Omer Ozturk

A new ranked set sample estimator of variance

Summary. We develop an unbiased estimator of the variance of a population based on a ranked set sample. We show that this new estimator is better than estimating the variance based on a simple random sample and more efficient than the estimator based on a ranked set sample proposed by Stokes. Also, a test to determine the effectiveness of the judgment ordering process is proposed.

Two-sample inference based on one-sample ranked set sample sign statistics

A two-sample sign test is developed for ranked set samples. It is shown that the testing procedure is distribution free but requires evaluation of the incomplete beta function. Efficiency and type I error, in general, depend on the measured observations and the ratio of cycle sizes. It is shown that there is a substantial gain in the efficiency of the test even when there are ranking errors. On the other hand, the type I error is inflated for imperfect ranking. The proposed test is superior to the two-sample Mann-Whitney-Wilcoxon ranked set sample test when the underlying probability model has heavy and long tail distribution, and the number of quantified observations in each cycle is small. We provide a simple way to implement the procedure.

Alternative ranked set sampling protocols for the sign test

In this paper we investigate the effects of different ranked set sampling protocols on the sign test statistic. Sampling protocols considered include sequential, mid-range and fixed sampling designs. We show that in all of these sampling protocols the introduction of any correlation structure on the quantified observations leads to a reduction in the Pitman efficiency of the sign test. In the fixed sampling protocol, design optimality is achieved when only the middle observation is quantified from each set.

Sampling from partially rank-ordered sets

In this paper we introduce a new sampling design. The proposed design is similar to a ranked set sampling (RSS) design with a clear difference that rankers are allowed to declare any two or more units are tied in ranks whenever the units can not be ranked with high confidence. These units are replaced in judgment subsets. The fully measured units are then selected from these partially ordered judgment subsets. Based on this sampling scheme, we develop unbiased estimators for the population mean and variance. We show that the proposed sampling procedure has some advantages over standard ranked set sampling.

Statistical inference under a stochastic ordering constraint in ranked set sampling

This paper introduces estimators for the judgment class and population cumulative distribution functions (CDF) under stochastic order restriction. The estimators are defined as the minimizer of a version of the Cramér-von Mises distance function. It is shown that the new estimators are strongly and uniformly consistent for the judgment class population distributions and have smaller integrated mean square errors than the integrated mean square errors of the empirical CDF estimators. The proposed estimators are used to calibrate the effect of imperfect ranking on statistical procedures. It is shown that this calibration works quite well in ranked set sample Mann–Whitney–Wilcoxon rank-sum and sign tests. The use of estimators and calibration procedure are illustrated on a ranked set sample data.

Nonparametric Ranked-set Sampling Confidence Intervals for Quantiles of a Finite Population

Ranked-set sampling from a finite population is considered in this paper. Three sampling protocols are described, and procedures for constructing nonparametric confidence intervals for a population quantile are developed. Algorithms for computing coverage probabilities for these confidence intervals are presented, and the use of interpolated confidence intervals is recommended as a means to approximately achieve coverage probabilities that cannot be achieved exactly. A simulation study based on finite populations of sizes 20, 30, 40, and 50 shows that the three sampling protocols follow a strict ordering in terms of the average lengths of the confidence intervals they produce. This study also shows that all three ranked-set sampling protocols tend to produce confidence intervals shorter than those produced by simple random sampling, with the difference being substantial for two of the protocols. The interpolated confidence intervals are shown to achieve coverage probabilities quite close to their nominal levels. Rankings done according to a highly correlated concomitant variable are shown to reduce the level of the confidence intervals only minimally. An example to illustrate the construction of confidence intervals according to this methodology is provided.

One- and two-sample sign tests for ranked set sample selective designs

Statistical inference based on a ranked set sample depends very much on the location of the quantified observations. A selective design which determines the location of the quantified observations in a ranked set sample is introduced. The paper investigates the effects of selective designs on one and two sample sign test statistics. The Pitman efficiencies of one- and two sample sign tests are calculated for selective designs and compared with ranked set samples of the same size. If the design quantifies observations at the center points, then the proposed procedure is superior to a ranked set sample of the same size in the sense of Pitman efficiency. Some practical problems are addressed for the two-sample sign test.

Estimation of population mean and variance in flock management : a ranked set sampling approach in a finite population setting

Ranked set sampling is a sampling technique that provides substantial cost efficiency in experiments where a quick, inexpensive ranking procedure is available to rank the units prior to formal, expensive and precise measurements. Although the theoretical properties and relative efficiencies of this approach with respect to simple random sampling have been extensively studied in the literature for the infinite population setting, the use of ranked set sampling methods has not yet been explored widely for finite populations. The purpose of this study is to use sheep population data from the Research Farm at Ataturk University, Erzurum, Turkey, to demonstrate the practical benefits of ranked set sampling procedures relative to the more commonly used simple random sampling estimation of the population mean and variance in a finite population. It is shown that the ranked set sample mean remains unbiased for the population mean as is the case for the infinite population, but the variance estimators are unbiased only with use of the finite population correction factor. Both mean and variance estimators provide substantial improvement over their simple random sample counterparts.

Nonparametric maximum-likelihood estimation of within-set ranking errors in ranked set sampling

A distribution-free statistical inference for the quality of within-set judgement ranking information is developed for ranked set samples. The judgement ranking information is modelled through Bohn–Wolfe (BW) model. The cumulative distribution function and the parameters of BW model are estimated by maximising nonparametric likelihood functions. A missing data model is introduced to construct an efficient computational algorithm. The advantages of the new estimators are that they require essentially no assumption on the underlying distribution function, which provides an estimate of the quality of within-set ranking information, and that they lead to a valid statistical inference even under imperfect ranking. The proposed estimators are applied to a water flow data set to estimate judgement ranking information and underlying distribution function.

Combining ranking information in judgment post stratified and ranked set sampling designs

Judgment post stratified (JPS) and ranked set sampling (RSS) designs rely on the ability of a ranker to assign ranks to potential observations on available experimental units. In many settings, there are often more than one rankers available and each of these rankers provide judgment ranks. This paper proposes two sampling schemes, one for JPS and the other for RSS, to combine the judgment ranks of these rankers to produce a strength of agreement measure for each fully measured unit. This strength measure is used to draw inference for the population mean and cumulative distribution function. The paper shows that the estimators constructed based on this strength measure provide a substantial improvement over the same estimators based on judgment ranking information of a single best ranker.