Abstract
Ranked set sampling (RSS) is an efficient method for estimating parameters when exact measurement of observation is difficult and/or expensive. In the current paper, several traditional and ad hoc estimators of the scale and shape parameters \(\theta \) and \(\alpha \) from the Pareto distribution \(p(\theta ,\alpha )\) will be respectively studied in cases when one parameter is known and when both are unknown under simple random sampling, RSS and some of its modifications such as extreme RSS(ERSS) and median RSS(MRSS). It is found for estimating of \(\theta \) from \(p(\theta ,\alpha )\) in which \(\alpha \) is known, the best linear unbiased estimator (BLUE) under ERSS is more efficient than the other estimators under the other sampling techniques. For estimating of \(\alpha \) from \(p(\theta ,\alpha )\) in which \(\theta \) is known, the modified BLUE under MRSS is more efficient than the other estimators under the other sampling techniques. For estimating of \(\theta \) and \(\alpha \) from \(p(\theta ,\alpha )\) in which both are unknown, the ad hoc estimators under ERSS are more efficient than the other estimators under the other sampling techniques. All efficiencies of these estimators are simulated under imperfect ranking. A real data set is used for illustration.
Similar content being viewed by others
References
Abu-Dayyeh WA, Al-Subh SA, Muttlak HA (2004) Logistic parameters estimation using simple random sampling and ranked set sampling data. Appl Math Comput 150(2):543–554
Abu-Dayyeh W, Assrhani A, Ibrahim K (2013) Estimation of the shape and scale parameters of pareto distribution using ranked set sampling. Stat Pap 54(1):207–225
Al-Saleh MF, Diab YA (2009) Estimation of the parameters of downton’s bivariate exponential distribution using ranked set sampling scheme. J Stat Plan Inference 139(2):277–286
Arnold BC (1983) Pareto Distributions. International Co-operative Publishing House, Fairland
Asrabadi BR (1990) Estimation in the pareto distribution. Metrika 37(1):199–205
Chen W, Xie M, Wu M (2013) Parametric estimation for the scale parameter for scale distributions using moving extremes ranked set sampling. Stat Probab Lett 83(9):2060–2066
Chen W, Xie M, Wu M (2016) Modified maximum likelihood estimator of scale parameter using moving extremes ranked set sampling. Commun Stat-Simul Comput 45(6):2232–2240
Chen W, Tian Y, Xie M (2017) Maximum likelihood estimator of the parameter for a continuous one-parameter exponential family under the optimal ranked set sampling. J Syst Sci Complex 30(6):1350–1363
Chen W, Tian Y, Xie M (2018) The global minimum variance unbiased estimator of the parameter for a truncated parameter family under the optimal ranked set sampling. J Stat Comput Simul 88(17):3399–3414
Dell DR, Clutter JL (1972) Ranked set sampling theory with order statistics background. Biometrics 28(2):545–555
Dey S, Salehi M, Ahmadi J (2017) Rayleigh distribution revisited via ranked set sampling. Metron 75(1):69–85
Esemen M, Grler S (2018) Parameter estimation of generalized rayleigh distribution based on ranked set sample. J Stat Comput Simul 88(4):615–628
Hassan AS (2013) Maximum likelihood and Bayes estimators of the unknown parameters for exponentiated exponential distribution using ranked set sampling. Int J Eng Res Appl 3(1):720–725
He X, Chen W, Qian W (2018) Maximum likelihood estimators of the parameters of the log-logistic distribution. Stat Pap. https://doi.org/10.1007/s00362-018-1011-3
Hogg RV, Klugman SA (1984) Loss distributions. Wiley, New York
Holmes JD, Moriarty WW (1999) Application of the generalized Pareto distribution to extreme value analysis in wind engineering. J Wind Eng Ind Aerodyn 83:1–10
Hussian MA (2014) Bayesian and maximum likelihood estimation for Kumaraswamy distribution based on ranked set sampling. Am J Math Stat 4(1):30–37
Lehmann EL (1983) Theory of point estimation. Wiley, New York
McIntyre GA (1952) A method for unbiased selective sampling, using ranked sets. Aust J Agric Res 3(4):385–390
Muttlak HA (1997) Median ranked set sampling. J Appl Stat Sci 6(4):245–255
Nadarajah S, Ali MM (2008) Pareto random variables for hydrological modeling. Water Resourc Manag 22(10):1381–1393
Rosen KT, Resnick M (1980) The size distribution of cities: an examination of the Pareto law and primacy. J Urban Econ 8(2):165–186
Saksena ASK, Johnson AM (1984) Best unbiased estimators for the parameters of a two-parameter pareto distribution. Metrika 31(1):77–83
Samawi HM, Ahmed MS, Abu-Dayyeh W (1996) Estimating the population mean using extreme ranked set sampling. Biometrical J 38(5):577–586
Shaibu AB, Muttlak HA (2004) Estimating the parameters of the normal, exponential and gamma distributions using median and extreme ranked set samples. Statistica 64(1):75–98
Sinha BK, Sinha BK, Purkayastha S (1996) On some aspects of ranked set sampling for estimation of normal and exponential parameters. Stat Decis 14(3):223–240
Stokes L (1995) Parametric ranked set sampling. Ann Inst Stat Math 47(3):465–482
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 21(1):249–255
Tripathi YM, Kumar S, Petropoulos C (2016) Estimating the shape parameter of a pareto distribution under restrictions. Metrika 79(1):91–111
Acknowledgements
The authors thank the referees for helpful comments that have led to 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.
This research was supported by the Fundamental Research Foundation of Xiangxi autonomous prefecture under Grant Nos. 2018SF5026.
Rights and permissions
About this article
Cite this article
Qian, W., Chen, W. & He, X. Parameter estimation for the Pareto distribution based on ranked set sampling. Stat Papers 62, 395–417 (2021). https://doi.org/10.1007/s00362-019-01102-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-019-01102-1