J., Probability with a View Toward Statistics, vol. II (1994), Chapman & Hall: Chapman & Hall New York · Zbl 0821.62003
[17] Huber, P., A robust version of the probability ratio test, Annals of Mathematical Statistics, 36, 1753-1758 (1965) · Zbl 0137.12702
[18] Huber, P., Robust Statistics (1981), Wiley: Wiley New York · Zbl 0536.62025
[19] Huber, P., 1997. Robust statistical procedures. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 68, second ed. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA.; Huber, P., 1997. Robust statistical procedures. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 68, second ed. SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA. · Zbl 0859.62003
[20] Huber, P.; Strassen, V., Minimax tests and the Neyman-Pearson lemma for capacities, The Annals of Statistics, 1, 251-263 (1973) · Zbl 0259.62008
[21] Hutter, M., Practical robust estimators for the imprecise Dirichlet model, International Journal of Approximate Reasoning, 50, 2, 231-242 (2009) · Zbl 1185.62067
[22] Kohl, M., 2005. Numerical contributions to the asymptotic theory of robustness. Ph.D. Thesis, Universität Bayreuth.; Kohl, M., 2005. Numerical contributions to the asymptotic theory of robustness. Ph.D. Thesis, Universität Bayreuth. · Zbl 1189.62051
[23] Kriegler, E., Held, H., 2003. Climate projections for the 21st century using random sets. In: Bernard, J., Seidenfeld, T., Zaffalon, M. (Eds.), ISIPTA’03, Proceedings of the Third International Symposium on Imprecise Probabilities and their Applications, Lugano. Carleton Scientific, Waterloo, pp. 345-360.; Kriegler, E., Held, H., 2003. Climate projections for the 21st century using random sets. In: Bernard, J., Seidenfeld, T., Zaffalon, M. (Eds.), ISIPTA’03, Proceedings of the Third International Symposium on Imprecise Probabilities and their Applications, Lugano. Carleton Scientific, Waterloo, pp. 345-360.
[24] Marazzi, A., 1993. Algorithms, routines, and S functions for robust statistics. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA. The FORTRAN library ROBETH with an interface to S-PLUS. With the collaboration of Johann Joss and Alex Randriamiharisoa. With a separately available computer disk.; Marazzi, A., 1993. Algorithms, routines, and S functions for robust statistics. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA. The FORTRAN library ROBETH with an interface to S-PLUS. With the collaboration of Johann Joss and Alex Randriamiharisoa. With a separately available computer disk. · Zbl 0777.62004
[25] Millar, P., Robust estimation via minimum distance methods, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 55, 1, 73-89 (1981) · Zbl 0461.62036
[26] Öztürk, Ö.; Hettmansperger, T., Simultaneous robust estimation of location and scale parameters: a minimum-distance approach, The Canadian Journal of Statistics, 26, 2, 217-229 (1998) · Zbl 0914.62020
[27] Parr, W.; Schucany, W., Minimum distance and robust estimation, Journal of the American Statistical Association, 75, 371, 616-624 (1980) · Zbl 0481.62031
[28] Quaeghebeur, E., de Cooman, G., 2005. Imprecise probability models for inference in exponential families. In: Cozman, F., Nau, R., Seidenfeld, T. (Eds.), ISIPTA’05, Proceedings of the Fourth International Symposium on Imprecise Probabilities and their Applications, Pittsburg. SIPTA, Manno, pp. 287-296.; Quaeghebeur, E., de Cooman, G., 2005. Imprecise probability models for inference in exponential families. In: Cozman, F., Nau, R., Seidenfeld, T. (Eds.), ISIPTA’05, Proceedings of the Fourth International Symposium on Imprecise Probabilities and their Applications, Pittsburg. SIPTA, Manno, pp. 287-296.
[29] Rieder, H., Robust Asymptotic Statistics (1994), Springer: Springer New York · Zbl 0927.62050
[30] Tukey, J., A survey of sampling from contaminated distributions, (Contributions to Probability and Statistics (1960), Stanford University Press: Stanford University Press Stanford, CA), 448-485
[31] van der Vaart, A., Asymptotic Statistics (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0910.62001
[32] van der Vaart, A., Wellner, J., 1996. Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York.; van der Vaart, A., Wellner, J., 1996. Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York. · Zbl 0862.60002
[33] Walley, P., Statistical Reasoning with Imprecise Probabilities (1991), Chapman & Hall: Chapman & Hall London · Zbl 0732.62004
[34] Walley, P., Inferences from multinomial data: learning about a bag of marbles, Journal of the Royal Statistical Society. Series B. Methodological, 58, 1, 3-57 (1996), with discussion and a reply by the author · Zbl 0834.62004
[35] Walter, G.; Augustin, T., Imprecision and prior-data conflict in generalized Bayesian inference, Journal of Statistical Theory and Practice, 3, 255-271 (2009) · Zbl 1211.62051
[36] Weichselberger, K., 2001. Elementare Grundbegriffe einer allgemeineren Wahrscheinlichkeitsrechnung I. Intervallwahrscheinlichkeit als umfassendes Konzept. Physica, Heidelberg.; Weichselberger, K., 2001. Elementare Grundbegriffe einer allgemeineren Wahrscheinlichkeitsrechnung I. Intervallwahrscheinlichkeit als umfassendes Konzept. Physica, Heidelberg. · Zbl 0979.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.