Estimating asymptotic variance of M-estimators in ranked set sampling
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Estimating asymptotic variance of M-estimators in ranked set sampling M. Mahdizadeh1
· Ehsan Zamanzade2
Received: 15 May 2019 / Accepted: 17 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract M-estimator for symmetric location families in ranked set sampling has been studied in the literature. Estimating asymptotic variance of this estimator has not been addressed yet. To fill this gap, we develop two consistent estimators in this article. The proposed estimators are then utilized to construct confidence intervals for the location parameter. Monte Carlo simulations are employed to assess performance of the intervals. Finally, an empirical study is presented for illustration. Keywords Judgment ranking · M-estimation · Robustness
1 Introduction In some situations, additional information on sample units is available in addition to characteristic of interest. This auxiliary information may be efficiently utilized to come up with more efficient inference methods. Ranked set sampling (RSS) is sampling technique that employs extra information, obtained through visual inspection or a covariate, so as to decrease likelihood of inclusion of unrepresentative units in sample. This is done by means of some preparatory rankings on the potential sample units before quantifying the final sample units. To draw a ranked set sample using set size k and cycle size m, the following procedure is repeated m times (cycles). First, k random samples, each of size k, are
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00180019-00946-3) contains supplementary material, which is available to authorized users.
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M. Mahdizadeh [email protected] Ehsan Zamanzade [email protected]; [email protected]
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Department of Statistics, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran
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Department of Statistics, University of Isfahan, Isfahan 81746-73441, Iran
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M. Mahdizadeh, E. Zamanzade
selected from the target population. The units in each sample are then ranked from smallest to largest, without any actual measurements. This could be based on subjective judgment, or a concomitant variable. Finally, the unit with rank r (r = 1, . . . , k) is measured from the r th sample. For brevity, the final sample is denoted by {X [r ] j : r = 1, . . . , k; j = 1, . . . , m}, where X [r ] j is the r th order statistic in the jth cycle. To refer to the fact that orderings need not be exact, the term “judgement” order statistic is usually used for X [r ] j ’s. Perfect ranking is the situation that the judgment order statistics coincide with the true ones. McIntyre (1952) was the first who introduced RSS in an agricultural experiment. Since then, it has found applications in a wide range of disciplines. Some recent examples include fishery (Ozturk 2011), auditing (Gemayel et al. 2012), agriculture (Mahdizadeh and Zamanzade 2018a, b, c; Ozturk 2019), reliability (Mahdizadeh and Zamanzade 2017a, c), medicine (Zamanzade and Mahdizadeh 2017, 2019; Mahdi
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