Abstract
In this paper we suggest several nonparametric quantile estimators based on Beta kernel. They are applied to transformed data by the generalized Champernowne distribution initially fitted to the data. A Monte Carlo based study has shown that those estimators improve the efficiency of the traditional ones, not only for light tailed distributions, but also for heavy tailed, when the probability level is close to 1. We also compare these estimators with the Extreme Value Theory Quantile applied to Danish data on large fire insurance losses.
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Azzalini, A.: A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika 68, 326–328 (1981)
Behrens, C.N., Lopes, H.F., Gamerman, D.: Bayesian analysis of extreme events with threshold estimation. Stat. Model. 3, 227–244 (2004)
Beirlant, J., Goegebeur, Y., Segers, J., Teugel, J.: Statistics of Extremes. Wiley Interscience, New York (2006)
Björk, T.: Arbitrage Theory in Continuous Time. Oxford University Press, London (1998)
Bolancé, C., Guillen, M., Nielsen, J.P.: Kernel density estimation of actuarial loss functions. Insur. Math. Econ. 32, 19–36 (2003)
Bouezmarni, T., Scaillet, O.: Consistency of asymmetric kernel density estimators and smoothed histograms with application to income data. Econom. Theory 21, 390–412 (2005)
Brodin, E.: On quantile estimation by bootstrap. Comput. Stat. Data Anal. 50, 1398–1406 (2006)
Buch-Larsen, T., Nielsen, J.P., Guillen, M., Bolance, C.: Kernel density estimation for heavy-tailed distribution using the Champernowne transformation. Statistics 6, 503–518 (2005)
Champernowne, D.G.: The Oxford meeting, September 25–29, by Brown P. Econometrica 5, 361–383 (1936)
Champernowne, D.G.: The graduation of income distributions. Econometrica 20, 591–615 (1952)
Charpentier, A., Fermanian, J.D., Scaillet, O.: The estimation of copulas: theory and practice. In: Copula Methods in Derivatives and Risk Management: From Credit Risk to Market Risk. Risk Book (2006)
Charpentier, A., Oulidi, A.: Estimating allocations for Value-at-Risk portfolio optimization. Math. Methods Oper. Res. (2009, forthcoming)
Chen, S.X.: A beta kernel estimator for density functions. Comput. Stat. Data Anal. 31, 131–145 (1999)
Chen, S.X., Tang, C.Y.: Nonparametric inference of values at risk for dependent financial returns. J. Financ. Econom. 3, 227–255 (2005)
Danielsson, J., de Haan, L., Peng, L., de Vries, C.G.: Using a bootstrap method to choose the sample fraction in tail index estimation. J. Multivar. Anal. 76, 226–248 (2001)
Devroye, L., Györfi, L.: Nonparametric Density Estimation: The L1 View. Wiley, New York (1985)
Embrechts, P., Klüppelberg, K., Mikosch, T.: Modeling Extremal Events for Insurance and Finance. Springer, Berlin (1997)
Fishman, G.S.: Monte Carlo: Concepts, Algorithms, and Applications. Springer Series in Operations Research. Springer, New York (1995)
Gençay, R., Selçuk, F.: Extreme value theory and value-At-Risk: Relative performance in emerging markets. Int. J. Forecast. 20, 287–303 (2004)
Gouriéroux, C., Laurent, J.P., Scaillet, O.: Sensitivity analysis of Values at Risk. J. Empir. Finance 7, 225–245 (2000)
Gouriéroux, C., Montfort, A.: (Non) Consistency of the Beta Kernel Estimator for Recovery Rate Distribution. CREST-DP 2006-32 (2006)
Gustafsson, J., Hagmann, M., Nielsen, J.P., Scaillet, O.: Local transformation kernel density estimation of loss. J. Bus. Econ. Stat. (2009, forthcoming)
Harrell, F.E., Davis, C.E.: A new distribution-free quantile estimators. Biometrika 69, 635–640 (1982)
Hass, C.N.: Importance of distribution form in characterizing inputs to Monte Carlo risk assessment. Risk Anal. 17(1), 107–113 (1997)
Hyndman, R.J., Fan, Y: Sample quantiles in statistical packages. Am. Stat. 50, 361–365 (1996)
Jones, M.C., Linton, O.B., Nielsen, J.P.: A simple and effective reduction method for density estimation. Biometrika 82, 93–1001 (1995)
Jorion, P.: Value-At-Risk. McGraw-Hill, New York (2000)
Lehmann, E.L.: Theory of Point Estimation. Wadsworth and Brooks/Cole, Belmont (1991)
Matthys, G., Beirlant, J.: Estimating the extreme value index and high quantiles with exponential regression models. Stat. Sin. 13, 853–880 (2003)
McNeil, A.J.: Estimating the tails of loss severity distributions using extreme value theory. In: 28th International ASTIN Colloquium (1997)
McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton (2005)
Modarres, R., Nayak, T.K., Gastwirth, J.L.: Estimation of upper quantiles under model and parameter uncertainty. Comput. Stat. Data Anal. 28, 529–554 (2002)
Nadaraya, E.A.: On estimating regression. Theory Probab. Appl. 10, 186–190 (1964)
Padgett, W.J.: A kernel-type estimator of a quantile function from right censored data. J. Am. Stat. Assoc. 81, 215–222 (1986)
Park, C.: Smooth nonparametric estimation of a quantile function under right censoring using beta kernels. Technical Report (TR 2006-01-CP), Department of Mathematical Sciences, Clemson University (2006)
Parzen, E.: Nonparametric statistical data modelling. J. Am. Stat. Assoc. 74, 105–131 (1979)
Ralescu, S.S., Sun, S.: Necessary and sufficient conditions for the asymptotic normality of perturbed sample quantiles. J. Stat. Plan. Inference 35, 55–64 (1993)
Rayner, G.D., MacGillivray, H.L.: Weighted quantile-based estimation for a class of transformation distributions. Comput. Stat. Data Anal. 39, 401–433 (2002)
Sheather, S.J., Marron, J.S.: Kernel quantile estimators. J. Am. Stat. Assoc. 85, 410–416 (1990)
Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Chapman & Hall, London (1986)
Wand, M., Jones, M.C.: Kernel Smoothing. Chapman & Hall, London (1995)
Wand, M.P., Marron, J.S., Ruppert, D.: Transformations in density estimation. J. Am. Stat. Assoc. 86, 343–361 (1991)
Wang, S.: Insurance pricing and increased limits ratemaking by proportional hazard transforms. Insur. Math. Econ. 17, 43–54 (1995)
Wang, S.: Premium calculation by transforming the layer premium density. ASTIN Bull. 26, 71–92 (1996)
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Charpentier, A., Oulidi, A. Beta kernel quantile estimators of heavy-tailed loss distributions. Stat Comput 20, 35–55 (2010). https://doi.org/10.1007/s11222-009-9114-2
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DOI: https://doi.org/10.1007/s11222-009-9114-2