Unbalanced Ranked Set Sampling Research Articles

Empirical likelihood inference for area under the receiver operating characteristic curve using ranked set samples.

The area under a receiver operating characteristic curve (AUC) is a useful tool to assess the performance of continuous-scale diagnostic tests on binary classification. In this article, we propose an empirical likelihood (EL) method to construct confidence intervals for the AUC from data collected by ranked set sampling (RSS). The proposed EL-based method enables inferences without assumptions required in existing nonparametric methods and takes advantage of the sampling efficiency of RSS. We show that for both balanced and unbalanced RSS, the EL-based point estimate is the Mann-Whitney statistic, and confidence intervals can be obtained from a scaled chi-square distribution. Simulation studies and two case studies on diabetes and chronic kidney disease data suggest that using the proposed method and RSS enables more efficient inference on the AUC.

Quantile estimation using near optimal unbalanced ranked set sampling

Quantile estimation using near optimal unbalanced ranked set sampling

On multiple imputation for unbalanced ranked set samples with applications in quantile estimation

We consider multiple imputation (MI) for unbalanced ranked set samples (URSS) by considering them as data sets with missing values. We replace each missing value with a set of plausible values drawn from a predictive distribution that represents the uncertainty about the appropriate value to impute. Using the structure of the MI dataset, we develop algorithms that imitate the structure of URSS to carry out the desired statistical inference. We provide results for the convergence of the empirical distribution functions of imputed samples to the population distribution function, under both URSS and simple random sampling (SRS). We obtain the variances of the imputed URSS, and the expected values of the variance estimators. We also study the problem of quantile estimation using an imputed URSS and propose a hybrid method based on the bootstrap and imputation of URSS data. We apply our results to estimate the mean and quantiles of the mercury in contaminated fish under perfect and imperfect URSS.

Simple Unequal Allocation Procedure for Ranked Set Sampling with Skew Distributions

A practical unbalanced Ranked Set Sampling (RSS) model is proposed to estimate the population mean of positively skewed distributions. The gains in the relative precisions of the population mean based on the proposed model for chosen distributions are uniformly higher than those based on balanced RSS and the t-model proposed in Kaur et al. (1997). The relative precisions of the simple unequal allocation model are, with one exception, better than (s, t)-model which is better than t-model. The relative precision of the proposed model is very close or equal to the optimal Neyman allocation model.

Bounds on the efficiency of unbalanced ranked-set sampling

Takahasi and Wakimoto (1968) derived a sharp upper bound on the efficiency of the balanced ranked-set sampling (RSS) sample mean relative to the simple random sampling (SRS) sample mean under perfect rankings. The bound depends on the set size and is achieved for uniform distributions. Here we generalize the Takahasi and Wakimoto (1968) result by finding a sharp upper bound in the case of unbalanced RSS. The bound depends on the particular unbalanced design, and the distributions where the bound is achieved can be highly nonuniform. The bound under perfect rankings can be exceeded under imperfect rankings.

Simple Unbalanced Ranked Set Sampling for Mean Estimation of Response Variable of Developmental Programs

Simple Unbalanced Ranked Set Sampling for Mean Estimation of Response Variable of Developmental Programs

An omnibus two-sample test for ranked-set sampling data

We develop an omnibus two-sample test for ranked-set sampling (RSS) data. The test statistic is the conditional probability of seeing the observed sequence of ranks in the combined sample, given the observed sequences within the separate samples. We compare the test to existing tests under perfect rankings, finding that it can outperform existing tests in terms of power, particularly when the set size is large. The test does not maintain its level under imperfect rankings. However, one can create a permutation version of the test that is comparable in power to the basic test under perfect rankings and also maintains its level under imperfect rankings. Both tests extend naturally to judgment post-stratification, unbalanced RSS, and even RSS with multiple set sizes. Interestingly, the tests have no simple random sampling analog.

An Unbalanced Ranked-Set Sampling Method to Get More Than One Sample from Each Set

An Unbalanced Ranked-Set Sampling Method to Get More Than One Sample from Each Set

Unbalanced ranked set sampling in cluster randomized studies

We consider the use of unbalanced ranked set sampling (URSS) with cluster randomized designs (CRDs), and extend nonparametric estimators and testing methods, previously developed by Wang et al. (2016) for the use of balanced RSS (BRSS) with CRDs, to account for unbalanced stratified structures under different ranking schemes. We study the optimality, finite-sample and asymptotic properties of the URSS estimators, and numerically quantify and compare the relative efficiency of the URSS vs. BRSS estimators. We also study and compare the power of the URSS tests vs. their BRSS counterparts via simulation. Further, we investigate the application of the proposed methods to unbalanced data from BRSS-structured CRDs due to missing observations and illustrate it with an example using educational data. Finally, based on our results, we offer recommendations about when to use URSS/BRSS with CRDs.

Tests of perfect judgment ranking using pseudo-samples

Ranked set sampling (RSS) is a sampling approach that can produce improved statistical inference when the ranking process is perfect. While some inferential RSS methods are robust to imperfect rankings, other methods may fail entirely or provide less efficiency. We develop a nonparametric procedure to assess whether the rankings of a given RSS are perfect. We generate pseudo-samples with a known ranking and use them to compare with the ranking of the given RSS sample. This is a general approach that can accommodate any type of raking, including perfect ranking. To generate pseudo-samples, we consider the given sample as the population and generate a perfect RSS. The test statistics can easily be implemented for balanced and unbalanced RSS. The proposed tests are compared using Monte Carlo simulation under different distributions and applied to a real data set.

Exponentially tilted empirical distribution function for ranked set samples

We study nonparametric estimation of the distribution function (DF) of a continuous random variable based on a ranked set sampling design using the exponentially tilted (ET) empirical likelihood method. We propose ET estimators of the DF and use them to construct new resampling algorithms for unbalanced ranked set samples. We explore the properties of the proposed algorithms. For a hypothesis testing problem about the underlying population mean, we show that the bootstrap tests based on the ET estimators of the DF are asymptotically normal and exhibit a small bias of order O(n−1). We illustrate the methods and evaluate the finite sample performance of the algorithms under both perfect and imperfect ranking schemes using a real data set and several Monte Carlo simulation studies. We compare the performance of the test statistics based on the ET estimators with those based on the empirical likelihood estimators.

Jackknife empirical likelihood inferences for the population mean with ranked set samples

Without requiring any easily violated assumptions needed by existing rank-based nonparametric methods for ranked set samples, we propose a nonparametric approach for interval estimation and hypothesis testing for the population mean and the difference between two population means with balanced and unbalanced ranked set samples using jackknife empirical likelihood.

Weighted Estimation of Quantiles Using Unbalanced Ranked Set Sampling

HeIn the case where the population distribution is unknown, the nonparametric weighted estimation of quantiles under unbalanced ranked set sampling is proposed because observations with different ranks are not identically distributed. It is shown analytically the optimal weighted version always improves the efficiency for all distributions whether ranking is perfect or not. Then, the sampling allocation that maximizes the estimation efficiency is identified and shown to not depend on the population distribution. Finally, results of computed relative efficiencies and an application to a real data set are presented to illustrate some of the theoretical findings.

Resampling Unbalanced Ranked Set Samples With Applications in Testing Hypothesis About the Population Mean

Ranked set sampling is a sampling approach that could lead to improved statistical inference when the actual measurement of the variable of interest is difficult or expensive to obtain but sampling units can be easily ordered by some means without actual quantification. In this paper, we consider the problem of bootstrapping an unbalanced ranked set sample (URSS) where the number of observations from each artificially created stratum can be unequal. We discuss resampling a URSS through transforming it into a balanced RSS and extending the existing algorithms. We propose two methods that are designed to obtain resamples from the given URSS. Algorithms are provided and several properties, including asymptotic normality of estimates, are discussed. The proposed methods are compared with the parametric bootstrap using Monte Carlo simulations for the problem of testing a hypothesis about the population mean.

Most powerful rank tests for perfect rankings

We consider the problem of testing for perfect rankings in ranked set sampling (RSS). By using a new algorithm for computing the probability that specified independent random variables have a particular ordering, we find most powerful rank tests of the null hypothesis of perfect rankings against fully specified alternatives. We compare the power of these most powerful rank tests to that of existing rank tests in the literature, and we find that the existing tests are surprisingly close to optimal over a wide range of alternatives to perfect rankings. This finding holds both for balanced RSS and for unbalanced RSS cases where the different ranks are not equally represented in the sample. We find that the best of the existing tests is the test that rejects when the null probability of the observed ranks is small, and we provide a new, more efficient R function for computing the test statistic.

Variance estimation using judgment post-stratification

We consider the problem of estimating the variance of a population using judgment post-stratification. By conditioning on the observed vector of ordered in-stratum sample sizes, we develop a conditionally unbiased nonparametric estimator that outperforms the sample variance except when the rankings are very poor. This estimator also outperforms the standard unbiased nonparametric variance estimator from unbalanced ranked-set sampling.

Ranked Set Sampling for a Population Proportion:Allocation of Sample Units to Each Judgment Order Statistic

Ranked set sampling is an alternative to simple random sampling that has been shown to outperform simple random sampling in many situations by reducing the variance of an estimator, thereby providing the same accuracy with a smaller sample size than is needed in simple random sampling. Ranked set sampling involves preliminary ranking of potential sample units on the variable of interest using judgment or an auxiliary variable to aid in sample selection. Ranked set sampling prescribes the number of units from each rank order to be measured. Balanced ranked set sampling assigns equal numbers of sample units to each rank order. Unbalanced ranked set sampling allows unequal allocation to the various ranks, but this allocation may be sensitive to the quality of information available to do the allocation. In this paper we use a simulation study to conduct a sensitivity analysis of optimal allocation of sample units to each of the order statistics in unbalanced ranked set sampling. Our motivating example comes from the National Survey of Families and Households.

Estimation of the scale parameter of a log-logistic distribution

In this work the ranked set sampling technique has been applied to estimate the scale parameter \(\alpha \) of a log-logistic distribution under a situation where the units in a sample can be ordered by judgement method without any error. We have evaluated the Fisher information contained in the order statistics arising from this distribution and observed that median of a random sample contains the maximum information about the parameter \(\alpha \). Accordingly we have used median ranked set sampling to estimate \(\alpha \). We have further carried out the multistage median ranked set sampling to estimate \(\alpha \) with improved precision. Suppose it is not possible to rank the units in a sample according to judgement method without error but the units can be ordered based on an auxiliary variable \(Z\) such that \((X, Z)\) has a Morgenstern type bivariate log-logistic distribution (MTBLLD). In such a situation we have derived the Fisher information contained in the concomitant of rth order statistic of a random sample of size \(n\) from MTBLLD and identified those concomitants among others which possess largest amount of Fisher information and defined an unbalanced ranked set sampling utilizing those units in the sample and thereby proposed an estimator of \(\alpha \) using the measurements made on those units in this ranked set sample.

Empirical Likelihood Inference for Population Quantiles with Unbalanced Ranked Set Samples

We consider nonparametric interval estimation for the population quantiles based on unbalanced ranked set samples. We derived the large sample distribution of the empirical log likelihood ratio statistic for the quantiles. Approximate intervals for quantiles are obtained by inverting the likelihood ratio statistic. The performance of the empirical likelihood interval is investigated and compared with the performance of the intervals based on the ranked set sample order statistics.

A note on ranked-set sampling using a covariate

Ranked-set sampling (RSS) and judgment post-stratification (JPS) use ranking information to obtain more efficient inference than is possible using simple random sampling. Both methods were developed with subjective, judgment-based rankings in mind, but the idea of ranking using a covariate has received a lot of attention. We provide evidence here that when rankings are done using a covariate, the standard RSS and JPS mean estimators no longer make efficient use of the available information. We first show that when rankings are done using a covariate, the standard nonparametric mean estimators in JPS and unbalanced RSS are inadmissible under squared error loss. We then show that when rankings are done using a covariate, nonparametric regression techniques yield mean estimators that tend to be significantly more efficient than the standard RSS and JPS mean estimators. We conclude that the standard estimators are best reserved for settings where only subjective, judgment-based rankings are available.