Statistical tests on tail index of a probability distribution

Metron

Volumn 61, Issue 2, 2003, Pages 151-190

  Jurečková, Jana ^a  

^a CHARLES UNIVERSITY   (Czech Republic)

Author keywords

Autoregressive time series; Empirical distribution function; Extreme regression quantile; Gumbel hypothesis; Heavy tailed distribution; Pareto tail index

Indexed keywords

EID: 33845644354     PISSN: 00261424     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (78)

1. Beirlant, J., Teugels, J.L., and Vynckier, P. (1996) Practical Analysis of Extreme Values, Leuven University Press.
2. Brockwell, P.J. and Davis, R.A. (1991) Time Series: Theory and Methods. 2nd Edition, Springer Verlag, New York.
3. Castillo, E. (1988) Extreme Value Theory in Engineering, Academic Press.
4. Castillo, E., Calambos, J., and Sarabia, J.M. (1989) The selection of the domain of attraction of an extreme value distribution from a data set. Extreme Value Theory (Proceedings of Oberwolfach 1987, J. Hüsler and R.-D. Reiss, eds.), Lecture Notes in Statistics 51, 181-190. Springer, New York.
5. Dasgupta, R. and Bhaumik, D.K. (1995) Upper and lower tolerance limits of athmospheric ozone level and extreme value distribution, Sankhya, B, 57, 182-199.
6. Davis, R.A., Knight, K., and Liu. J. (1992) M-estimation for autoregressions with infinite variance, J. Stochastic Process. Appl., 40, 145-180.
7. Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance, Springer-Verlag.
8. Epstein, B. (1948) Application to the theory of extreme values in fracture problems, J. Amer. Statist. Assoc., 43, 403-412.
9. Falk, M. (1995a) On testing the extreme value index via the POT-method, Ann. Statist., 23, 2013- 2035.
10. Falk, M. (1995b) LAN of extreme order statistics, Ann. Inst. Statist. Math., 47, 693-717.
11. Fialov á, A., Jurečková, J., and Picek J. (2003) Estimating Pareto tail index using all observations, submitted.
12. Fuller, W.E. (1914) Flood flows, Trans. Amer. Soc. Civil Engineers, 77, 564.
13. Galambos, J. (1982) A statistical test for extreme value distribution. Non-parametric Statistical Inference, (B.V. Gnedenko et al., eds.), pp. 221-230. North Holland, Amsterdam.
14. Gomes, M. I. (1989) Comparison od extremal models through statistical choice in multidimensional background. Extreme Value Theory, (Proceedings of Oberwolfach 1987, J. Hüsler and R.-D. Reiss, eds.). Lecture Notes in Statistics 51, 191-203. Springer, New York.
15. Gomes, M.I. and Alpuim, M.T. (1986) Inference in a multivariate generalized extreme value model - asymptotic properties of two test statistics, Scand. J. Statist., 13, 291-300.
16. Griffith, A.A. (1920) The phenomena of rupture and flow in solids, Philos. Trans. Roy. Soc. London A, 221, 163-198.
17. Gumbel, E.J. (1941) The return period of flood flows, Ann. Math. Statist., 12, 163-190.
18. Gumbel, E.J. (1944) On the plotting pf flood discharges, Trans. Amer. Geophys. Union, 25, 699-719.
19. Gumbel, E.J. (1945) Floods estimated by probability methods, Engrg. News Record, 143, 97-101.
20. Gumbel, E. J. (1949a) The statistical forecast of floods, Ohio Water Resources Board Bulletin, 15, 1-21.
21. Hájek, J. (1970) Miscellaneous problems of rank test theory. Nonparametric Technique in Statiatical Inference, (M. L. Puri, ed.), pp. 3-19. Cambridge University Press.
22. Hannan, E.J. and Kanter, M. (1977) Autoregressive processes with infinite variance, J. Appl. Probability, 14, 411-415.
23. Hasofer, A.M. and Wang, Z. (1992) A test for extreme value domain of attraction, J. Amer. Statist. Assoc., 87, 171-177.
24. He, X., Jurečková, J., Koenker, R., and Portnoy, S. (1990) Tail behavior of regression estimators and their breakdown points, Econometrica, 58, 1195-1214.
25. Hosking, J.R.M. (1984) Testing whether the shape parameter is zero in generalized extreme value distribution, Biometrika, 71, 367-374.
26. Jurečková, J. (1999) Tests of tails based on extreme regression quantiles, Statist. & Probab. Letters, 49, 53-61.
27. Jurečková, J., Koul, H.L., and Picek, J. (2003) Testing the tail index in autoregressive models, submitted.
28. Jurečková, J. and Picek, J. (2001) A class of tests on the tail index, Extremes, 4:2, 165-183.
29. King, J.R. (1959) Summary of extreme-value theory and its relation to reliability analysis, Proc. 12th Annual Conference of the American Society for Quality Control, 13, 163-167.
30. Knight, K. (1987) Rate of convergence of centred estimates of autoregressive parameters for infinite variance autoregressions, J. Time Ser. Anal., 8, 51-60.
31. Koenker, R. and Bassett, G. (1978) Regression Quantiles, Econometrica, 46, 33-50.
32. Kotz, S. and S. Nadarajah (2000) Extreme Value Distributions. Theory and Applications, Imperial College Press, London (2nd Edition 2002).
33. Koul, H.L. (2002) Weighted empirical processes in dynamic nonlinear models, Lecture Notes in Statistics 166, Springer-Verlag.
34. Marohn, F. (1994) On testing exponential and Gumbel distribution, Extreme Value Theory, (J. Galambos et al., eds.) 159-174. Kluwer, Dordrecht.
35. Marohn, F. (1998a) Testing the Gumbel hypothesis via the POT-method, Extremes, 1, 191-213.
36. Marohn, F. (1998b) An adaptive efficient test for Gumbel domain of attraction, Scand. J. Statist., 25, 311-324.
37. Montfort, M.A.J. van and Gomes, M.I. (1985) Statistical choice of extremal models for complete and censored data, J. Hydrology, 77, 77-87.
38. Montfort, M.A.J. van and Witter, J.V. (1985) Testing exponentially against generalized Pareto distribution, J. Hydrology, 78, 305-315.
39. Nordquist, J.M. (1945) Theory of largest values, applied to earthquake magnitudes, Trans. Amer. Geophys. Union, 26, 29-31.
40. Tiago de Oliveira, J. (1984) Univariate extremes: Statistical choice. Statistical Extremes and Applications, (J. Tiago de Oliveira, ed.), 91-107. Reidel, Dodrecht.
41. Picek, J. and Jurečková, J. (2001) A class of tests on the tail index using the modified extreme regression quantiles, In ROBUST'2000 (J. Antoch, G. Dohnal, Eds.), Union of Czech Mathematicians and Physicists, Prague, 217-226.
42. Posner, E.C. (1965) The application of extreme-value theory to error-free communication, Technometrics, 7, 517-529.
43. Potter, W.D. (1949) Normalcy test of precipitation and frequency studies of runoff on small watersheds, U.S. Department of Agriculture Technical Bulletin, No. 985.
44. Portnoy, S. and Jurečková, J. (1999) On extreme regression quantiles, Extremes, 2:3, 227-243.
45. Resnick, S.I. and Feigin, P.D. (1994) Limit distributions for linear programming time series estimators, J. Stochast. Process. & their Appl., 51, 135-165.
46. Rantz, S.F. and Riggs, H.C. (1949) Magnitude and frequency of floods in the Columbia river basin, U.S. Geological Survey, Water Supply Paper 1080, 317-476.
47. Smith, R. (1994) Nonregular regression, Biometrika, 81, 173-183.
48. Stephens, M.A. (1977) Goodness-of-fit for the extreme value distribution, Biometrika, 64, 583-588.
49. Teugels, J.L. and Beirlant, J. (1993) Extremes in insurance. Conference on Extreme Value Theory and its Applications, May 2-7, 1993, NIST, Gaitershurg, MD.
50. Weibull, W. (1939) A statistical theory of the strength of the material, Ing. Vet. Akad. Handlingar, 151.
51. Weibull, W. (1939) The phenomenon of rupture in solids, Ing. Vet. Akad. Handlingar, 152, 2.
52. Wiggins, J.B. (1991) Empirical tests of the bias and efficiency of the extreme-value variance estimator for common stocks, J. Business Univ. Chicago, 64, 417-432.
53. Beirlant, J., Dierckx, G., Goegebeur, Y., and Matthys, G. (1999) Tail index estimation and an exponential regression model, Extremes, 2, 177-200.
54. Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution, Annals of Statistics, 3, 1163-1174.
55. Barbe, Ph. and Broniatowski, M. (2003) Influence fraction of a sample and loyal estimators, preprint.
56. Berlant, J., Broniatowski, M., Teugels, J., and Vickers, L. (1995) The mean residual life function at great age: application to tail estimation, JSPI, 45, 21-48.
57. Broniatowski, M. and Fuchs, A. (1995) Tauberian Theorems, Chernoff Inequality and the tail behaviour of finite convolutions of distribution functions, Advances in Mathematics, 116, 1, 12-33.
58. Broniatowski, M. and Weber, M. (1997) Strong laws for sums of extreme values, avec M.Weber, Theory of Probability and Applications, 3, 553-563.
59. Barbe Ph. and Broniatowski, M. (2003) Influence fraction of a sample and loyal estimators, preprint.
60. Feller, W. (1971) Introduction to probability theory and its applications, Wiley.
61. Jurečková, J. and Sen, P.K. (1996) Robust statistical procedures. Asymptotics and interrrelations, Wiley.
62. Kusnier, J. and Mizera, I. (1999) Tail behavior and breakdown properties of equivariant estimator of location, preprint.
63. Jurečková, J. and Picek, J. (2001) A class of tests on the tail index, Extremes, 4:2, 165-183.
64. Antoch, J. (1984) Behaviour of L-estimators of location from the point of view of large deviations, Robustness in Statist. Methods and Nonpar. Statist., (D.Rasch and M.L.Tiku, eds.), pp. 5-8.
65. Beirlant, J., Dierckx, G., Goegebeur, Y., and Matthys, G. (1999) Tail index estimation and an exponential regression model, Extremes, 2, 177-200.
66. Beirlant, J., Broniatowski, M., Teugles, J. L., and Vynckier, P. (1995) The mean residual life function at great age: Application to tail estimation, J. Statist. Planning Infer., 45, 21-48.
67. Gumbel, E. (1962) Statistics of Extremes, Columbia University Press, NewYork. (Russian Translation 1965).
68. Feuerverger, A. and Hall, P. (1999) Estimating a tail exponent by modelling departure form a Pareto distribution, Ann. Statist., 27, 760-781.
69. He, X., Jurečková, J., Koenker, R. and Portnoy, S. (1990) Tail behavior of regression estimators and their breakdown points, Econometrica, 58, 1195-1214.
70. Jurečková, J. (1979) Finite-sample comparison of L-estimators of location, Comment. Math. Univ. Carolinae, 20, 507-518.
71. Jurečková, J. (1981) Tail behavior of location estimators, Ann. Statist., 9, 578-585.
72. Jurečková, J., Koenker, R. and Portnoy, S. (2001) Tail behavior of the least squares estimator, Statist. & Probab. Letters, 55, 377-384.
73. Jurečková, J., Koul, H. L., and Picek, J. (2003) Testing the tail index in autoregressive models, submitted
74. Jurečková, J. and Picek J. (2001) A class of tests on the tail index, Extremes, 4:2, 165-183.
75. Kusnier, J. and Mizera, I. (1999) Tail behavior and breakdown properties of equivariant estimators of location, Ann. Inst. Statist. Math.
76. Matthys, G. and Beirlant, J. (2001) Extreme quantile estimation for heavy-tailed distributions, preprint.
77. Mizera, I. and Müller, C. H. (1999) Breakdown points and variation exponents of robust Mestimators in linear models, Ann. Statist., 27, 1164-1177.
78. Segers, J. (2001) Abelian and Tauberian theorems on the bias of the Hill estimator, preprint.