Abstract
This paper deals with the asymptotic independence of the normalized kth upper- and rth lower-order statistics and their locations, defined on some strictly stationary sequences \(\left\{ X_n\right\} _{n\ge 1}\) admitting clusters of both high and low values. The main result is the asymptotic independence of the joint locations of the k-largest extremes and the joint locations of the r-smallest extremes of \(\left\{ X_{n}\right\} _{n\ge 1}\), which allows us to censor a sample, by ensuring that the set of observations that we selected contains the k-largest and r-smallest order statistics of the stationary sequence \(\left\{ X_{n}\right\} _{n\ge 1}\) with a predetermined probability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balakrishnan, N., Cohen, A.: Order Statistics and Inference. Academic Press Inc, London (1991)
Davis, R.: Maxima and minima of stationary sequences. Ann. Probab. 7, 453–460 (1979)
Davis, R.: Limit laws for the maximum and minimum of stationary sequences. Z. Wharscheinlichkeits Theorie verw. Gebiete 61, 31–42 (1982)
Davis, R.: Limit laws for upper and lower extremes from stationary mixing sequences. J. Multivar. Anal. 13, 273–286 (1983)
Davis, R.: On upper and lower extremes in stationary sequences. In: Tiago Oliveira, J. (ed.) Statistical Extremes and Applications, NATO ASI Series, vol. 131, pp. 443–460. Springer, Netherlands (1984)
Ferreira, H., Scotto, M.: On the asymptotic locations of high values of a stationary sequence. Stat. Probab. Lett. 60, 475–482 (2002)
Hsing, T., Husler, J., Leadbetter, M.R.: On the exceedance point process for a stationary sequence. Probab. Theory Relat. Fields 78, 97–112 (1988)
Hsing, T.: On the extreme order statistics for a stationary sequence. Stoch. Process. Appl. 29, 155–169 (1988)
Jakubowski, A.: Asymptotic (\(\text{ r }-1\))-dependent representation for \(r\)th order statistic from a stationary sequence. Stoch. Process. Appl. 46, 29–46 (1993)
Kallenberg, O.: Random Measures. Akademische Verlag, Berlin (1975)
Leadbetter, M.R.: On extreme values in stationary sequences. Wahrscheinlichkeitstheorie and Verwandte Gebiete 2(8), 289–303 (1974)
Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin (1983)
Nandagopalan, S.: Multivariate extremes and estimation of the extremal index. Ph.D. Thesis. University of North Carolina at Chapel Hill (1990)
Pereira, L., Ferreira, H.: The asymptotic locations of the maximum and minimum of stationary sequences. J. Stat. Plan. Inference 104, 287–295 (2002)
Pereira, L.: The asymptotic location of the maximum of a stationary random field. Stat. Probab. Lett. 79, 2166–2169 (2009)
Pereira, L.: Asymptotic location of largest values of a stationary random field. Commun. Stat. Theory Methods 42, 4513–4524 (2013)
Pereira, L.: On the maximum and minimum of a stationary random field. In: Lita da Silva, J., Caeiro, F., Natário, I., Braumann C. (eds.) Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications. Studies in Theoretical and Applied Statistics, pp. 337–345. Springer, Berlin, Heidelberg (2013)
Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975)
Acknowledgements
The author would like to thank the referees for several corrections and important suggestions which significantly improved this paper. This work was partially supported by National Foundation of Science and Technology through UID/MAT/00212/2013.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pereira, L. On the Asymptotic Locations of the Largest and Smallest Extremes of a Stationary Sequence. J Theor Probab 31, 853–866 (2018). https://doi.org/10.1007/s10959-017-0742-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-017-0742-8