In this study, an application of empirical distribution function (EDF) estimators based on ranked set sampling (RSS) using real-life data set (body fat data) is illustrated. In this application, three variables which are percentage of body fat (Y), abdomen circumference (X1) and age (X2) are used. Age and abdomen circumference are separately used in ranking process as auxiliary variables which have correlation 0.813 (for perfect ranking) and 0.291 (for imperfect ranking) with the percentage of body fat, respectively. Ranked set samples are constructed by using three different sampling designs which are level-0 level-1 and level-2. The effects of perfect and imperfect ranking on the estimators of the sampling designs are investigated. Relative efficiencies of the EDF estimators are obtained by using their mean squared errors (MSE) and integrated mean squared errors (IMSE), numerically. For both perfect and imperfect ranking, these EDF estimators based on sampling designs have outperformance against EDF estimator based on SRS.
Keywords: Ranked set sampling; sampling designs; empirical distribution function; mean squared error; integrated mean squared error; body fat data
Bu çalışmada, sıralı küme örneklemesine dayalı ampirik dağılım fonksiyonu kestiricilerinin gerçek bir veri seti (vücut yağı verisi) kullanılarak uygulaması gerçekleştirildi. Bu uygulamada, vücut yağı yüzdesi (Y), karın çevresi (X1) ve yaş (X2) olmak üzere 3 değişken kullanılmıştır. Vücut yağ yüzdesiyle korelasyonları sırasıyla 0.813 (kusursuz sıralama) ve 0.291 (kusurlu sıralama) olan yaş ve karın çevresi, sıralama prosedüründe ayrı ayrı yardımcı değişkenler olarak kullanılmıştır. Sıralı küme örneklemleri, düzey0, düzey-1 ve düzey-2 olan 3 farklı örneklem tasarımı kullanılarak elde edilmiştir. Kusursuz ve kusurlu sıralamanın örneklem tasarımlarının kestiricileri üzerine etkileri araştırılmıştır. Ampirik dağılım fonksiyonu kestiricilerinin göreceli etkinlikleri, hata kareler ortalamaları ve integrallenmiş hata kareler ortalamaları kullanılarak sayısal olarak elde edilmiştir. Kusurlu ve kusursuz sıralama için örneklem tasarımlarına dayalı ampirik dağılım fonksiyonu kestiricilerinin, basit rasgele örneklemeye dayalı ampirik dağılım kestiricisi karşısında daha iyi bir performansa sahip olduğu görülmüştür.
Anahtar Kelimeler: Sıralı küme örneklemesi; örneklem tasarımları; ampirik dağılım fonksiyonu; hata kareler ortalaması; integrallenmiş hata kareler ortalaması; vücut yağı verisi
- McIntyre GA. A method for unbiased selective sampling, using ranked sets. Aust J Agr Res. 1952;3(4):385-90. [Crossref]
- Takahasi K, Wakimoto K. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics. 1968; 20(1):1-31. [Crossref]
- Dell T, Clutter J. Ranked set sampling theory with order statistics background. Biometrics. 1972;28(2):545-55. [Crossref]
- Stokes SL. Inferences on the correlation coefficient in bivariate normal populations from ranked set samples. J Am Stat Assoc. 1980;75(372):989-95. [Crossref]
- Stokes SL. Estimation of variance using judgement ordered ranked set samples. Biometrics. 1980;36(1):35-42. [Crossref]
- Neerchal NK, Sinha BK, Lacayo H. Ranked set sampling from a dichotomous population. Journal of Applied Statistical Science. 1998;11(1):83-90.
- Samawi HM, Ahmed MS, Abu-Dayyeh W. Estimating the population mean using extreme ranked set sampling. Biometrical J. 1996;38(5):577-86. [Crossref]
- Muttlak HA. Median ranked set sampling. Journal of Applied Statistical Science. 1997;6(4):245-55.
- Muttlak HA. Modified ranked set sampling methods. Pak J Statist. 2003;19(3):315-23.
- Stokes SL. Ranked set sampling with concomitant variables. Comm Stat Theor Meth. 1977;6(12):1207-11. [Crossref]
- Stokes SL, Sager TW. Characterization of a ranked-set-sample with application to estimating distribution functions. J Am Stat Assoc. 1988;83(402):374-81. [Crossref]
- Samawi HM, Al-Sagheer OA. On the estimation of the distribution function using extreme and median ranked set sampling. Biometrical J. 2001;43(3):357-73. [Crossref]
- Al-Subh S, Alodat M, Ibrahim K, Jemain A. EDF goodness of fit tests of logistic distribution under selective order statistics. Pak J Statist. 2009;25(3):265-74.
- Nazari S, Jafari Jozani M, Kharrati-Kopaei M. On distribution function estimation with partially rank-ordered set samples: estimating mercury level in fish using length frequency data. Statistics. 2016;50(6):1387-410. [Crossref]
- Sevil YC, Ozkal Yildiz T. Power comparison of the Kolmogorov-Smirnov test under ranked set sampling and simple random sampling. J Stat Comput Sim. 2017;87(11):2175-85. [Crossref]
- Ozkal Yildiz T, Sevil YC. Performances of some goodness-of-fit tests for sampling designs in ranked set sampling. J Stat Comput Sim. 2018;88(9):1702-16. [Crossref]
- Ozkal Yildiz T, Sevil YC. Empirical distribution function estimators based on sampling designs in a finite population using single auxiliary variable. J Appl Stat. 2019;46(16):2962-74. [Crossref]
- Samawi HM, Ababneh FM. On regression analysis using ranked set sample. Journal of Statistical Research. 2001;35(2):93-105.
- Chen H, Stasny EA, Wolfe DA. Improved procedures for estimation of disease prevalence using ranked set sampling. Biometrical J. 2007;49(4):530-8. [Crossref] [PubMed]
- Gory J, Ozturk O. Analysis of the NHANES III data set using ranked set and judgement post-stratified samples. Adv Appl Stat. 2015;47(1):65-89. [Crossref]
- Göçoğlu A, Demirel N. Estimating the population proportion in modified ranked set sampling methods. J Stat Comput Sim. 2019;89(14):2694-710. [Crossref]
- Deshpande JV, Frey J, Ozturk O. Nonparametric ranked-set sampling confidence intervals for quantiles of a finite population. Environ Ecol Stat. 2006;13(1):25-40. [Crossref]
- Al-Saleh MF, Samawi HM. A note on inclusion probability in ranked set sampling and some of its variations. Test. 2007;16(1):198-209. [Crossref]
- Ozdemir YA, Gokpinar F. A generalized formula for inclusion probabilities in ranked set sampling. Hacettepe Journal of Mathematics and Statistics. 2007;36(1):89-99.
- Ozdemir YA, Gokpinar F. A new formula for inclusion probabilities in median ranked set sampling. Comm Stat Theor Meth. 2008;37(13):2022-33. [Crossref]
- Gokpinar F, Ozdemir YA. Generalization of inclusion probabilities in ranked set sampling. Hacettepe Journal of Mathematics and Statistics. 2010;39(1):89-95.
- Frey J. Recursive computation of inclusion probabilities in ranked set sampling. J Stat Plan Infer. 2011;141(11):3632-9. [Crossref]
- Jafari Jozani M, Johnson BC. Design based estimation for ranked set sampling in finite population. Environ Ecol Stat. 2011;18(4):663-85. [Crossref]
- Jafari Jozani M, Johnson BC. Randomized nomination sampling in finite populations. J Stat Plan Infer. 2012;142(7):2103-15. [Crossref]
- Ozturk O, Jozani MJ. Inclusion probabilities in partially rank ordered set sampling. Computational Statistics & Data Analysis. 2014;69:122-32. [Crossref]
- Ozturk O. Estimating of population mean and total in a finite population setting using multiple auxiliary variables. J Agr Biol Envir St. 2014;19(2):161-84. [Crossref]
- Penrose KW, Nelson AG, Fisher AG. Generalized body composition prediction equation for men using simple measurement techniques. Medicine & Science in Sports & Exercise. 1985;17(2):189. [Crossref]
- Wang X, Wang K, Lim J. Isotonized CDF estimation from judgement poststratification data with empty strata. Biometrics. 2012;68(1):194-202. [Crossref] [PubMed]
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