Asymptotic behavior of ordered random variables in mixture of two Gaussian sequences with random index

H. M. Barakat; M. H. Dwes; H. M. Barakat; M. H. Dwes

doi:10.3934/math.20221060

AIMS Mathematics

2022, Volume 7, Issue 10: 19306-19324. doi: 10.3934/math.20221060

Previous Article Next Article

Research article

Asymptotic behavior of ordered random variables in mixture of two Gaussian sequences with random index

H. M. Barakat ¹,
M. H. Dwes ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
2.
Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt

Received: 22 June 2022 Revised: 07 August 2022 Accepted: 11 August 2022 Published: 31 August 2022
MSC : Primary 62E20; Secondary 62E15, 62G30

When the random sample size is assumed to converge weakly and to be independent of the basic variables, the asymptotic distributions of extreme, intermediate, and central order statistics, as well as record values, for a mixture of two stationary Gaussian sequences under an equi-correlated setup are derived. Furthermore, sufficient conditions for convergence are derived in each case. An interesting fact is revealed that in several cases, the limit distributions of the aforementioned statistics are the same when the sample size is random and non-random. e.g., when one mixture component has a correlation that converges to a non-zero value.
- mixture distribution,
- Gaussian sequences,
- random index,
- order statistics,
- record values
Citation: H. M. Barakat, M. H. Dwes. Asymptotic behavior of ordered random variables in mixture of two Gaussian sequences with random index[J]. AIMS Mathematics, 2022, 7(10): 19306-19324. doi: 10.3934/math.20221060

Related Papers:

Abstract

When the random sample size is assumed to converge weakly and to be independent of the basic variables, the asymptotic distributions of extreme, intermediate, and central order statistics, as well as record values, for a mixture of two stationary Gaussian sequences under an equi-correlated setup are derived. Furthermore, sufficient conditions for convergence are derived in each case. An interesting fact is revealed that in several cases, the limit distributions of the aforementioned statistics are the same when the sample size is random and non-random. e.g., when one mixture component has a correlation that converges to a non-zero value.

References

[1]	H. M. Barakat, A new method for adding two parameters to a family of distributions with application to the normal and exponential families, Stat. Methods Appl., 24 (2015), 359–372. https://doi.org/10.1007/s10260-014-0265-8 doi: 10.1007/s10260-014-0265-8
[2]	H. M. Barakat, O. M. Khaled, Towards the establishment of a family of distributions that best fits any data set, Commun. Stat. Simul. Comput., 46 (2017), 6129–6143. https://doi.org/10.1080/03610918.2016.1197245 doi: 10.1080/03610918.2016.1197245
[3]	H. M. Barakat, M. A. Abd Elgawad, M. A. Alawady, Asymptotic order statistics of mixtures of distributions, Rocky Mountain J. Math., 50 (2020), 429–443. https://doi.org/10.1216/rmj.2020.50.429 doi: 10.1216/rmj.2020.50.429
[4]	H. M. Barakat, O. M. Khaled, N. K. Rakha, Modern techniques in data analysis, with application to the water pollution, Proceedings of the Latvian Academy of Sciences. Section B. Natural, Exact, and Applied Sciences, 72 (2018), 184–192. https://doi.org/10.2478/prolas-2018-0005 doi: 10.2478/prolas-2018-0005
[5]	H. M. Barakat, A. W. Aboutahoun, N. N. El-kadar, A new extended mixture skew normal distribution, with applications, Rev. Colomb. Estad., 42 (2019), 167–183. https://doi.org/10.15446/rce.v42n2.70087 doi: 10.15446/rce.v42n2.70087
[6]	H. M. Barakat, A. W. Aboutahoun, N. N. El-Kadar, On some generalized families arising from mixture normal distribution with applications, Commun. Stat. Simul. Comput., 50 (2021), 198–216. https://doi.org/10.1080/03610918.2018.1554110 doi: 10.1080/03610918.2018.1554110
[7]	F. Z. Doğru, O. Arslan, Robust mixture regression based on the skew t distribution, Rev. Colomb. Estad., 40 (2017), 45–64. https://doi.org/10.15446/rce.v40n1.53580 doi: 10.15446/rce.v40n1.53580
[8]	D. M. Titterington, A. F. M. Smith, U. E. Makov, Statistical analysis of finite mixture distributions, New York: Wiley, 1985.
[9]	C. Bernard, S. Vanduffel, Quantile of a mixture, 2014, arXiv: 1411.4824.
[10]	H. M. Barakat, O. M. Khaled, H. A. Ghonem, New method for prediction of future order statistics, Qual. Technol. Quant. Manag., 18 (2021), 101–116. https://doi.org/10.1080/16843703.2020.1782087 doi: 10.1080/16843703.2020.1782087
[11]	H. M. Barakat, M. H. Dwes, Limit distributions of ordered random variables in mixture of two Gaussian sequences, J. Math., 2022, in press.
[12]	M. A. Abd Elgawad, H. M. Barakat, S. Xiong, Limit distributions of random record model in a stationary Gaussian sequence, Commun. Stat. Theor. M., 49 (2020), 1099–1119. https://doi.org/10.1080/03610926.2018.1554131 doi: 10.1080/03610926.2018.1554131
[13]	E. O. Abo Zaid, H. M. Barakat, E. M. Nigm, Limit distributions of generalized order statistics in a stationary Gaussian sequence, Quaest. Math., 41 (2018), 629–642. https://doi.org/10.2989/16073606.2017.1394395 doi: 10.2989/16073606.2017.1394395
[14]	H. M. Barakat, M. A. Abd Elgawad, Asymptotic behavior of the record values in a stationary Gaussian sequence with applications, Math. Slovaca, 69 (2019), 707–720. https://doi.org/10.1515/ms-2017-0259 doi: 10.1515/ms-2017-0259
[15]	H. M. Barakat, E. M. Nigm, E. O. Abo Zaid, Limit distributions of order statistics with random indices in a stationary Gaussian sequence, Commun. Stat. Theor. M., 46 (2017), 7099–7113. https://doi.org/10.1080/03610926.2016.1148732 doi: 10.1080/03610926.2016.1148732
[16]	R. Vasudeva, A. Y. Moridani, Limit distributions of the extremes of a random number of random variables in a stationary Gaussian sequence, ProbStat Forum, 3 (2010), 78–90.
[17]	H. M. Barakat, Asymptotic properties of bivariate random extremes, J. Stat. Plan. Infer., 61 (1997), 203–217. https://doi.org/10.1016/S0378-3758(96)00157-7 doi: 10.1016/S0378-3758(96)00157-7
[18]	H. M. Barakat, M. A. El-Shandidy, On the limit distribution of the extreme of a random number of independent random variables, J. Stat. Plan. Infer., 26 (1990), 353–361. https://doi.org/10.1016/0378-3758(90)90137-J doi: 10.1016/0378-3758(90)90137-J
[19]	H. M. Barakat, M. A. El-Shandidy, On general asymptotic behaviour of order statistics with random index, Bull. Malays. Math. Sci. Soc. (2), 27 (2004), 169–183.
[20]	H. M. Barakat, E. M. Nigm, The mixing property of order statistics with some applications, Bull. Malays. Math. Sci. Soc. (2), 19 (1996), 39–52.
[21]	J. Galambos, The asymptotic theory of extreme order statistics, 2 Eds., Florida: Krieger Pub. Co., 1987.
[22]	O. Barndorff-Nielsen, On the limit distribution of the maximum of a random number of independent random variables, Acta Mathematica Academiae Scientiarum Hungaricae, 15 (1964), 399–403. https://doi.org/10.1007/BF01897148 doi: 10.1007/BF01897148
[23]	B. V. Gnedenko, L. Senusi-Bereksi, On a property of the limit distributions for the maximum and minimum terms of a variational series, (Russian), Dokl. Akad. Nauk SSSR, 267 (1982), 1039–1040.
[24]	B. C. Arnold, N. Balakrishnan, H. N. Nagaraja, Records, New York: Wiley, 1998. https://doi.org/10.1002/9781118150412
[25]	M. R. Leadbetter, G. Lindgren, H. Rootzén, Extremes and related properties of random sequences and processes, New York: Springer, 1983. https://doi.org/10.1007/978-1-4612-5449-2
[26]	N. V. Smirnov, Limit distribution for terms of a variational series, (Russian), Tr. Mat. Inst. Steklova, 25 (1949), 3–60.
[27]	D. M. Chibisov, On limit distributions for order statistics, Theor. Probab. Appl., 9 (1964), 142–148. https://doi.org/10.1137/1109021 doi: 10.1137/1109021
[28]	L. de Haan, On regular variation and its application to the weak convergence of sample extremes, Amsterdam: Mathematisch Centrum, 1970.
[29]	B. Cheng, The limiting distributions of order statistics, Acta Math. Sin., 14 (1964), 694–714.
[30]	K. N. Chandler, The distribution and frequency of record values, Journal of the Royal Statistical Society: Series B (Methodological), 14 (1952), 220–228. https://doi.org/10.1111/j.2517-6161.1952.tb00115.x doi: 10.1111/j.2517-6161.1952.tb00115.x
[31]	S. I. Resnick, Limit laws for record values, Stoch. Proc. Appl., 1 (1973), 67–82. https://doi.org/10.1016/0304-4149(73)90033-1 doi: 10.1016/0304-4149(73)90033-1
[32]	G. Louchard, H. Prodinger, The number of elements close to near-records in geometric samples, Quaest. Math., 29 (2006), 447–470. https://doi.org/10.2989/16073600609486175 doi: 10.2989/16073600609486175

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)