Abstract
Nonparametric estimation of the density of a heavy-tailed probability distribution is investigated. The initial data are transformed to a bounded interval and the distribution density is determined by an inverse transformation of the distribution density estimate of transformed data. An adaptive data transformation is studied, in which the order of decay of the tail of the true distribution density is preserved and stable estimation of the deviation in tail index estimates is guaranteed. In classification, the empirical risk of erroneous classification by the Bayes empirical classifier is used as a measure for the quality of distribution density estimates.
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REFERENCES
Sigman, K., Appendix: A Primer on Heavy-Tailed Distributions, Queueing Syst., 1999, no. 33, pp. 261–275.
Gnedenko B.V., Sur la distribution limite du terme maximum d'une serie ale'atoire, Ann. Math., 1943, no. 44, pp. 423–453.
Devroye, L. and Györfi, L., Nonparametric Density Estimation: The L 1 View, New York: Wiley, 1985.
Wand, M.P., Marron, J.S., and Ruppert, D., Transformations in Density Estimation, J. Am. Statist. Ass., 1991, no. 86(414), pp. 343–353.
Hill, B.M., A Simple General Approach to Inference about the Tail of a Distribution, Ann. Statist., 1975, no. 3, pp. 1163–1174.
Pickands, J., Statistical Inference using Extreme Order Statistics, Ann. Statist., 1975, no. 3, pp. 119–131.
Novak, S.Y., Berry-Esseen Inequalities for a Ratio of Sums of Random Variables, Univ. of Sussex, Research Report No. 98/07 SMS/CSSM, 1998.
Tarasenko, F.P., Neparametricheskaya statistika (Nonparametric Statistics), Tomsk: Tomsk. Gos. Univ., 1976.
Berlinet, A., Vajda, I., and van der Meulen, E.C., About the Asymptotic Accuracy of Barron Density Estimates, IEEE Trans. Inf. Theory, 1998, no. 44, pp. 999–1009.
Markovich, N.M., Experimental Analysis of Nonparametric Probability Density Estimates and Methods for Smoothing Them, Autom. Remote Control, 1989, no. 50, pp. 941–948.
Caers, J. and Van Dyck, J., Nonparametric Tail Estimation using a Double Bootstrap Method, Comput. Stat. Data Anal., 1999, no. 29, pp. 191–211.
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Markovich, N.M. Transformed Estimates of Densities of Heavy-Tailed Distributions and Classification. Automation and Remote Control 63, 627–640 (2002). https://doi.org/10.1023/A:1015182232410
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DOI: https://doi.org/10.1023/A:1015182232410