Smooth tail-index estimation

Journal of Statistical Computation and Simulation

Volumn 79, Issue 9, 2009, Pages 1155-1167

  Müller, Samuel ^a   Rufibach, Kaspar ^b  

^a UNIVERSITY OF SYDNEY   (Australia)

^b UNIVERSITY OF ZURICH   (Switzerland)

Author keywords

'Extreme value' theory; Log concave density estimation; Negative Hill estimator; Pickands estimator; Small sample performance; Tail index estimation

Indexed keywords

EID: 70349440808     PISSN: 00949655     EISSN: 15635163     Source Type: Journal    
DOI: 10.1080/00949650802142667     Document Type: Article

References (40)

1. M.M. Dacorogna, U.A. Müller, O.V. Pictet, and C.R. de Vries, The distribution of extremal foreign exchange rate returns in extremely large data sets, Discussion Paper 95-70, Tinbergen Institute, Netherlands, 1995.
2. V. Brazauskas and R. Serfling, Small sample performance of robust estimators of tail parameters for Pareto and exponential models, J. Stat. Comput. Simulation 70 (2001), pp. 1-19.
3. R. Huisman, K.G. Koedijk, C.J.M. Kool, and F. Palm, Tail-index estimates in small samples, J. Bus. Econom. Stat. 19 (2001), pp. 208-216.
4. N. Wagner and T.A. Marsh, Tail index estimation in small samples. Simulation results for independent and ARCH-type financial return models, Stat. Pap. 45 (2004), pp. 545-561.
5. B.V. Gnedenko, Sur la distribution limite du terme maximum d'une série aléatoire, Ann. Math. 44 (1943), pp. 423-453.
6. J. Pickands, Statistical inference using extreme order statistics, Ann. Stat. 3 (1975), pp. 119-131.
7. S. Müller and K. Rufibach, On the max-domain of attraction of distributions with log-concave densities (2008). Stat. Probab. Lett. 78 (2), pp. 1440-1444.
8. S. Csörgö and D.M. Mason, Simple estimators of the endpoint of a distribution, in Extreme Value Theory, J. Hüsler and R.-D. Reiss, eds., Lecture Notes in Statistics, Springer New York, 1989, pp. 132-147.
9. A. Ferreira, L. de Haan, and L. Peng, On optimising the estimation of high quantiles of a probability distribution, Statistics 37 (2003), pp. 401-434.
10. P. Hall, On estimating the endpoint of a distribution, Ann. Stat. 10 (1982), pp. 556-568.
11. P. Hall and J.Z. Wang, Estimating the end-point of a probability distribution using minimum-distance methods, Bernoulli 5 (1999), pp. 177-189.
12. K. Aarssen and L. de Haan, On the maximal life span of humans, Math. Popul. Stud. 4 (1994), pp. 259-281.
13. M. Mayer and I. Molchanov, Limit theorems for the diameter of a random sample in the unit ball, Extremes 10 (2007), pp. 151-174.
14. M.J. Farrell, The measurement of productive efficiency, J. Roy. Stat. Soc. Ser. A 120 (1957), pp. 253-281.
15. S. Kotz and S. Nadarajah, Extreme Value Distributions. Theory and Applications, Imperial College Press, London, 2000.
16. D. Li and L. Peng, Still fit generalized Pareto distributions?, Preprint (2008). Available at http://www.math.gatech.edu/peng/GPD.pdf.
17. B.M. Hill, A simple general approach to inference about the tail of a distribution, Ann. Statist. 3 (1975), pp. 1163-1174.
18. S.G. Coles and M.J. Dixon, Likelihood-based inference for extreme value models, Extremes 2 (1999), pp. 5-23.
19. R.L. Smith, Maximum likelihood estimation in a class of nonregular cases, Biometrika 72 (1985), pp. 67-90.
20. R.L. Smith, Estimating tails of probability distributions, Ann. Stat. 15 (1987), pp. 1174-1207.
21. R.L. Smith and I. Weissman, Maximum likelihood estimation of the lower tail of a probability distribution, J. Roy. Stat. Soc. Ser. B 47 (1985), pp. 285-298.
22. A.L.M. Dekkers, L. de Haan, and J.H.H. Einmahl, A moment estimator for the index of an extreme-value distribution, Ann. Stat. 17 (1989), pp. 1833-1855.
23. M. Falk, Extreme quantile estimation in δ-neighborhoods of generalized Pareto distributions, Stat. Probab. Lett. 20 (1994), pp. 9-21.
24. M. Falk, Some best parameter estimates for distributions with finite endpoint, Statistics 27 (1995), pp. 115-125.
25. S. Müller and J. Hüsler, Iterative estimation of the extreme value index, Methodol. Comput. Appl. Probab. 7 (2005), pp. 139-148.
26. H. Drees, Refined Pickands estimator of the extreme value index, Ann. Stat. 23 (1995), pp. 2059-2080.
27. J. Segers, Generalized Pickands estimators for the extreme value index, J. Stat. Plan. Inference 128 (2005), pp. 381-396.
28. S. Müller, Tail estimation based on numbers of near m-extremes, Methodol. Comput. Appl. Probab. 5 (2003), pp. 197-210.
29. J. Hüsler, D. Li, and S. Müller, Weighted least squares estimation of the extreme value index, Stat. Probab. Lett. 76 (2006), pp. 920-930.
30. J.R.M. Hosking, J.R. Wallis, and E.F. Wood, Estimation of the generalized extreme-value distribution by the method of probability-weighted moments, Technometrics 27 (1985), pp. 251-261.
31. P. Groeneboom, H.P. Lopuhaä, and P.P. de Wolf, Kernel-type estimators for the extreme value index, Ann. Statist. 31 (2003), pp. 1956-1995.
32. S. Csörgö, P. Deheuvels, and D.M. Mason, Kernel estimates of the tail index of a distribution, Ann. Statist. 13 (1985), pp. 1050-1077.
33. L. Dümbgen and K. Rufibach, Maximum likelihood estimation of a log-concave density and its distribution function: basic properties and uniform consistency, Bernoulli, to appear.
34. K. Rufibach, Log-concave density estimation and bump hunting for i.i.d. observations, Ph.D. diss., Universities of Bern and Göttingen, 2006. Available at http://www.stub.unibe.ch/download/eldiss/06rufibach_k.pdf.
35. L. Dümbgen, A. Hüsler, and K. Rufibach, Active set and EM algorithms for log-concave densities based on complete and censored data, preprint (2007). Available at http://arxiv.org/abs/0707.4643.
36. F. Balabdaoui, K. Rufibach, and J.A. Wellner, Limit distribution theory for maximum likelihood estimation of a log-concave density, Ann. Statist., to appear.
37. P. Groeneboom, G. Jongbloed, and J.A. Wellner, Estimation of a convex function: characterization and asymptotic theory, Ann. Statist. 29 (2001), pp. 1653-1698.
38. K. Rufibach and S. Müller, Smoothtail (Smooth estimation of GPD shape parameter), R package version 1.1, 2006; software available at http://www.r-project.org.
39. K. Rufibach and L. Dümbgen, Logcondens (Estimate a log-concave probability density from iid observations), R package version 1.3, 2006; software available at http://www.r-project.org.
40. K. Rufibach, Computing maximum likelihood estimators of a log-concave density function, J. Stat. Comput. Simul. 77 (2007), pp. 561-574.