Recursive computation of inclusion probabilities in ranked-set sampling

We derive recursive algorithms for computing first-order and second-order inclusion probabilities for ranked-set sampling from a finite population. These algorithms make it practical to compute inclusion probabilities even for relatively large sample and population sizes. As an application, we use the inclusion probabilities to examine the performance of Horvitz-Thompson estimators under different varieties of balanced ranked-set sampling. We find that it is only for balanced Level 2 sampling that the Horvitz-Thompson estimator can be relied upon to outperform the simple random sampling mean estimator.     

MODEL-UNBIASEDNESS AND THE HORVITZ-THOMPSON ESTIMATOR IN FINITE POPULATION SAMPLING

Under the, notion of superpopulation models, the concept of minimum expected variance is adopted as an optimality criterion for design-unbiased estimators, i.e. unbiased under repeated sampling. In this article, it is shown that the Horvitz-Thompson estimator is optimal among such estimators if and only if it is model-unbiased, i.e. unbiased under the model. The family of linear models is considered and a sample design is suggested to preserve the model-unbiasedness (and hence the optimality) of the Horvitz-Thompson estimator. It is also shown that under these models the Horvitz-Thompson estimator together with the suggested sample design is optimal among design-unbiased estimators with any sample design (of fixed size n ) having non-zero probabilities of inclusion for all population units.     

On the Convergence of the Horvitz-Thompson Estimator

We investigate the degree to which the Horvitz-Thompson estimator approximates the population mean in survey sampling. Specifically, conditions for (mean square) consistency are given, along with rates of convergence. The conditions involve only first and second-order inclusion probabilities, and for various sampling designs they are easy to check.     

Remainder Markov systematic sampling

Systematic sampling is the simplest and easiest of the most common sampling methods. However, when the population size N cannot be evenly divided by the sampling size n, systematic sampling cannot be performed. Not only is it difficult to determine the sampling interval k equivalent to the sampling probability of the sampling unit, but also the sample size will be inconstant and the sample mean will be a biased estimator of the population mean. To solve this problem, this paper introduces an improved method for systematic sampling: the remainder Markov systematic sampling method. This new method involves separately finding the first-order and second-order inclusion probabilities. This approach uses the Horvitz-Thompson estimator as an unbiased estimator of the population mean to find the variance of the estimator. This study examines the effectiveness of the proposed method for different super-populations.     

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.     

A SIMPLE PROCEDURE FOR SAMPLING πpswor1

The author's 1963 procedure for selecting two units per stratum πpswor is generalized to any feasible fixed number of sample units. Simple closed expressions are given for the working probabilities at each draw but the formula for the joint probabilities of inclusion of pairs of units is recursive and tedious to apply. The stability of the variance estimator is believed to be close to the optimum that can be obtained using the Horvitz-Thompson estimator of total in situations where that estimator itself is optimal. Some suggestions are made for rotation of the sample.     

A two-phase sampling scheme and πps designs

A two-phase approach for sampling with unequal inclusions probabilities and fixed sample size is presented. The expansion estimator using target inclusion probabilities is suggested for estimation of population parameters. As an alternative, the estimator for two-phase sampling can be used for estimation. Inclusion probabilities are shown to be asymptotically equivalent to the targeted inclusion probabilities. By means of simulation associated estimators are shown to work well with respect to bias and precision.     

Optimal inclusion probabilities for balanced sampling

When auxiliary information is available at the design stage, samples may be selected by means of balanced sampling. The variance of the Horvitz-Thompson estimator is then reduced, since it is approximately given by that of the residuals of the variable of interest on the balancing variables. In this paper, a method for computing optimal inclusion probabilities for balanced sampling on given auxiliary variables is studied. We show that the method formerly suggested by Tillé and Favre (2005) enables the computation of inclusion probabilities that lead to a decrease in variance under some conditions on the set of balancing variables. A disadvantage is that the target optimal inclusion probabilities depend on the variable of interest. If the needed quantities are unknown at the design stage, we propose to use estimates instead (e.g., arising from a previous wave of the survey). A limited simulation study suggests that, under some conditions, our method performs better than the method of Tillé and Favre (2005).     

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.     

A New Formula for Inclusion Probabilities in Median-Ranked Set Sampling

This article develops a new generalized formula to compute the inclusion probabilities of a median-ranked set sample in a finite population setting. The use of this formula is illustrated in a numerical example. Furthermore, the inclusion probabilities of a median-ranked set sample is compared with the inclusion probabilities of ranked set and simple random samples.     

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.     

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。     

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.     

Limited information likelihood analysis of survey data

Analysts of survey data are often interested in modelling the population process, or superpopulation, that gave rise to a 'target' set of survey variables. An important tool for this is maximum likelihood estimation. A survey is said to provide limited information for such inference if data used in the design of the survey are unavailable to the analyst. In this circumstance, sample inclusion probabilities, which are typically available, provide information which needs to be incorporated into the analysis. We consider the case where these inclusion probabilities can be modelled in terms of a linear combination of the design and target variables, and only sample values of these are available. Strict maximum likelihood estimation of the underlying superpopulation means of these variables appears to be analytically impossible in this case, but an analysis based on approximations to the inclusion probabilities leads to a simple estimator which is a close approximation to the maximum likelihood estimator. In a simulation study, this estimator outperformed several other estimators that are based on approaches suggested in the sampling literature.     

Improving the best linear unbiased estimator for the scale parameter of symmetric distributions by using the absolute value of ranked set samples

Ranked set sampling is a cost efficient sampling technique when actually measuring sampling units is difficult but ranking them is relatively easy. For a family of symmetric location-scale distributions with known location parameter, we consider a best linear unbiased estimator for the scale parameter. Instead of using original ranked set samples, we propose to use the absolute deviations of the ranked set samples from the location parameter. We demonstrate that this new estimator has smaller variance than the best linear unbiased estimator using original ranked set samples. Optimal allocation in the absolute value of ranked set samples is also discussed for the estimation of the scale parameter when the location parameter is known. Finally, we perform some sensitivity analyses for this new estimator when the location parameter is unknown but estimated using ranked set samples and when the ranking of sampling units is imperfect.     

Estimators for a Poisson parameter using ranked set sampling

Using ranked set sampling, a viable BLUE estimator is obtained for estimating the mean of a Poisson distribution. Its properties, such as efficiency relative to the ranked set sample mean and to the maximum likelihood estimator, have been calculated for different sample sizes and values of the Poisson parameter. The estimator (termed the normal modified r.s.s. estimator is more efficient than both the ranked set sample mean and the MLE. It is recommended as a reasonable estimator of the Poisson mean ( λ) to be used in a ranked set sampling environment.     

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.     

RANKED SET SAMPLING FROM LOCATION-SCALE FAMILIES OF SYMMETRIC DISTRIBUTIONS

Statistical inference based on ranked set sampling has primarily been motivated by nonparametric problems. However, the sampling procedure can provide an improved estimator of the population mean when the population is partially known. In this article, we consider estimation of the population mean and variance for the location-scale families of distributions. We derive and compare different unbiased estimators of these parameters based on rindependent replications of a ranked set sample of size n.Large sample properties, along with asymptotic relative efficiencies, help identify which estimators are best suited for different location-scale distributions.     

A Further Study on Reliable Life Estimation under Ranked Set Sampling

This article considers nonparametric estimation of reliable life based on ranked set sampling and its properties. It is proven analytically that the large sample efficiency of the reliable life estimator under the balanced ranked set sampling is higher than that under the simple random sampling of the same size, but the relative efficiency damps away as the reliable life moves away from the median on both directions. To improve the efficiency for the estimation of extreme reliable life, we then propose a reliable life estimator under a modified ranked set sampling protocol, its strong consistency and asymptotic normality are established. The proposed sampling is shown to be superior to the balanced ranked set sampling, and the relative advantage improves as the reliable life moves away from median. Finally, results of simulation studies for small sample as well as an application to a real data set are presented to illustrate some of the theoretical findings.     

On the consistency between model-and design-based estimators in survey sampling

In finite population sampling, often a distinction is made between model-and design-based estimators of the parameters of interest (like the population total, population variance, etc.). The model-based estimators depend on the (known) parameters of the model, while the design-based estimators depend on the (known) selection probabilities of the different units in the population. It is shown in this paper that the two approaches are not necessarily incompatible, and indeed can often lead to the same estimator. Our ideas are illustrated with the Horvitz-Thompson, and the generalized Horvitz-Thompson estimator. These estimators are identified as hierarchical Bays estimators. Also, certain “stepwise-Bayes” estimators of Vardeman and Meeden (J. Stat. Inf. (1983), V7, pp 329-341) are unified from a hierarchical Bayes point of view.