On the performance of estimation methods under ranked set sampling
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On the performance of estimation methods under ranked set sampling Cesar Augusto Taconeli1
· Wagner Hugo Bonat1
Received: 18 May 2019 / Accepted: 6 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Maximum likelihood estimation (MLE) applied to ranked set sampling (RSS) designs is usually based on the assumption of perfect ranking. However, it may suffers of lack of efficiency when ranking errors are present. The main goal of this article is to investigate the performance of six alternative estimation methods to MLE for parameter estimation under RSS. We carry out an extensive simulation study and measure the performance of the maximum product of spacings, ordinary and weighted least-squares, Cramér-von-Mises, Anderson–Darling and right-tail Anderson–Darling estimators, along with the maximum likelihood estimators, through the Kullback–Leibler divergence from the true and estimated probability density functions. Our simulation study considered eight continuous probability distributions, six sample sizes and six levels of correlation between the interest and concomitant variables. In general, our results show that the Anderson–Darling method outperforms its competitors and that the maximum likelihood estimators strongly depends on perfect ranking for accurate estimation. Finally, we present an illustrative example using a data set concerning the percent of body fat. R code is available in the supplementary material. Keywords Anderson–Darling statistic · Kullback–Leibler divergence · Maximum likelihood estimation · Monte Carlo simulation · Statistical efficiency
1 Introduction Ranked set sampling (RSS) is a cost-efficient sampling design for situations where the actual measurements of sample units are difficult to obtain. For instance, the data collection can be expensive, destructive or time-consuming, however, it may be
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Cesar Augusto Taconeli [email protected] Wagner Hugo Bonat [email protected]
1
Department of Statistics, Federal University of Paraná, Curitiba, Paraná, Brazil
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C. A. Taconeli and W. H. Bonat
possible ranking sample units in small sets using some accessible and inexpensive ranking criterion. The ranking process can be based on values of some concomitant variable, personal judgment or visual comparison of the sample units, etc. We say that the ranking is perfect, if the sample units can be ranked without errors, and imperfect, otherwise. As expected, the accuracy of the estimators provided by RSS and its extensions become lower when the ranking process is more susceptible to errors. The ranked set sampling design can be described as follows: Step 1 Identify n 2 elements from the target population and divide them randomly into n sets of size n; Step 2 Rank the sample units within each set according to some easy and inexpensive ranking criterion; Step 3 Select the unit ranked in position i from the ith set, for i = 1, . . . , n, to compose the final sample. Only these observations must be effectively measured for the variable of interest; Step 4 Steps 1–3 c
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