Limit Theorems for Bivariate Generalised Order Statistics in Stationary Gaussian Sequences with Random Sample Sizes

Essawe, Fatma Hashem; Elgawad, Mohamed Abd; Barakat, Haroon Mohamed; Zhao, Hui

doi:10.2478/prolas-2019-0080

Abstract

In this paper, we study the limit distribution functions of the (lower-lower), (upper-upper) and (lower-upper) extreme and central-central m-generalised order statistics (m–GOS) of stationary Gaussian sequences under an equi-correlated set up, when the random sample size is assumed to converge weakly and independent of the basic variables. Moreover, sufficient conditions for a weak convergence of generalised quasi-range with random indices are obtained.

References

Abd Elgawad, M. A., Barakat, H. M. ,Yan, T. (2019a). Bivariate limit theorems for record values based on random sample sizes. Sankhya A: Indian J. Statist. Available from: https://doi.org/10.1007/s13171-019-00167-2 (accessed 25.10.2019).10.1007/s13171-019-00167-2(accessed25.10.2019)
Open DOI Search in Google Scholar Back to article
Abd Elgawad, M. A., Barakat, H. M., Xiong, S. (2019b). Limit distributions of random record model in a stationary Gaussian sequence. Comm. Statist.- Theory Meth.https://doi.org/10.1080/03610926.2018.1554131 (accessed 25.10.2019).10.1080/03610926.2018.1554131(accessed25.10.2019)
Open DOI Search in Google Scholar Back to article
Barakat, H. M. (2007). Limit theory of generalized order statistics. J. Statist. Plann. Inf., 137 (1), 1–11.10.1016/j.jspi.2005.10.003
Search in Google Scholar Back to article
Barakat, H. M. (2018). The asymptotic dependence structural and some functions of generalized order statistics in a stationary Gaussian sequence. Math. Rep. Available from: https://www.researchgate.net/publication/329268250_The_asymptotic_dependence_structure_and_some_functions_of_generalized_order_statistics_in_a_stationary_Gaussian_sequence
Search in Google Scholar Back to article
Barakat, H. M., Abd Elgawad, M. A. (2019). Asymptotic behavior of the record values in a stationary Gaussian sequence, with applications. Math. Slovaca, 69 (3), 707–720.10.1515/ms-2017-0259
Search in Google Scholar Back to article
Barakat, H. M., Nigm, E. M. (1991). Weak limits of random quasimidranges. Microelectron. Reliab., 31 (5), 941–953.10.1016/0026-2714(91)90039-A
Open DOI Search in Google Scholar Back to article
Barakat, H. M., Nigm, E. M., Abd Elgawad, M. A. (2014). Limit theory for joint generalized order statistics. REVSTAT. Stat. J., 12 (3), 1–22.
Search in Google Scholar Back to article
Barakat, H. M., Nigm, E. M., Abo Zaid, E. O. (2016a). Limit distributions of order statistics with random indices in a stationary Gaussian sequence. Comm. Statist.Theory Meth., 46 (14), 7099–7113.10.1080/03610926.2016.1148732
Search in Google Scholar Back to article
Barakat, H. M., Nigm, E. M., Abo Zaid, E. O. (2016b). Generalized order statistics with random indices in a stationary Gaussian sequence. Appl. Math. Inf. Sci., 10 (3), 1081–1090.10.18576/amis/100326
Open DOI Search in Google Scholar Back to article
Cramer, E. (2003). Contributions to Generalized Order Statistics. Habililationsschrift, Reprint, University of Oldenburg.
Search in Google Scholar Back to article
De Haan, L. (1970). On Regular Variation and its Application to the Week Convergence of Sample Extremes. Mathematisch Centrum, Amsterdam. 124 pp.
Search in Google Scholar Back to article
Kamps, U. (1995). A Concept of Generalized Order Statistics. Teubner, Stuttgart. 210 pp.10.1007/978-3-663-09196-7
Search in Google Scholar Back to article
Nasri-Roudsari, D. (1996). Extreme value theory of generalized order statistics. J. Statist. Plann. Inf., 55, 281–297.10.1016/S0378-3758(95)00200-6
Search in Google Scholar Back to article
Nasri-Roudsari, D., Cramer, E. (1999). On the convergence rates of extreme generalized order statistics. Extremes, 2, 421–447.10.1023/A:1009904316589
Search in Google Scholar Back to article
Vasudeva, R., Moridani, A. Y. (2010). Limit distributions of the extremes of a random number of random variables in a stationary Gaussian sequence. ProbStat Forum, 3, 78–90.
Search in Google Scholar Back to article

Limit Theorems for Bivariate Generalised Order Statistics in Stationary Gaussian Sequences with Random Sample Sizes

Abstract

Paradigm

My account