Shrinkage Estimators for the Location Parameter of the Normal Distribution in Median Ranked Set Sampling

Year 2022, , 502 - 514, 20.12.2022

Kübra Gürsoy , Meral Ebegil , Yaprak Özdemir , Fikri Gökpınar

https://doi.org/10.19113/sdufenbed.1121320

Abstract

Unbiased estimators of the population parameters are often used to make an inference about the population. In cases where unbiased estimators have large variance, biased estimators such as shrinkage estimators may be preferred. In this study, shrinkage estimators of the location parameter of the normal distribution were obtained under ranked set sampling and median ranked set sampling. In addition, mean square errors of shrinkage estimators were obtained theoretically under ranked set sampling and median ranked set sampling. In order to examine the efficiency of the estimators, the mean square errors were calculated under different conditions using Monte Carlo simulation study. According to the results, it was observed that the shrinkage estimators obtained under median ranked set sampling were more efficient than the shrinkage estimators obtained under ranked set sampling and simple random sampling.

Keywords

Ranked Set Sampling, Median Ranked Set Sampling, Mean Square Error, Shrinkage Estimator

Medyan Sıralı Küme Örneklemesinde Normal Dağılımın Konum Parametresi İçin Shrinkage Tahmin Edicileri

Abstract

Yığına ilişkin bir çıkarsama yapabilmek için genellikle yığın parametrelerinin sapmasız tahmin edicileri kullanılır. Sapmasız tahmin edicilerin büyük varyansa sahip olmaları durumunda, shrinkage tahmin edicileri gibi sapmalı tahmin ediciler tercih edilebilir. Bu çalışmada, normal dağılımın konum parametresi için shrinkage tahmin edicileri, sıralı küme örneklemesi ve medyan sıralı küme örneklemesi kullanılarak elde edilmiştir. Ayrıca sıralı küme örneklemesi ve medyan sıralı küme örneklemesi altında elde edilen shrinkage tahmin edicilerinin ortalama hata kareleri teorik olarak elde edilmiştir. Önerilen tahmin edicilerin etkinliklerini incelemek amacıyla farklı durumlar altında Monte Carlo simülasyon çalışması ile ortalama hata kareleri hesaplanmıştır. Elde edilen sonuçlara göre, medyan sıralı küme örneklemesi kullanılarak elde edilen shrinkage tahmin edicilerinin sıralı küme örneklemesi ve basit tesadüfi örnekleme altında elde edilen shrinkage tahmin edicilerinden daha etkin olduğu gözlemlenmiştir.

Keywords

Sıralı küme örneklemesi, Medyan sıralı küme örneklemesi, Ortalama hata kare, Shrinkage tahmin edicisi, Ranked Set Sampling, Median Ranked Set Sampling, Mean Square Error, Shrinkage Estimator

References

• [1] Thompson, J. R. (1968). Some shrinkage techniques for estimating the mean. Journal of the American Statistical Association, 63(321), 113-122.
• [2] Mehta, J. S., & Srinivasan, R. (1971). Estimation of the mean by shrinkage to a point. Journal of the American Statistical Association, 66(333), 86-90.
• [3] Jani, P. N. (1991). A class of shrinkage estimators for the scale parameter of the exponential distribution. IEEE Transactions on Reliability, 40(1), 68-70.
• [4] Kourouklis, S. (1994). Estimation in the 2-parameter exponential distribution with prior information. IEEE Transactions on Reliability, 43(3), 446-450.
• [5] Özdemir, Ş. (2012). Shrinkage tahmin ediciler sınıfı üzerine bir çalışma. Gazi Üniversitesi Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi, 98s, Ankara.
• [6] Özdemir, Y. A. (2005). Sıralı Küme Örneklemesiyle Doğrusal Regresyon Modelinde Parametre Tahminlerinin İncelenmesi, Doktora Tezi, Gazi Üniversitesi, Fen Bilimleri Enstitüsü, 186s, Ankara.
• [7] McIntyre, G. A. (1952). A method of unbiased selective sampling, using ranked sets. Aust. J. Agric. Res. 3, 385-90.
• [8] Takahasi, K., & Wakimoto, K. (1968). On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics, 20(1), 1-31.
• [9] Dell, T. R., & Clutter, J. L. (1972). Ranked set sampling theory with order statistics background. Biometrics, 545-555.
• [10] MacEachern, S. N., Öztürk, Ö., Wolfe, D. A., & Stark, G. V. (2002). A new ranked set sample estimator of variance. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(2), 177-188.
• [11] Muttlak, H. A. (1997). Median ranked set sampling. J Appl Stat Sci, 6, 245-255.
• [12] Muttlak, H. A. (1998). Median ranked set sampling with concomitant variables and a comparison with ranked set sampling and regression estimators. Environmetrics: The official journal of the International Environmetrics Society, 9(3), 255-267.
• [13] Chen, Z., Bai, Z., & Sinha, B. (2003). Ranked set sampling: theory and applications (Vol. 176). Springer Science & Business Media.
• [14] Jemain, A. A., Al-Omari, A., & Ibrahim, K. (2008). Some variations of ranked set sampling. Electronic Journal of Applied Statistical Analysis, 1(1), 1-15.
• [15] Tseng, Y., Wu, S., (2007). Ranked- Set- Sample- based Tests for Normal and Exponential Means. Communication in Statistics: Simulation and Computation.36: 761-782.
• [16] Muttlak, H. A., Ahmed, S. E., & Al-Momani, M. (2010). Shrinkage estimation in replicated median ranked set sampling. Journal of Statistical Computation and Simulation, 80(11), 1185-1196.
• [17] Koyuncu, N. (2018). Regression estimators in ranked set, median ranked set and neoteric ranked set sampling. Pakistan Journal of Statistics and Operation Research, 89-94.
• [18] Ebegil, M., Özdemir, Y. A., & Gökpinar, F. (2021). Some Shrinkage estimators based on median ranked set sampling. Journal of Applied Statistics, 1-26.
• [19] Koyuncu, N., & Al-Omari, A. I. (2021). Generalized robust-regression-type estimators under different ranked set sampling. Mathematical Sciences, 15(1), 29-40.
• [20] Özdemir, Y. A., Ebegil, M. and Gökpinar, F. (2017). A test statistic based on ranked set sampling for two normal means. Communications in Statistics-Simulation and Computation, 46(10), 8077-8085.
• [21] Gürsoy, K. (2019), Medyan sıralı küme örneklemesi kullanılarak shrinkage tahmini, Gazi Üniversitesi Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi, 98s, Ankara.