Parameter estimation in the Farlie–Gumbel–Morgenstern bivariate Bilal distribution via multistage ranked set sampling

S. P. Arun; M. R. Irshad; R. Maya; Amer I. Al-Omari; Shokrya S. Alshqaq; S. P. Arun; M. R. Irshad; R. Maya; Amer I. Al-Omari; Shokrya S. Alshqaq

doi:10.3934/math.2025098

AIMS Mathematics

2025, Volume 10, Issue 2: 2083-2097. doi: 10.3934/math.2025098

Previous Article Next Article

Research article Special Issues

Parameter estimation in the Farlie–Gumbel–Morgenstern bivariate Bilal distribution via multistage ranked set sampling

S. P. Arun ^{1
,
,},
M. R. Irshad ^{2
,},
R. Maya ^{2
,},
Amer I. Al-Omari ^{3
,
,},
Shokrya S. Alshqaq ^{4
,}

1.
Kerala University Library, Research Centre, University of Kerala, Thiruvananthapuram 695034, India
2.
Department of Statistics, Cochin University of Science and Technology, Cochin 682 022, Kerala, India; irshadmr@cusat.ac.in, publicationsofmaya@gmail.com
3.
Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 251113, Jordan
4.
College of Science, Jazan University, P.O.Box. 114, Jazan 45142, Kingdom of Saudi Arabia; salshekak@jazanu.edu.sa

Received: 17 October 2024 Revised: 03 January 2025 Accepted: 17 January 2025 Published: 08 February 2025
MSC : 62G30, 62F10

Ranked set sampling is a well-known and efficient method compared to simple random sampling for estimating population parameters. In this study, we focus on the challenge of estimating the scale parameter of the primary variable $ Z $ using a multistage ranked set sample obtained by ordering the marginal observations of an auxiliary variable $ W $, where the pair $ (W, Z) $ follows the Farlie–Gumbel–Morgenstern bivariate Bilal distribution. Assuming that the dependence parameter $ \phi $ is known, we introduce the best linear unbiased estimator for the scale parameter of the primary variable, utilizing a multistage ranked set sample. We also compare the efficiency of the proposed estimator with that of the maximum likelihood estimator based on the same number of measured units. It is found that the suggested estimators are more efficient than the classical estimators considered in this study.
Citation: S. P. Arun, M. R. Irshad, R. Maya, Amer I. Al-Omari, Shokrya S. Alshqaq. Parameter estimation in the Farlie–Gumbel–Morgenstern bivariate Bilal distribution via multistage ranked set sampling[J]. AIMS Mathematics, 2025, 10(2): 2083-2097. doi: 10.3934/math.2025098

Related Papers:

Abstract

Ranked set sampling is a well-known and efficient method compared to simple random sampling for estimating population parameters. In this study, we focus on the challenge of estimating the scale parameter of the primary variable $ Z $ using a multistage ranked set sample obtained by ordering the marginal observations of an auxiliary variable $ W $, where the pair $ (W, Z) $ follows the Farlie–Gumbel–Morgenstern bivariate Bilal distribution. Assuming that the dependence parameter $ \phi $ is known, we introduce the best linear unbiased estimator for the scale parameter of the primary variable, utilizing a multistage ranked set sample. We also compare the efficiency of the proposed estimator with that of the maximum likelihood estimator based on the same number of measured units. It is found that the suggested estimators are more efficient than the classical estimators considered in this study.

References

[1]	A. M. Abd-Elrahman, Utilizing ordered statistics in lifetime distributions production: a new lifetime distribution and applications, J. Probab. Stat. Sci., 11 (2013), 153–164.
[2]	D. Hinkley, On quick choice of power transformation, Appl. Stat., 26 (1977), 67–69. https://doi.org/10.2307/2346869 doi: 10.2307/2346869
[3]	M. E. Ghitany, B. Atieh, S. Nadarajah, Lindley distribution and its application, Math. comput. simul., 78 (2008), 493–506. https://doi.org/10.1016/j.matcom.2007.06.007 doi: 10.1016/j.matcom.2007.06.007
[4]	A. M. Abd-Elrahman, S. F. Niazi, Approximate Bayes estimators applied to the Bilal model, J. Egypt. Math. Soc., 25 (2017), 65–70. https://doi.org/10.1016/j.joems.2016.05.001 doi: 10.1016/j.joems.2016.05.001
[5]	A. M. Abd-Elrahman, A new two-parameter lifetime distribution with decreasing, increasing or upside-down bathtub-shaped failure rate, Commun. Stat.-Theory Methods, 46 (2017), 8865–8880. https://doi.org/10.1080/03610926.2016.1193198 doi: 10.1080/03610926.2016.1193198
[6]	R. Maya, M. R. Irshad, M. Ahammed, C. Chesneau, The Harris extended Bilal distribution with applications in hydrology and quality control, AppliedMath, 3 (2023), 221–242. https://doi.org/10.3390/appliedmath3010013 doi: 10.3390/appliedmath3010013
[7]	M. R. Irshad, M. Ahammed, R. Maya, A. I. Al-Omari, Marshal-Olkin Bilal distribution with associated minification process and accptance sampling plans, Hacet. J. Math. Stat., 53 (2023), 201–229. https://doi.org/10.15672/hujms.1143156 doi: 10.15672/hujms.1143156
[8]	R. Maya, M. R. Irshad, S. P. Arun, Farlie–Gumbel–Morgenstern bivariate Bilal distribution and its inferential aspects using concomitants of order statistics, J. Probab. Stat. Sci., 19 (2021), 1–20.
[9]	S. P. Arun, C. Chesneau, R. Maya, M. R. Irshad, Farlie–Gumbel–Morgenstern bivariate moment exponential distribution and its inferences based on concomitants of order statistics, Stats, 6 (2023), 253–267. https://doi.org/10.3390/stats6010015 doi: 10.3390/stats6010015
[10]	R. A. Johnson, D. W. Wichern, Applied multivariate statistical analysis, Pearson College Div, Subsequent edition, 2002.
[11]	G. A. McIntyre, A method for unbiased selective sampling using ranked sets, Australian J. Agr. Res., 3 (1952), 385–390. https://doi.org/10.1071/AR9520385 doi: 10.1071/AR9520385
[12]	K. Takahasi, K. Wakimoto, On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Stat. Math., 20 (1968), 1–31.
[13]	S. L. Stokes, Ranked set sampling with concomitant variables, Commun. Stat.-Theory Methods, 6 (1977), 1207–1211. https://doi.org/10.1080/03610927708827563 doi: 10.1080/03610927708827563
[14]	H. A. David, H. N. Nagaraja, Order statistics, 3 Eds., John Wiley and Sons, New York, 2003.
[15]	M. R. Irshad, R. Maya, A. I. Al-Omari, A. A. Hanadeh, S. P. Arun, Estimation of a parameter of Farlie–Gumbel–Morgenstern bivariate Bilal distribution by ranked set sampling, Reliability: Theory Appl., 18 (2023), 129–140.
[16]	M. F. Al-Saleh, M. Al-Kadiri, Double-ranked set sampling, Stat. Probabil. Lett., 48 (2000), 205–212. https://doi.org/10.1016/S0167-7152(99)00206-0
[17]	M. F. Al-Saleh, A. I. Al-Omari, Multistage ranked set sampling, J. Stat. plan. Infer., 102 (2002), 273–286. https://doi.org/10.1016/S0378-3758(01)00086-6 doi: 10.1016/S0378-3758(01)00086-6
[18]	M. F. Al-Saleh, Steady-state ranked set sampling and parametric estimation, J. Stat. plan. Infer., 123 (2004), 83–95. https://doi.org/10.1016/S0378-3758(03)00139-3 doi: 10.1016/S0378-3758(03)00139-3
[19]	A. A. Jemain, A. I. Al-Omari, Multistage percentile ranked set samples, Adv. Appl. Stat., 7 (2007), 127–139.
[20]	A. A. Jemain, A. I. Al-Omari, Multistage quartile ranked set samples, Pakistan J. Stat., 23 (2007), 11–22.
[21]	A. A. Jemain, A. I. Al-Omari, K. Ibrahim, Multistage median ranked set sampling for estimating the population median, J. Math. Stat., 3 (2007), 58–64.
[22]	A. I. Al-Omari, C. N. Bouza, Review of ranked set sampling: modifications and applications, Revista Investigacion Operacional, 35 (2014), 215–240.
[23]	A. Haq, J. Brown, E. Moltchanova, A. I. Al-Omari, Varied L ranked set sampling scheme, J. Stat. Theory Pract., 9 (2015), 741–767. https://doi.org/10.1080/15598608.2015.1008606 doi: 10.1080/15598608.2015.1008606
[24]	N. Koyuncu, A. I. Al-Omari, Generalized robust-regression-type estimators under different ranked set sampling, Math. Sci., 15 (2021), 29–40. https://doi.org/10.1007/s40096-020-00360-7 doi: 10.1007/s40096-020-00360-7
[25]	E. Zamanzade, E. Zamanzade, A. Parvardeh, Estimation of a decreasing mean residual life based on ranked set sampling with an application to survival analysis, Int. J. Biostat., 20 (2024), 571–583. https://doi.org/10.1515/ijb-2023-0051 doi: 10.1515/ijb-2023-0051
[26]	E. Zamanzade, M. Mahdizadeh, H. M. Samawi, Nonparametric estimation of mean residual lifetime in ranked set sampling with a concomitant variable, J. Appl. Stat., 51 (2024), 2512–2528. https://doi.org/10.1080/02664763.2023.2301334 doi: 10.1080/02664763.2023.2301334
[27]	E. Zamanzade, M. Asadi, A. Parvardeh, E. Zamanzade, A ranked-based estimator of the mean past lifetime with an application, Stat. Papers, 64 (2023), 161–177. https://doi.org/10.1007/s00362-022-01314-y doi: 10.1007/s00362-022-01314-y

Reader Comments

Your name:*

Email:*
© 2025 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)