Abstract
In this article, we consider the problem of estimating cumulative distribution function (CDF) and a reliability parameter using moving extreme ranked set sampling (MERSS). Two different CDF estimators are described and compared with their competitors in simple random sampling (SRS) and ranked set sampling (RSS). It turns out the CDF estimators in MERSS can be more efficient than their competitors in SRS and RSS at a point in a particular tail of the distribution when the quality of rankings is sufficiently good. Motivated by this efficiency gain, we develop some estimators for the stress-strength probability using MERSS. The suggested estimators are then compared with their counterparts in the literature via Monte Carlo simulation. Finally, a real dataset is used to show the applicability of the developed procedures.
Similar content being viewed by others
Notes
Access date: August 2019.
References
Abu-Dayyeh, W., Al-Sawi, E.: Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling. Stat. Pap. 50, 249–259 (2009)
Al-Odat, M.T., Al-Saleh, M.F.: A variation of ranked set sampling. J. Appl. Stat. Sci. 10, 137–146 (2000)
Al-Saleh, M.F., Al-Ananbeh, A.M.: Estimation of the means of the bivariate normal using moving extreme ranked set sampling with concomitant variable. Stat. Pap. 48, 179–195 (2007)
Al-Saleh, M.F., Al-Hadhrami, S.A.: Estimation of the mean of the exponential distribution using moving extremes ranked set sampling. Stat. Pap. 44, 367–382 (2003)
Al-Saleh, M.F., Samawi, H.: On estimating the odds using moving extreme ranked set sampling. Stat. Methodol. 7, 133–140 (2010)
Bamber, D.C.: The area above the ordinal dominance graph and the area below the receiver operating characteristic graph. J. Math. Psychol. 12, 387–415 (1975)
Chen, Z.: Ranked set sampling: its essence and some new applications. Environ. Ecol. Stat. 14, 355–363 (2007)
Chen, H., Stasny, E.A., Wolfe, D.A.: Ranked set sampling for efficient estimation of a population proportion. Stat. Med. 24, 3319–3329 (2005)
Chen, H., Stasny, E.A., Wolfe, D.A.: Improved procedures for estimation of disease prevalence using ranked set sampling. Biom. J. 49, 530–538 (2007)
Dell, T.R., Clutter, J.L.: Ranked set sampling theory with order statistics background. Biometrics 28(2), 545–555 (1972)
Duembgen, L., Zamanzade, E.: Inference on a distribution function from ranked set samples. Ann. Inst. Stat. Math. 72, 157–185 (2020)
Fligner, M.A., MacEachern, S.N.: Nonparametric two-sample methods for ranked-set sample data. J. Am. Stat. Assoc. 101(475), 1107–1118 (2006)
Frey, J.: A note on ranked-set sampling using a covariate. J. Stat. Plan. Infer. 141, 809–816 (2011)
Huang, J.: Asymptotic properties of the NPMLE of a distribution function based on ranked set samples. Ann. Stat. 25(3), 1036–1049 (1997)
Kotz, S., Lumelskii, Y., Pensky, M.: The Stress-Strength Model and Its Generalizations. Theory and Applications. World Scientific, Singapore (2003)
Kvam, P.H.: Ranked set sampling based on binary water quality data with covariates. J. Agric. Biol. Environ. Stat. 8, 271–279 (2003)
Kvam, P.H., Samaniego, F.J.: Nonparametric maximum likelihood estimation based on ranked set samples. J. Am. Stat. Assoc. 89, 526–537 (1994)
McIntyre, G.A.: A method for unbiased selective sampling using ranked set sampling. Aust. J. Agric. Res. 3, 385–390 (1952)
MacEachern, S.N., Ozturk, O., Wolfe, D.A., Stark, G.V.: A new ranked set sample estimator of variance. J. R. Stat. Soc. B 64, 177–188 (2002)
Mahdizadeh, M., Zamanzade, E.: Efficient body fat estimation using multistage pair ranked set sampling. Stat. Methods Med. Res. 28, 223–234 (2019)
Ozturk, O., Balakrishnan, N.: An exact control-versus-treatment comparison test based on ranked set samples. Biometrics 65, 1213–1222 (2009)
Samawi, H.M., Al-Sagheer, O.A.: On the estimation of the distribution function using extreme and median ranked set samples. Biom. J. 43(3), 357–373 (2001)
Samawi, H.M., Al-Saleh, M.F.: Valid estimation of odds ratio using two types of moving extreme ranked set sampling. J. Korean Stat. Soc. 42, 17–24 (2013)
Samawi, H.M., Rochani, H., Linder, D., Chatterjee, A.: More efficient logistic analysis using moving extreme ranked set sampling. J. Appl. Stat. 44(4), 753–766 (2017)
Samawi, H.M., Helu, A., Rochani, H., Yin, J., Yu, L., Vogel, R.: Reducing sample size needed for accelerated failure time model using more efficient sampling methods. Stat. Theory Pract. 12(3), 530–541 (2018)
Samawi, H.M., Yu, L., Rochani, H., Vogel, R.: Reducing sample size needed for Cox-proportional hazards model analysis using more efficient sampling method. Commun. Stat. Theory Methods 49, 1281–1298 (2020)
Stokes, S.L.: Estimation of variance using judgment ordered ranked set samples. Biometrics 36, 35–42 (1980)
Stokes, S.L., Sager, T.W.: Characterization of a ranked-set sample with application to estimating distribution functions. J. Am. Stat. Assoc. 83, 374–381 (1988)
Takahasi, K., Wakimoto, K.: On unbiased estimates of the population mean based on the sample stratified by means of ordering. Ann. Inst. Stat. Math. 20, 1–31 (1968)
Wang, X., Lim, J., Stokes, L.: Using ranked set sampling with cluster randomized designs for improved inference on treatment effects. J. Am. Stat. Assoc. 111, 1576–1590 (2017)
Zamanzade, E., Mahdizadeh, M.: A more efficient proportion estimator in ranked set sampling. Stat. Probab. Lett. 129, 28–33 (2017)
Zamanzade, E., Mahdizadeh, M.: Estimating the population proportion in ranked set sampling with application to air quality monitoring. J. Appl. Stat. 45(3), 426–437 (2018)
Zamanzade, E., Mahdizadeh, M.: Using ranked set sampling with extreme ranks in estimating the population proportion. Stat. Methods Med. Res. 29, 165–177 (2020)
Zamanzade, E., Wang, X.: Estimation of population proportion for judgment post-stratification. Comput. Stat. Data Anal. 112, 257–269 (2017)
Acknowledgements
The authors thank the reviewers for helpful comments that have resulted in an improved paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zamanzade, E., Mahdizadeh, M. & Samawi, H.M. Efficient estimation of cumulative distribution function using moving extreme ranked set sampling with application to reliability. AStA Adv Stat Anal 104, 485–502 (2020). https://doi.org/10.1007/s10182-020-00368-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10182-020-00368-3