Semiparametric lower bounds for tail index estimation |
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| Institution: | 1. Department Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Leuven, Belgium;2. Institut de Statistique, Université Libre de Bruxelles, Campus de la Plaine, CP210, B-1050 Bruxelles, Belgium;3. Finance and Econometrics group, Center, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands;1. Department of Statistics, Pennsylvania State University, PA, 16802, USA;2. School of Finance and Statistics, Hunan University, Changsha, 410082, China;1. Department of Mathematics, University of Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria;2. Statistical Inverse Problems in Biophysics, Max Planck Institute for Biophysical Chemistry, Am Faßberg 11, D-37077 Göttingen, Germany;3. Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstraße 7, D-37077 Göttingen, Germany;1. Instituto de Cálculo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Buenos Aires, Argentina;2. Departamento de Ciencias Exactas, Ciclo Básico Común, Universidad de Buenos Aires, Ciudad Universitaria, Buenos Aires, Argentina;1. Department of Statistics, Zhejiang Agriculture and Forestry University, Hangzhou, 311300, China;2. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China;3. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China |
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| Abstract: | We consider estimation of the tail index parameter from i.i.d. observations in Pareto and Weibull type models, using a local and asymptotic approach. The slowly varying function describing the non-tail behavior of the distribution is considered as an infinite dimensional nuisance parameter. Without further regularity conditions, we derive a local asymptotic normality (LAN) result for suitably chosen parametric submodels of the full semiparametric model. From this result, we immediately obtain the optimal rate of convergence of tail index parameter estimators for more specific models previously studied. On top of the optimal rate of convergence, our LAN result also gives the minimal limiting variance of estimators (regular for our parametric model) through the convolution theorem. We show that the classical Hill estimator is regular for the submodels introduced with limiting variance equal to the induced convolution theorem bound. We also discuss the Weibull model in this respect. |
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