Abstract
In this paper, we consider three sampling methods that are ranked set sampling (RSS), generalized modified ranked set sampling (GMRSS) and extreme ranked set sampling (ERSS). RSS and ERSS are well-known sampling schemes. In GMRSS procedure, a single ranked unit is selected for full measurement. This paper improves estimators based on RSS, GMRSS and ERSS for association parameter of type-I Gumbel’s bivariate exponential distribution (GBED-I) by using maximum likelihood (ML) estimation. To determine whether there is a statistically significant association between the components, we investigate likelihood ratio tests based on RSS, MRSS and ERSS. To examine the performances of suggested estimators and tests with respect to their counterparts of simple random sampling (SRS), we provide an extensive Monte Carlo simulation. According to the results, it appears that GMRSS provides a more efficient ML estimator than the other sampling methods when only the maximum ranked units are selected. Also, it is observed that the test statistic based on GMRSS has the highest power among the test statistics when the sample is obtained from the maximum ranked units. Considering that GMRSS can be obtained with less effort than other samples, these results become even more meaningful.
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References
Abo-Eleneen Z, Nagaraja H (2002) Fisher information in an order statistic and its concomitant. Ann Inst Stat Math 54(3):667–680
Al-Saleh MF, Samawi HM (2005) Estimation of the correlation coefficient using bivariate ranked set sampling with application to the bivariate normal distribution. Commun Stat-Theory Methods 34(4):875–889
Balakrishnan N, Lai CD (2009) Bivariate exponential and related distributions. In: Continuous bivariate distributions. Springer, pp 401–475
Balasubramanian K, Beg M (1998) Concomitant of order statistics in Gumbel’s bivariate exponential distribution. Sankhyā Indian J Stat Ser B, pp 399–406
Barnett V (1985) The bivariate exponential distribution; a review and some new results. Stat Neerl 39(4):343–356
Chen Z (2000) The efficiency of ranked-set sampling relative to simple random sampling under multi-parameter families. Statistica Sinica, pp 247–263
Devroye L (1986) Non-uniform random variate generation. Springer, New York
Gumbel EJ (1960) Bivariate exponential distributions. J Am Stat Assoc 55(292):698–707
Hanandeh AA, Al-Saleh MF (2013) Inference on downtons bivariate exponential distribution based on moving extreme ranked set sampling. Aust J Stat 42(3):161–179
He Q (2007) Inference on correlation from incomplete bivariate samples. PhD thesis, The Ohio State University
He Q, Nagaraja H, Wu C (2013) Efficient simulation of complete and censored samples from common bivariate exponential distributions. Comput Stat 28(6):2479–2494
Hussian MA (2014) Bayesian and maximum likelihood estimation for Kumaraswamy distribution based on ranked set sampling. Am J Math Stat 4(1):30–37
Khamnei HJ, Abusaleh S (2017) Estimation of parameters in the generalized logistic distribution based on ranked set sampling. Int J Nonlinear Sci 24(3):154–160
Lehmann EL (1959) Testing statistical hypotheses. Wiley, New York
McIntyre G (1952) A method for unbiased selective sampling, using ranked sets. Aust J Agric Res 3(4):385–390
Modarres R, Zheng G (2004) Maximum likelihood estimation of dependence parameter using ranked set sampling. Stat Probab Lett 68(3):315–323
Moore R, Clarke R (1981) A distribution function approach to rainfall runoff modeling. Water Resour Res 17(5):1367–1382
R Core Team (2020) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, https://www.R-project.org/
Samawi HM, Ahmed MS, Abu-Dayyeh W (1996) Estimating the population mean using extreme ranked set sampling. Biom J 38(5):577–586
Stokes SL (1980) Inferences on the correlation coefficient in bivariate normal populations from ranked set samples. J Am Stat Assoc 75(372):989–995
Stokes SL (1995) Parametric ranked set sampling. Ann Inst Stat Math 47(3):465–482
Taconeli CA, Bonat WH (2020) On the performance of estimation methods under ranked set sampling. Comput Stat, pp 1–22
Taconeli CA, Giolo SR (2020) Maximum likelihood estimation based on ranked set sampling designs for two extensions of the lindley distribution with uncensored and right-censored data. Comput Stat, pp 1–25
Takahasi K, Wakimoto K (1968) On unbiased estimates of the population mean based on the sample stratified by means of ordering. Ann Inst Stat Math 20(1):1–31
Weer D, Basu A (1980) Testing for indepedence in multivariate exponential distributions\(^{2}\). Aust J Stat 22(3):276–288
Zheng G, Al-Saleh MF (2002) Modified maximum likelihood estimators based on ranked set samples. Ann Inst Stat Math 54(3):641–658
Zheng G, Modarres R (2006) A robust estimate of the correlation coefficient for bivariate normal distribution using ranked set sampling. J Stat Plan Inference 136(1):298–309
Acknowledgements
The authors thank the reviewers and the editor for helpful comments that have improved the paper. The first author is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) 2211/A National PhD scholarship program and the Higher Education Council of Turkey (YÖK) 100/2000 PhD scholarship program.
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Sevil, Y.C., Yildiz, T.O. Gumbel’s bivariate exponential distribution: estimation of the association parameter using ranked set sampling. Comput Stat 37, 1695–1726 (2022). https://doi.org/10.1007/s00180-021-01176-2
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DOI: https://doi.org/10.1007/s00180-021-01176-2