Imperfect Ranking Research Articles

Partial ranked set sampling design

In many environmental studies, the main focus is on observational economy, that is, to obtain data on the basis of cost‐effective and efficient sampling methods. In this paper, we propose a partial ranked set sampling (PRSS) method for estimation of population mean, median and variance. On the basis of perfect and imperfect rankings, Monte Carlo simulations from symmetric and asymmetric distributions are used to evaluate the effectiveness of the proposed estimators. It is found that the estimators under PRSS are more efficient than the estimators based on simple random sampling. The procedure is illustrated with a case study using a real data set. Copyright © 2013 John Wiley & Sons, Ltd.

On the Effect of Imperfect Ranking on the Amount of Fisher Information in Ranked Set Samples

It is well known that a ranked set sample under perfect ranking provides more information than an i.i.d. sample of the same size. Then it may be interesting to study how much information is lost due to imperfect ranking. In this article, we consider some ranking mechanisms and study the loss of the Fisher information according to the degree of imperfect ranking. Then we continue to discuss the optimal combination of the sample size and number of strata in terms of maximizing the Fisher information for the bivariate normal and exponential distributions.

Parametric estimation of location and scale parameters in ranked set sampling

This paper develops parametric inference for the parameters of location-scale family of distributions based on a ranked set sample. Likelihood function incorporates within-set ranking errors into the model through a missing data mechanism. The maximum likelihood estimators of the location-scale and missing data model parameters are constructed and an EM-algorithm is provided. It is shown that the proposed estimator is robust against imperfect ranking error and provides higher efficiency over its competitors.

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.

Robust Inference Against Imperfect Ranking in Ranked Set Sampling

This article develops robust statistical inference against imperfect ranking in a ranked set sample data obtained from a family of discrete distributions. We provide a class of disparity estimators that minimizes the disparities between the empirical and model distributions. It is shown that the Hellinger disparity estimator in this class is the most stable estimator in the neighborhood of an epsilon-contaminated judgment ranking model. When the true model is equal to assumed model and there is no ranking error, all disparity estimators are asymptotically efficient. A simulation study is provided to discuss the finite sample properties of the estimator. The proposed procedure is applied to a ranked set sample data in a water quality study.

Unbalanced Ranked Set Sampling for Estimating A Population Proportion Under Imperfect Rankings

The application of unbalanced ranked set sampling (RSS) to estimation of a population proportion has been studied for the perfect ranking situation. When the rankings are not perfect, the probabilities of success ranks for the judgment order statistics incorporate information on ranking errors as well as ranks. The objective of this article is to investigate the ranking errors effect of imperfection in rankings on unbalanced RSS for binary variables and provide methods to obtain estimates for the probabilities of success for the judgment order statistics using training samples so that Neyman allocation can be implemented. We also use a substantial data set, the NHANES III data, to demonstrate the feasibility and benefits of Neyman allocation in RSS for binary variables in the case of imperfect rankings.

Ranked set sampling for ordered categorical variables

AbstractThe authors show how ranked set sampling, both balanced and unbalanced, can be extended to ordered categorical variables with the goal of estimating the probabilities of all categories. They use ordinal logistic regression to aid in the ranking of the ordinal variable of interest. They also propose an optimal allocation scheme and methods for implementing it under either perfect or imperfect rankings. Results from a simulation study using data from the third National Health and Nutrition Examination Survey indicate that the use of ordinal logistic regression in ranking leads to substantial gains in precision for estimation of cell probabilities.

Unbiased estimation of P(X>Y) using ranked set sample data

The problem considered is that of an unbiased estimation of P[X>Y] using ranked set sample data for two independent random variables X and Y with unknown probability distributions. Postulating a model for imperfect ranking, it is proved that the ranked set samples provide an unbiased estimator with smaller variance as compared with simple random samples of same sizes, even when the rankings are imperfect. It is further shown that the ranked set sampling provides maximum efficiency when the rankings are perfect.

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 Two-Sample Methods for Ranked-Set Sample Data

A new collection of procedures is developed for the analysis of two-sample, ranked-set samples, providing an alternative to the Bohn–Wolfe procedure. These procedures split the data based on the ranks in the ranked-set sample and lead to tests for the centers of distributions, confidence intervals, and point estimators. The advantages of the new tests are that they require essentially no assumptions about the mechanism by which rankings are imperfect, that they maintain their level whether rankings are perfect or imperfect, that they lead to generalizations of the Bohn–Wolfe procedure that can be used to increase power in the case of perfect rankings, and that they allow one to analyze both balanced and unbalanced ranked-set samples. A new class of imperfect ranking models is proposed, and the performance of the procedure is investigated under these models. When rankings are random, a theorem is presented which characterizes efficient data splits. Because random rankings are equivalent to iid samples, this theorem applies to a wide class of statistics and has implications for a variety of computationally intensive methods.

New imperfect rankings models for ranked set sampling

Ranked set sampling is a sampling approach that leads to improved statistical inference in situations where the units to be sampled can be ranked relative to each other prior to formal measurement. This ranking may be done either by subjective judgment or according to an auxiliary variable, and it need not be completely accurate. In fact, results in the literature have shown that no matter how poor the quality of the ranking, procedures based on ranked set sampling tend to be at least as efficient as procedures based on simple random sampling. However, efforts to quantify the gains in efficiency for ranked set sampling procedures have been hampered by a shortage of available models for imperfect rankings. In this paper, we introduce a new class of models for imperfect rankings, and we provide a rigorous proof that essentially any reasonable model for imperfect rankings is a limit of models in this class. We then describe a specific, easily applied method for selecting an appropriate imperfect rankings model from the class.

Unbiased variance estimation in a simple exponential population using ranked set samples

The problem considered in this paper is that of unbiased estimation of the variance of an exponential distribution using a ranked set sample (RSS). We propose some unbiased estimators each of which is better than the non-parametric minimum variance quadratic unbiased estimator based on a balanced ranked set sample as well as the uniformly minimum variance unbiased estimator based on a simple random sample (SRS) of the same size. Relative performances of the proposed estimators and a few other properties of the estimators including their robustness under imperfect ranking have also been studied.

Statistical quality control based on ranked set sampling

Different quality control charts for the sample mean are developed using ranked set sampling (RSS), and two of its modifications, namely median ranked set sampling (MRSS) and extreme ranked set sampling (ERSS). These new charts are compared to the usual control charts based on simple random sampling (SRS) data. The charts based on RSS or one of its modifications are shown to have smaller average run length (ARL) than the classical chart when there is a sustained shift in the process mean. The MRSS and ERSS methods are compared with RSS and SRS data, it turns out that MRSS dominates all other methods in terms of the out-of-control ARL performance. Real data are collected using the RSS, MRSS, and ERSS in cases of perfect and imperfect ranking. These data sets are used to construct the corresponding control charts. These charts are compared to usual SRS chart. Throughout this study we are assuming that the underlying distribution is normal. A check of the normality for our example data set indicated that the normality assumption is reasonable.

Controlled sampling using ranked set sampling

Based on ranked set sampling procedure, a technique to obtain a controlled preferable sample is introduced. Using this technique it is possible to obtain an approximate preferable sample from a population of interest. In particular, the technique can be used to obtain sample points that tend to be from a specific portion of a distribution such as the lower or upper quartiles. The performance of the technique is investigated under perfect and imperfect ranking. Some other related topics are also discussed.

Parametric estimation for the location parameter for symmetric distributions using moving extremes ranked set sampling with application to trees data

AbstractA modification of ranked set sampling (RSS) called moving extremes ranked set sampling (MERSS) is considered parametrically, for the location parameter of symmetric distributions. A maximum likelihood estimator (MLE) and a modified MLE are considered and their properties are studied. Their efficiency with respect to the corresponding estimators based on simple random sampling (SRS) are compared for the case of normal distribution. The method is studied under both perfect and imperfect ranking (with error in ranking). It appears that these estimators can be real competitors to the MLE using (SRS). The procedure is illustrated using tree data. Copyright © 2003 John Wiley & Sons, Ltd.

Estimation of the mean of the exponential distribution using moving extremes ranked set sampling

Moving Extremes Ranked Set Sampling (MERSS) is a useful modification of Ranked Set Sampling (RSS). Unlike RSS, MERSS allows for an increase of set size without introducing too much ranking error. The method is considered parametrically under exponential distribution. Maximum likelihood estimator (MLE), and a modified MLE are considered and their properties are studied. The method is studied under both perfect and imperfect ranking (with error in ranking). It appears that these estimators can be real competitors to the MLE using the usual simple random sampling (SRS).

Investigating the estimation of the population mean using random ranked set samples

Ranked set sampling (RSS) uses fixed set size and number of cycles (or replications). In real life however, we may encounter problems that requiring random set size or number of cycles or both. In dealing with such problems we suggest several unbiased estimators of the population mean using random ranked set sampling (RRSS) method in case of perfect ranking and imperfect ranking. The efficiencies of the estimators of the population mean under RRSS and RSS are compared in the case of perfect ranking. The results show, under certain conditions, the efficiency of estimators is improved by using RRSS in the case of perfect ranking. Finally the asymptotic properties of the proposed estimators of the population mean are discussed for prefect and imperfect ranking.

On the theory of ranked-set sampling and its ramifications

We consider in this article ranked-set sampling (RSS) and its ramifications including RSS with imperfect ranking, RSS by ranking a concomitant variable and RSS with multivariate samples, etc. We deal with a unified sampling scheme which is referred to as generalized ranked-set sampling (GRSS) and which includes RSS and its ramifications as special cases. We develop a general theory for GRSS in both parametric and nonparametric settings. In a parametric setting, it is shown that the Fisher information matrix about the unknown parameters of a GRSS sample minus that of an SRS sample of the same size is always positive definite. In a nonparametric setting, a particular model, the smooth-function-of-means model, is considered and it is proved that the method-of-moment estimates of parameters based on a GRSS sample will always have smaller asymptotic variances than those based on an SRS sample of the same size. An example for RSS with multivariate samples is treated in detail and a simulation study is reported. Some other issues and open problems such as those involving optimal designs for the GRSS are also discussed.

U-Statistics and imperfect ranking in ranked set sampling

Ranked set sampling has attracted considerable attention as an efficient sampling design, particularly for environmental and ecological studies. A number of authors have noted a gain in efficiency over ordinary random sampling when specific estimators and tests of hypotheses are applied to rank set sample data. We generalize such results by deriving the asymptotic distribution for random sample U-statistics when applied to ranked set sample data. Our results show that the ranked set sample procedure is asymptotically at least as efficient as the random sample procedure, regardless of the accuracy of judgement ranking. Some errors in the ranked set sampling literature are also revealed, and counterexamples provided. Finally, application of majorization theory to these results shows when perfect ranking can be expected to yield greater efficiency than imperfect ranking.

A performance indicator for ranked set sampling using ranking error probability matrix

Ranked set sampling (RSS) is a sampling procedure that has been shown to provide more efficient procedures than simple random sampling, in particular the Mann-Whitney-Wilcoxon (MWW) statistic and the empirical distribution function (EDF). We briefly review the work of Bohn (1992) and Stokes and Sager (1988) on the effect of imperfect ranking on the RSS-based MWW test and on the RSS-based EDF, respectively. We propose a model for a ranking error probability matrix which we hope will become a useful tool for evaluating RSS-based statistical procedures