Calibration estimator of population mean under stratified ranked set sampling design

Calibration method adjusts the original design weights to improve the estimates by using auxiliary information. In this article we have proposed new calibration estimators under stratified ranked set sampling design and derive the estimator of variance of calibration estimator. A simulation study is carried out to see the performance of proposed estimators.     

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.     

A Horvitz-Thompson Estimator of the Population Mean Using Inclusion Probabilities of Ranked Set Sampling

In this study, we define the Horvitz-Thompson estimator of the population mean using the inclusion probabilities of a ranked set sample in a finite population setting. The second-order inclusion probabilities that are required to calculate the variance of the Horvitz-Thompson estimator were obtained. The Horvitz-Thompson estimator, using the inclusion probabilities of ranked set sample, tends to be more efficient than the classical ranked set sampling estimator especially in a positively skewed population with small sizes. Also, we present a real data example with the volatility of gasoline to illustrate the Horvitz-Thompson estimator based on ranked set sampling.     

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.     

Two-stage cluster sampling with hybrid ranked set sampling in the secondary sampling frame

In surveys of natural resources in agriculture, ecology, fisheries, forestry, environmental management, etc., cost-effective sampling methods are of major concern. In this paper, we propose a two-stage cluster sampling (TSCS) in integration with the hybrid ranked set sampling (HRSS)—named TSCS-HRSS—in the second stage of sampling for estimating the population mean. The TSCS-HRSS scheme encompasses several existing ranked set sampling (RSS) schemes and may help in selecting a smaller number of units to rank. It is shown both theoretically and numerically that the TSCS-HRSS provides an unbiased estimator of the population mean and it is more precise than the mean estimators based on TSCS with SRS and RSS schemes. An unbiased estimator of the variance of the proposed mean estimator is also derived. A similar trend is observed when studying the impact of imperfect rankings on the performance of the TSCS-HRSS based mean estimator.     

Estimating variances of strata in ranked set sampling

In RSS, the variance of observations in each ranked set plays an important role in finding an optimal design for unbalanced RSS and in inferring the population mean. The empirical estimator (i.e., the sample variance in a given ranked set) is most commonly used for estimating the variance in the literature. However, the empirical estimator does not use the information in the entire data over different ranked sets. Further, it is highly variable when the sample size is not large enough, as is typical in RSS applications. In this paper, we propose a plug-in estimator for the variance of each set, which is more efficient than the empirical one. The estimator uses a result in order statistics which characterizes the cumulative distribution function (CDF) of the rth order statistics as a function of the population CDF. We analytically prove the asymptotic normality of the proposed estimator. We further apply it to estimate the standard error of the RSS mean estimator. Both our simulation and empirical study show that our estimators consistently outperform 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.     

Variance Estimators of the Gini Coefficient—Probability Sampling

An estimator of the Gini coefficient (the well-known income inequality measure) of a finite population is defined for an arbitrary probability sampling design, taking the sampling design into consideration. Alternative estimators of the variance of the estimated Gini coefficient are introduced. The sampling performance of the Gini coefficient estimator and its variance estimators is studied by means of a Monte Carlo study, using stratified sampling from a miniature population of Swedish households with authentic income data.     

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.     

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.     

Some calibration estimators for finite population mean in two-stage stratified random sampling

Abstract

In the present article, an effort has been made to develop calibration estimators of the population mean under two-stage stratified random sampling design when auxiliary information is available at primary stage unit (psu) level. The properties of the developed estimators are derived in-terms of design based approximate variance and approximate consistent design based estimator of the variance. Some simulation studies have been conducted to investigate the relative performance of calibration estimator over the usual estimator of the population mean without using auxiliary information in two-stage stratified random sampling. Proposed calibration estimators have outperformed the usual estimator without using auxiliary information.     

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.     

Estimation of the population mean and median using truncation-based ranked set samples

In this paper, a new sampling method is suggested, namely truncation-based ranked set samples (TBRSS) for estimating the population mean and median. The suggested method is compared with the simple random sampling (SRS), ranked set sampling (RSS), extreme ranked set sampling (ERSS) and median-ranked set sampling (MRSS) methods. It is shown that for estimating the population mean when the underlying distribution is symmetric, TBRSS estimator is unbiased and it is more efficient than the SRS estimator based on the same number of measured units. For asymmetric distributions considered in this study, TBRSS estimator is more efficient than the SRS for all considered distributions except for exponential distribution when the selection coefficient gets large. When compared with ERSS and MRSS methods, TBRSS performs well with respect to ERSS for all considered distributions except for U(0, 1) distribution, while TBRSS efficiency is higher than that of MRSS for U(0, 1) distribution. For estimating the population median, the TBRSS estimators have higher efficiencies when compared with SRS and ERSS. A real data set is used to illustrate the suggested method.     

Combining multi‐observer information in partially rank‐ordered judgment post‐stratified and ranked set samples

This paper develops two sampling designs to create artificially stratified samples. These designs use a small set of experimental units to determine their relative ranks without measurement. In each set, the units are ranked by all available observers (rankers), with ties whenever the units cannot be ranked with high confidence. The rankings from all the observers are then combined in a meaningful way to create a single weight measure. This weight measure is used to create judgment strata in both designs. The first design constructs the strata through judgment post‐stratification after the data has been collected. The second design creates the strata before any measurements are made on the experimental units. The paper constructs estimators and confidence intervals, and develops testing procedures for the mean and median of the underlying distribution based on these sampling designs. We show that the proposed sampling designs provide a substantial improvement over their competitor designs in the literature. The Canadian Journal of Statistics 41: 304–324; 2013 © 2013 Statistical Society of Canada     

Ratio estimators based on a ranked set sample in a finite population setting

This paper considers the ratio estimator in a finite population setting in a ranked set sampling (RSS) design, where the sample is constructed either with or without replacement policies. It is shown that the proposed ratio estimator is slightly biased, but the amount of bias is smaller than the amount of bias of a simple random sample (SRS) ratio estimator. For the proposed ratio estimator, the paper provides explicit expressions for its mean square error and precision relative to the other competing estimators. It is shown that the new estimator has a substantial amount of improvement in efficiency with respect to SRS estimator. The proposed estimator is applied to two different finite population settings to estimate population mean.     

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.     

Model‐Based Non‐parametric Variance Estimation for Systematic Sampling

Abstract. Systematic sampling is frequently used in surveys, because of its ease of implementation and its design efficiency. An important drawback of systematic sampling, however, is that no direct estimator of the design variance is available. We describe a new estimator of the model‐based expectation of the design variance, under a non‐parametric model for the population. The non‐parametric model is sufficiently flexible that it can be expected to hold at least approximately in many situations with continuous auxiliary variables observed at the population level. We prove the model consistency of the estimator for both the anticipated variance and the design variance under a non‐parametric model with a univariate covariate. The broad applicability of the approach is demonstrated on a dataset from a forestry survey.     

Estimation of the distribution function under hybrid ranked set sampling

Recently, a hybrid ranked set sampling (HRSS) scheme has been proposed in the literature. The HRSS scheme encompasses several existing ranked set sampling (RSS) schemes, and it is a cost-effective alternative to the classical RSS and double RSS schemes. In this paper, we propose an improved estimator for estimating the cumulative distribution function (CDF) using HRSS. It is shown, both theoretically and numerically, that the CDF estimator under HRSS scheme is unbiased and its variance is always less than the variance of the CDF estimator with simple random sampling (SRS). An unbiased estimator of the variance of CDF estimator using HRSS is also derived. Using Monte Carlo simulations, we also study the performances of the proposed and existing CDF estimators under both perfect and imperfect rankings. It turns out that the proposed CDF estimator is by far a superior alternative to the existing CDF estimators with SRS, RSS and L-RSS schemes. For a practical application, a real data set is considered on the bilirubin level of babies in neonatal intensive care.     

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。     

Finite Population Variance Estimation Under LSS with Multiple Random Starts

In this article, an unbiased estimator for finite population variance is developed under linear systematic sampling with two random starts and an explicit expression for its variance is also obtained. The study is supported by two real life situations. A detailed numerical comparative study has been carried out to compare its average variance with the average variance of the conventional unbiased estimator for finite population variance under simple random sampling for a wide variety of populations. Results based on the study strongly favor the use of the developed estimator for such populations.