Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T07:22:21.376Z Has data issue: false hasContentIssue false
  • English
  • Français

EXTREME VALUE DISTRIBUTIONS FOR BIASED SAMPLES

Published online by Cambridge University Press:  20 January 2015

George Tzavelas  and
Polychronis Economou
George Tzavelas*
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou str., 185 34 Piraeus, Greece
Polychronis Economou
Affiliation:
Department of Civil Engineering, University of Patras, Rion-Patras, Greece E-mail: peconom@upatras.gr
*
E-mail: tzafor@unipi.gr (Corresponding author)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work, the relationship between the extreme value distributions of the parent and its weighted counterpart distribution is studied. Sufficient conditions are provided in terms of the weight w(x) under which the weighted and parent distributions belong to the same attractor (Fréchet, Weibull, or Gumbel) and the relation of the corresponding shape parameters of the limiting distributions is presented. Additionally, a biased sampling corrected extreme value index is proposed, when the extreme value index is estimated from a biased sample. Finally, some simulation results are presented that suggest the superiority of a biased sample in estimating the extreme value index.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 2 , April 2015 , pp. 277 - 290
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Embrechts, P., Klüppelberg, C. & Mikosch, Th. (2001). Modelling Extremal Events for Insurance and Finance. New York: Springer-Verlag.Google Scholar
2.Finkenstädt, B. & Rootzén, H. (2004). Extreme Values in Finance, Telecommunications, and the Environment. Chapman & Hall/CRC.Google Scholar
3.Gnedenko, B. (1943). Sur la distribution limite du terme maximum d'une serie aleatoire. The Annals of Mathematics 44(3): 423–453.CrossRefGoogle Scholar
4.Gumbel, E.J. (1958). Statistics of Extremes. New York: Columbia University Press.CrossRefGoogle Scholar
5.Gupta, R.C. & Keating, J.P. (1986). Relations for reliability measures under length biased sampling. Scandinavian Journal of Statistics 13(1): 49–56.Google Scholar
6.Gusev, S.I. (1976). Asymptotic expansions associated with some statistical estimators in the smooth case II. Expansions of moments and distributions. Theory of Probability and its Applications XXI(1): 14–33.CrossRefGoogle Scholar
7.Hernändez, A., Guenni, L. & Sanso, B. (2009). Extreme limit distribution of truncated models for daily rainfall. Environmetrics 20: 962–980.CrossRefGoogle Scholar
8.Liang, S. (2005). Generalized linear and additive models with weighted distribution. Ph.D. thesis, National University of Singapore.Google Scholar
9.Pickands, J. III (1975). Statistical inference using extreme order statistics. Annals of Statistics 3(1): 119–131.Google Scholar
10.Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. New York: Springer-Verlag.CrossRefGoogle Scholar
11.Simon, R. (1980). Length-biased sampling in etiological studies. Am. J. Epidemiol. 111: 444–452.CrossRefGoogle Scholar
12.Zelen, M. & Feinleib, M., (1969). On the theory of screening for chronic diseases. Biometrika 56: 601–614.CrossRefGoogle Scholar