Hable, Robert A minimum distance estimator in an imprecise probability model – computational aspects and applications. (English) Zbl 1204.62025 Int. J. Approx. Reasoning 51, No. 9, 1114-1128 (2010). Summary: The article considers estimating a parameter \(\theta \) in an imprecise probability model \((\overline{P}_\theta )_{\theta \in \Omega }\) which consists of coherent upper previsions \(\overline{P}_\theta \). After the definition of a minimum distance estimator in this setup and a summarization of its main properties, the focus lies on applications. It is shown that approximate minimum distances on the discretized sample space can be calculated by linear programming. After a discussion of some computational aspects, the estimator is applied in a simulation study consisting of two different models. Finally, the estimator is applied on a real data set in a linear regression model. Cited in 2 Documents MSC: 62F10 Point estimation 62F86 Parametric inference and fuzziness 90C05 Linear programming 62F99 Parametric inference 05C90 Applications of graph theory 65C60 Computational problems in statistics (MSC2010) Keywords:imprecise probability; coherent lower prevision; minimum distance estimator; empirical measure; R project for statistical computing Software:lp_solve; imprProbEst; R; lpSolve; CRAN PDFBibTeX XMLCite \textit{R. Hable}, Int. J. Approx. Reasoning 51, No. 9, 1114--1128 (2010; Zbl 1204.62025) 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] T. Augustin, R. Hable, On the impact of robust statistics on imprecise probability models: a review, Structural Safety (2010), in press.; T. Augustin, R. Hable, On the impact of robust statistics on imprecise probability models: a review, Structural Safety (2010), in press. [4] M. Berkelaar, lpSolve: interface to Lp_solve v. 5.5 to solve linear/integer programs, 2008, Contributed R-Package on CRAN, Version 5.6.3, 2008-05-06; maintainer S. Buttrey.; M. Berkelaar, lpSolve: interface to Lp_solve v. 5.5 to solve linear/integer programs, 2008, Contributed R-Package on CRAN, Version 5.6.3, 2008-05-06; maintainer S. Buttrey. [5] M. Bickis, U. Bickis, Predicting the next pandemic: An exercise in imprecise hazards, in: G. deCooman, J. Vejnarová, M. Zaffalon (Eds.), Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications, SIPTA, Prague, 2007, pp. 41-46.; M. Bickis, U. Bickis, Predicting the next pandemic: An exercise in imprecise hazards, in: G. deCooman, J. Vejnarová, M. Zaffalon (Eds.), Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications, SIPTA, Prague, 2007, pp. 41-46. [6] Dunford, N.; Schwartz, J. T., Linear Operators. I. General Theory (1958), Wiley Interscience Publishers: Wiley Interscience Publishers New York [7] R.Hable, imprProbEst: minimum distance estimation in an imprecise probability model, 2008, Contributed R-Package on CRAN, Version 1.0, 2008-10-23; maintainer R. Hable.; R.Hable, imprProbEst: minimum distance estimation in an imprecise probability model, 2008, Contributed R-Package on CRAN, Version 1.0, 2008-10-23; maintainer R. Hable. [8] R.Hable, Data-based decisions under complex uncertainty, Ph.D. Thesis, Ludwig-Maximilians-Universität (LMU) Munich, 2009. http://edoc.ub.uni-muenchen.de/9874/; R.Hable, Data-based decisions under complex uncertainty, Ph.D. Thesis, Ludwig-Maximilians-Universität (LMU) Munich, 2009. http://edoc.ub.uni-muenchen.de/9874/ [9] Hable, R., Discretization of sample spaces in data-based decision theory under imprecise probabilities, International Journal of Approximate Reasoning, 50, 1115-1128 (2009) · Zbl 1185.62022 [10] Hable, R., Minimum distance estimation in imprecise probability models, Journal of Statistical Planning and Inference, 140, 461-479 (2010) · Zbl 1177.62028 [11] Hutter, M., Practical robust estimators for the imprecise Dirichlet model, International Journal of Approximate Reasoning, 50, 231-242 (2009) · Zbl 1185.62067 [12] Kriegler, E.; Held, H., Climate projections for the 21st century using random sets, (Bernard, J. M.; Seidenfeld, T.; Zaffalon, M., ISIPTA’03, Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, Lugano (2003), Carleton Scientific: Carleton Scientific Waterloo), 345-360 [13] Levi, I., The Enterprise of Knowledge (1980), MIT Press: MIT Press London [14] E. Quaeghebeur, G. deCooman, Imprecise probability models for inference in exponential families, in F.G. Cozman, R. Nau, T. Seidenfeld (Eds.), ISIPTA’05, Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, Pittsburg, SIPTA, Manno, 2005, pp. 287-296.; E. Quaeghebeur, G. deCooman, Imprecise probability models for inference in exponential families, in F.G. Cozman, R. Nau, T. Seidenfeld (Eds.), ISIPTA’05, Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, Pittsburg, SIPTA, Manno, 2005, pp. 287-296. [15] R Development Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2008, ISBN 3-900051-07-0.; R Development Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2008, ISBN 3-900051-07-0. [16] van der Vaart, A. W., Asymptotic Statistics (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0910.62001 [17] Walley, P., Statistical Reasoning with Imprecise Probabilities (1991), Chapman & Hall: Chapman & Hall London · Zbl 0732.62004 [18] 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) · Zbl 0834.62004 [19] Walter, G.; Augustin, T., Imprecision and prior-data conflict in generalized Bayesian inference, Journal of Statistical Theory and Practice: Special Issue on Imprecision, 3, 255-271 (2009) · Zbl 1211.62051 [20] Weichselberger, K., The theory of interval-probability as a unifying concept for uncertainty, International Journal of Approximate Reasoning, 24, 149-170 (2000) · Zbl 0995.68123 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.