×
Yan, Jigao

The uniform convergence and precise asymptotics of generalized stochastic order statistics. (English) Zbl 1135.62037

J. Math. Anal. Appl. 343, No. 2, 644-653 (2008).
Summary: Let \(X(i,n,m,k), i=1,\dots ,n\), be generalized order statistics based on \(F\). For fixed \(r\in \mathcal N\), and a suitable counting process \(N(t), t>0\), we mainly discuss the precise asymptotics of the generalized stochastic order statistics \(X(N(n) - r+1,N(n),m,k)\). This not only makes the results of J. Yan, Y. Wang and F. Y. Cheng [Precise asymptotics for order statistics of a non-random sample and a random sample. J. Syst. Sci. Math. Sci. 26, No. 2, 237–244 (2006; Zbl 1091.60506)] a special case of our results, and presents many groups of weighted functions and boundary functions, but also permits a unified approach to several models of ordered random variables.

Cited in 4 Documents

MSC:

62G30 Order statistics; empirical distribution functions
62G20 Asymptotic properties of nonparametric inference

Keywords:

generalized order statistics

Citations:

Zbl 1091.60506
Cite Review PDF
Full Text: DOI

References:

[1] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0617.26001
[2] Baum, L. E.; Katz, M., Convergence rates in the law of large numbers, Trans. Amer. Math. Soc., 120, 108-123 (1965) · Zbl 0142.14802
[3] Chen, R., A remark on the tail probability of a distribution, J. Multivariate Anal., 8, 328-333 (1978) · Zbl 0376.60033
[4] Embrechts, P.; Klüppelberg, C.; Minkosch, T., Modelling Extremal Events for Insurance and Finance (1997), Springer: Springer Berlin · Zbl 0873.62116
[5] Erdös, P., On a theorem of Hsu and Robbins, Ann. Math. Statist., 20, 286-291 (1949) · Zbl 0033.29001
[6] Erdös, P., Remark on my paper on a theorem of Hus and Robbins, Ann. Math. Statist., 21, 138 (1950) · Zbl 0035.21403
[7] Galambos, J., The Asymptotics Theory of Extreme Order Statistics (2001), Krieger Publishing Co.
[8] Gut, A., Precise asymptotics for record times and the associated counting process, Stochastic Process. Appl., 101, 233-239 (2002) · Zbl 1075.60055
[9] Gut, A.; Spătaru, A., Precise asymptotics in the Baum-Katz and Davis law of large numbers, J. Math. Anal. Appl., 248, 233-246 (2000) · Zbl 0961.60039
[10] Gut, A.; Spătaru, A., Precise asymptotics in the law of the iterated logarithm, Ann. Probab., 28, 1870-1883 (2000) · Zbl 1044.60024
[11] Heyde, C. C., A supplement to the strong law of large numbers, J. Appl. Probab., 12, 173-175 (1975) · Zbl 0305.60008
[12] Hsu, P. L.; Robbins, H., Complete convergence and the strong law of large numbers, Proc. Natl. Acad. Sci. USA, 33, 25-31 (1947) · Zbl 0030.20101
[13] Kamps, U., A concept of generalized order statistics, J. Statist. Plann. Inference, 48, 1-23 (1995) · Zbl 0838.62038
[14] Marohn, F., Strong domain of attraction of extreme generalized order statistics, Extremes, 5, 369-386 (2002) · Zbl 1035.60052
[15] Nasri-Roudsari, D., Extreme value theory of generalized order statistics, J. Statist. Plann. Inference, 55, 281-297 (1996) · Zbl 0863.60027
[16] Nasri-Roudsari, D.; Cramer, E., On the convergence rates of extreme generalized order statistics, Extremes, 2, 421-447 (1999) · Zbl 0972.62029
[17] Spătaru, A., Precise asymptotics in Spitzer’s law of large numbers, J. Theoret. Probab., 12, 811-819 (1999) · Zbl 0932.60026
[18] Wang, Y. B.; Yang, Y., A general law of precise asymptotics for the counting process of record times, J. Math. Anal. Appl., 286, 753-764 (2003) · Zbl 1031.60037
[19] Yan, J. G.; Wang, Y. B.; Cheng, F. Y., Precise asymptotics for order statistics of a non-random sample and a random sample, J. Systems Sci. Math. Sci., 26, 2, 237-244 (2006) · Zbl 1091.60506
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.