Abstract
Small samples are a challenge in extreme value theory. Asymptotic results do not apply and many estimation techniques, e.g. maximum likelihood, are unstable. In such situations, imposing qualitative constraints on the empirical distribution function is known to greatly reduce variability. Distribution functions typically appearing in the extreme-value theory, e.g. the generalized extreme-value distribution or the generalized Pareto distribution, have monotone upper tails. Applying monotone density estimation to parts of initial kernel density estimators leads to partially smooth estimated distribution functions. Particularly in small samples, replacing the order statistics in tail-index estimators by their corresponding quantiles from partially smooth estimated distribution functions leads to improved tail-index estimators. Monte Carlo simulations demonstrate that the partially smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean-squared error.
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References
Beirlant J, Goegebeur Y, Segers J, Teugels J (2004) Statistics of extremes: theory and applications. Wiley, England
Birke M (2009) Shape constrained kernel density estimation. J Statist Plann Inference 139(8): 2851–2862
Birke M, Dette H (2008) A note on estimating a smooth monotone regression by combining kernel and density estimates. J Nonparametr Stat 20(8): 679–691
Chhay H (2009) Tail-index and quantile estimation for small samples. Honours thesis, University of Sydney, School of Mathematics and Statistics
Csörgő S, Deheuvels P, Mason D (1985) Kernel estimates of the tail index of a distribution. Ann Statist 13: 1050–1077
Dacorogna MM, Müller UA, Picted OV, de Vries CR (1995) The distribution of extremal foreign exchange rate returns in extremely large data sets. Discussion Paper 95–70, Tinbergen Institute, Netherlands
Deidda R, Puliga M (2009) Performances of some parameter estimators of the generalized pareto distribution over rounded-off samples. Phys Chem Earth 34: 626–634
Dette H, Neumeyer N, Pilz KF (2006) A simple nonparametric estimator of a strictly monotone regression function. Bernoulli 12: 469–490
Epanechnikov VA (1969) Nonparametric estimates of a multivariate probability density. Theor Probab Appl 14: 153–158
Furrer R, Naveau P (2007) Probability weighted moments properties of small samples. Statist Probab Lett 77: 190–195
Gasser T, Müller HG (1979) Kernel estimates of regression functions. In: Gasser T, Rosenblatt M (eds) Smoothing techniques for curve estimation, Lecture Notes in Math. 757. Springer, Berlin
Gnedenko BV (1943) Sur la distribution limite du terme maximum dune série aléatoire. Ann of Math 2(44): 423–453
Greenwood JA, Landwehr JM, Matalas NC, Wallis JR (1979) Probability weighted moments: definition and relation to parameters of several distributions expressable in inverse form. Water Resour Res 15: 1049–1054
Grenander U (1956) On the theory of mortality measurement. ii. Skand Aktuarietidskr 39: 125–153
Groeneboom P, Lopuhaä HP, de Wolf PP (2003) Kernel-type estimators for the extreme value index. Ann Statist 31: 1956–1995
Hosking JRM, Wallis JR (1987) Parameter and quantile estimation for the generalized pareto distribution. Technometrics 29: 339–349
Hosking JRM, Wallis JR, Wood EF (1985) Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27: 251–261
Hosking JRM (1990) L-Moments: Analysis and estimation of distributions using linear combinations of order statistics. J Roy Statist Soc Ser B 52: 105–124
Huisman R, Koedijk KG, Kool CJM, Palm F (2001) Tail-index estimation in small samples. J Business Econ Statist 19: 208–216
Kotz S, Nadarajah S (2000) Extreme value distributions. Imperial College Press, London
Landwehr JM, Matalas NC, Wallis JR (1979) Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resour Res 15: 1055–1064
Müller HG (1985) Kernel estimators of zeros and of location and size of extrema of regression functions. Scand J Statist 12: 221–232
Müller S, Rufibach K (2008) On the max-domain of attraction of distributions with log-concave densities. Statist Probab Lett 78(12): 1440–1444
Müller S, Rufibach K (2009) Smooth tail-index estimation. J Stat Comput Simulat 79: 1155–1167
Pickands J (1975) Statistical inference using extreme order statistics. Ann Statist 3: 119–131
Reiss RD, Thomas M (2007) Statistical analysis of extreme values. With applications to insurance, Finance, hydrology and other fields, 3rd edn. Birkhäuser Verlag AG, Basel
Scheder R (2009) monoProc: strictly monotone and smooth regression estimation in R. R package version 1.0-6
Scott DW (2008) Multivariate density estimation: theory, practice, and visualization. Wiley, Hoboken
Segers J (2005) Generalized pickands estimators for the extreme value index. J Statist Plann Inference 128(2): 381–396
Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall, London
Smith RL (1985) Maximum likelihood estimation in a class of nonregular cases. Biometrika 72: 67–90
Wand MP, Jones MC (1995) Kernel smoothing. Chapman & Hall, London
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Müller, S., Chhay, H. Partially smooth tail-index estimation for small samples. Comput Stat 26, 491–505 (2011). https://doi.org/10.1007/s00180-010-0221-5
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DOI: https://doi.org/10.1007/s00180-010-0221-5