On distribution function estimation with partially rank-ordered set samples: estimating mercury level in fish using length frequency data

We study the non-parametric estimation of a continuous distribution function F based on the partially rank-ordered set (PROS) sampling design. A PROS sampling design first selects a random sample from the underlying population and uses judgement ranking to rank them into partially ordered sets, without measuring the variable of interest. The final measurements are then obtained from one of the partially ordered sets. Considering an imperfect PROS sampling procedure, we first develop the empirical distribution function (EDF) estimator of F and study its theoretical properties. Then, we consider the problem of estimating F, where the underlying distribution is assumed to be symmetric. We also find a unique admissible estimator of F within the class of nondecreasing step functions with jumps at observed values and show the inadmissibility of the EDF. In addition, we introduce a smooth estimator of F and discuss its theoretical properties. Finally, we expand on various numerical illustrations of our results via several simulation studies and a real data application and show the advantages of PROS estimates over their counterparts under the simple random and ranked set sampling designs.     

Mixture Model Analysis of Partially Rank‐Ordered Set Samples: Age Groups of Fish from Length‐Frequency Data

We present a novel methodology for estimating the parameters of a finite mixture model (FMM) based on partially rank‐ordered set (PROS) sampling and use it in a fishery application. A PROS sampling design first selects a simple random sample of fish and creates partially rank‐ordered judgement subsets by dividing units into subsets of prespecified sizes. The final measurements are then obtained from these partially ordered judgement subsets. The traditional expectation–maximization algorithm is not directly applicable for these observations. We propose a suitable expectation–maximization algorithm to estimate the parameters of the FMMs based on PROS samples. We also study the problem of classification of the PROS sample into the components of the FMM. We show that the maximum likelihood estimators based on PROS samples perform substantially better than their simple random sample counterparts even with small samples. The results are used to classify a fish population using the length‐frequency data.     

On Entropy-Based Test of Exponentiality in Ranked Set Sampling

When the sampling units can be easily ranked than quantified, ranked set sampling (RSS) is a viable alternative to the traditional simple random sampling (SRS). Much effort has been made for modifying basic RSS protocol with the aim of deriving more efficient estimators of the population attributes. Entropy has been seminal in developing measures of distributional disparities as a tool for statistical inference. This article is concerned with testing exponentiality based on sample entropy under some RSS-based designs. A simulation study shows that the proposed tests possess good power properties against several alternatives as compared with the ordinary test based on SRS.     

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.     

Robust extreme ranked set sampling

In this paper, a robust extreme ranked set sampling (RERSS) procedure for estimating the population mean is introduced. It is shown that the proposed method gives an unbiased estimator with smaller variance, provided the underlying distribution is symmetric. However, for asymmetric distributions a weighted mean is given, where the optimal weights are computed by using Shannon's entropy. The performance of the population mean estimator is discussed along with its properties. Monte Carlo simulations are used to demonstrate the performance of the RERSS estimator relative to the simple random sample (SRS), ranked set sampling (RSS) and extreme ranked set sampling (ERSS) estimators. The results indicate that the proposed estimator is more efficient than the estimators based on the traditional sampling methods.     

Quantile inference based on partially rank-ordered set samples

This paper develops statistical inference for population quantiles based on a partially rank-ordered set (PROS) sample design. A PROS sample design is similar to a ranked set sample with some clear differences. This design first creates partially rank-ordered subsets by allowing ties whenever the units in a set cannot be ranked with high confidence. It then selects a unit for full measurement at random from one of these partially rank-ordered subsets. The paper develops a point estimator, confidence interval and hypothesis testing procedure for the population quantile of order p. Exact, as well as asymptotic, distribution of the test statistic is derived. It is shown that the null distribution of the test statistic is distribution-free, and statistical inference is reasonably robust against possible ranking errors in ranking process.     

L Ranked Set Sampling: A Generalization Procedure for Robust Visual Sampling

In this article, a robust ranked set sampling (LRSS) scheme for estimating population mean is introduced. The proposed method is a generalization for many types of ranked set sampling that introduced in the literature for estimating the population mean. It is shown that the LRSS method gives unbiased estimator for the population mean with minimum variance providing that the underlying distribution is symmetric. However, for skewed distributions a weighted mean is given, where the optimal weights is computed by using Shannon's entropy. The performance of the population mean estimator is discussed along with its properties. Monte Carlo comparisons for detecting outliers are made with the traditional simple random sample and the ranked set sampling for some distributions. The results indicate that the LRSS estimator is superior alternative to the existing methods.     

Efficiency of ranked set sampling in entropy estimation and goodness-of-fit testing for the inverse Gaussian law

When measuring units are expensive or time consuming, while ranking them is relatively easy and inexpensive, it is known that ranked set sampling (RSS) is preferable to simple random sampling (SRS). Many authors have suggested several extensions of RSS. As a variation, Al-Saleh and Al-Kadiri [Double ranked set sampling, Statist. Probab. Lett. 48 (2000), pp. 205–212] introduced double ranked set sampling (DRSS) and it was extended by Al-Saleh and Al-Omari [Multistage ranked set sampling, J. Statist. Plann. Inference 102 (2002), pp. 273–286] to multistage ranked set sampling (MSRSS). The entropy of a random variable (r.v.) is a measure of its uncertainty. It is a measure of the amount of information required on the average to determine the value of a (discrete) r.v.. In this work, we discuss entropy estimation in RSS design and aforementioned extensions and compare the results with those in SRS design in terms of bias and root mean square error (RMSE). Motivated by the above observed efficiency, we continue to investigate entropy-based goodness-of-fit test for the inverse Gaussian distribution using RSS. Critical values for some sample sizes determined by means of Monte Carlo simulations are presented for each design. A Monte Carlo power analysis is performed under various alternative hypotheses in order to compare the proposed testing procedure with the existing methods. The results indicate that tests based on RSS and its extensions are superior alternatives to the entropy test based on SRS.     

Nonparametric Estimation for Random Censored Data Based on Ranking Set Sampling

In the case where the population distribution is unknown, the Kaplan–Meier estimator of the reliability function based on a ranked set sample with random right-censored data is first proposed. It is shown to be a unique self-consistent estimator. Then, the censored RSS estimator of the population mean is constructed. A simulation study is conducted to compare the performance of the proposed estimators with the corresponding estimators based on a simple random sample. It is shown that the ranked set sampling has higher efficiency. Finally, the proposed method is applied to a renal carcinoma study.     

On the Kaplan–Meier estimator based on ranked set samples

When quantification of all sampling units is expensive but a set of units can be ranked, without formal measurement, ranked set sampling (RSS) is a cost-efficient alternate to simple random sampling (SRS). In this paper, we study the Kaplan–Meier estimator of survival probability based on RSS under random censoring time setup, and propose nonparametric estimators of the population mean. We present a simulation study to compare the performance of the suggested estimators. It turns out that RSS design can yield a substantial improvement in efficiency over the SRS design. Additionally, we apply the proposed methods to a real data set from an environmental study.     

Environmental sampling with a concomitant variable: A comparison between ranked set sampling and stratified simple random sampling

In many environmental sampling situations, the variable of interest is either not easily observable or is too expensive to observe. Under such circumstances, the need arises to observe another variable, related to the variable of interest, so as to estimate the population parameters of interest. We study the performance of two different sampling procedures, i.e. ranked set sampling and stratified simple random sampling, when both stratification and ranking are accomplished on the basis of such a concomitant variable. The relative precision of the two methods is obtained and expressed as a function of population variance, between-stratum and between-rank variation, and the correlation coefficient between the variable of interest and the concomitant variable. The relative precision is computed for several important families of distributions that occur frequently in environmental and ecological work. Under equal allocation of sampling units, stratified simple random sampling is found to perform better than ranked set sampling, when the costs incurred to obtain sample measurements are ignored. When optimum allocation is considered for both methods, ranked set sampling performs better than stratified simple random sampling, when the concomitant variable is not highly correlated with the variable of interest. Furthermore, when the costs of sampling and the costs of measurement are incorporated into the assessment of the relative precision, the ranked set sampling is seen to be more efficient than stratified simple random sampling, particularly when the cost of stratification is high compared with that of ranking. This is generally the case in practice.     

Estimation of mean based on modified robust extreme ranked set sampling

In this paper, double robust extreme ranked set sampling (DRERSS) and its properties for estimating the population mean are considered. It turns out that, when the underlying distribution is symmetric, DRERSS gives unbiased estimators of the population mean. Also, it is found that DRERSS is more efficient than the simple random sampling (SRS), ranked set sampling (RSS), and extreme ranked set sampling (ERSS) methods. For asymmetric distributions considered in this study, the DRERSS has a small bias and it is more efficient than SRS, RSS, and ERSS. A real data set is used to illustrate the DRERSS method.     

Environmental sampling with a concomitant variable: a comparison between ranked set sampling and stratified simple random sampling

In many environmental sampling situations, the variable of interest is either not easily observable or is too expensive to observe. Under such circumstances, the need arises to observe another variable, related to the variable of interest, so as to estimate the population parameters of interest. We study the performance of two different sampling procedures, i.e. ranked set sampling and stratified simple random sampling, when both stratification and ranking are accomplished on the basis of such a concomitant variable. The relative precision of the two methods is obtained and expressed as a function of population variance, between-stratum and between-rank variation, and the correlation coefficient between the variable of interest and the concomitant variable. The relative precision is computed for several important families of distributions that occur frequently in environmental and ecological work. Under equal allocation of sampling units, stratified simple random sampling is found to perform better than ranked set sampling, when the costs incurred to obtain sample measurements are ignored. When optimum allocation is considered for both methods, ranked set sampling performs better than stratified simple random sampling, when the concomitant variable is not highly correlated with the variable of interest. Furthermore, when the costs of sampling and the costs of measurement are incorporated into the assessment of the relative precision, the ranked set sampling is seen to be more efficient than stratified simple random sampling, particularly when the cost of stratification is high compared with that of ranking. This is generally the case in practice.     

Ranked set sampling with random subsamples

A ranked sampling procedure with random subsamples is proposed to estimate the population mean. Four methods of obtaining random subsamples are described. Several estimators of the mean of the population based on random subsamples in ranked set sampling are proposed. These estimators are compared with the mean of a simple random sample for estimating the mean of symmetric and skew distributions. Extensive simulation under several subsampling distributions, sample sizes, and symmetric and skew distributions shows that the estimators of the mean based on random subsamples are more accurate than existing methods.     

Estimating the population proportion in pair ranked set sampling with application to air quality monitoring

In this paper, we consider the problem of estimating the population proportion in pair ranked set sampling design. An unbiased estimator for the population proportion is proposed, and its theoretical properties are studied. It is shown that the estimator is more (less) efficient than its counterpart in simple random sampling (ranked set sampling). Asymptotic normality of the estimator is also established. Application of the suggested procedure is illustrated using a data set from an environmental study.     

Robust extreme double ranked set sampling

As a well-known method for selecting representative samples of populations, ranked set sampling (RSS) has been considered increasingly in recent years. This (RSS) method has proved to be more efficient than the usual simple random sampling (SRS) for estimating most of the population parameters. In order to have a more efficient estimate of the population mean, a new sampling scheme called as robust extreme double ranked set sampling (REDRSS) is introduced and investigated in this paper. A simulation study shows that using REDRSS scheme gives more efficient estimates of population mean with smaller variance than the usual SRS, RSS and most other sampling schemes based on RSS estimators in non-uniform (symmetric or non-symmetric) distributions.     

Increased Fisher’s information for parameters of association in count regression via extreme ranks

The article details a sampling scheme which can lead to a reduction in sample size and cost in clinical and epidemiological studies of association between a count outcome and risk factor. We show that inference in two common generalized linear models for count data, Poisson and negative binomial regression, is improved by using a ranked auxiliary covariate, which guides the sampling procedure. This type of sampling has typically been used to improve inference on a population mean. The novelty of the current work is its extension to log-linear models and derivations showing that the sampling technique results in an increase in information as compared to simple random sampling. Specifically, we show that under the proposed sampling strategy the maximum likelihood estimate of the risk factor’s coefficient is improved through an increase in the Fisher’s information. A simulation study is performed to compare the mean squared error, bias, variance, and power of the sampling routine with simple random sampling under various data-generating scenarios. We also illustrate the merits of the sampling scheme on a real data set from a clinical setting of males with chronic obstructive pulmonary disease. Empirical results from the simulation study and data analysis coincide with the theoretical derivations, suggesting that a significant reduction in sample size, and hence study cost, can be realized while achieving the same precision as a simple random sample.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

On stratified bivariate ranked set sampling for regression estimators

We investigate the relative performance of stratified bivariate ranked set sampling (SBVRSS), with respect to stratified simple random sampling (SSRS) for estimating the population mean with regression methods. The mean and variance of the proposed estimators are derived with the mean being shown to be unbiased. We perform a simulation study to compare the relative efficiency of SBVRSS to SSRS under various data-generating scenarios. We also compare the two sampling schemes on a real data set from trauma victims in a hospital setting. The results of our simulation study and the real data illustration indicate that using SBVRSS for regression estimation provides more efficiency than SSRS in most cases.     

基于排序集样本和辅助变量中位数的均值比率估计方法

排序集抽样下利用辅助变量中位数构建了总体均值的改进比率估计模型,分析了该比率估计量的偏差和均方误差,并与简单随机抽样下的比率估计比较,证明了改进后的比率估计均方误差更小。以农作物播种面积和产量为研究对象进行实例分析,研究表明,基于排序集样本和辅助变量中位数的比率估计方法可以有效提高估计精度,验证了该构造方法的可行性。