Mixed ranked set sampling design

The main focus of agricultural, ecological and environmental studies is to develop well designed, cost-effective and efficient sampling designs. Ranked set sampling (RSS) is one method that leads to accomplish such objectives by incorporating expert knowledge to its advantage. In this paper, we propose an efficient sampling scheme, named mixed RSS (MxRSS), for estimation of the population mean and median. The MxRSS scheme is a suitable mixture of both simple random sampling (SRS) and RSS schemes. The MxRSS scheme provides an unbiased estimator of the population mean, and its variance is always less than the variance of sample mean based on SRS. For both symmetric and asymmetric populations, the mean and median estimators based on SRS, partial RSS (PRSS) and MxRSS schemes are compared. It turns out that the mean and median estimates under MxRSS scheme are more precise than those based on SRS scheme. Moreover, when estimating the mean of symmetric and some asymmetric populations, the mean estimates under MxRSS scheme are found to be more efficient than the mean estimates with PRSS scheme. An application to real data is also provided to illustrate the implementation of the proposed sampling scheme.     

A new sampling method for estimating the population mean

A double L ranked set sampling (DLRSS) method is suggested for estimating the population mean. The DLRSS is compared with the simple random sampling (SRS), ranked set sampling (RSS) and L ranked set sampling (LRSS) methods based on the same number of measured units. The conditions for which the suggested estimator performs better than the other estimators are derived. It is found that, the suggested DLRSS estimator is an unbiased of the population mean, and is more efficient than its counterparts using SRS, RSS, and LRSS methods. Real data sets are used for illustration.     

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.     

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.     

Stein-type estimation using ranked set sampling

In this study, we consider the application of the James–Stein estimator for population means from a class of arbitrary populations based on ranked set sample (RSS). We consider a basis for optimally combining sample information from several data sources. We succinctly develop the asymptotic theory of simultaneous estimation of several means for differing replications based on the well-defined shrinkage principle. We showcase that a shrinkage-type estimator will have, under quadratic loss, a substantial risk reduction relative to the classical estimator based on simple random sample and RSS. Asymptotic distributional quadratic biases and risks of the shrinkage estimators are derived and compared with those of the classical estimator. A simulation study is used to support the asymptotic result. An over-riding theme of this study is that the shrinkage estimation method provides a powerful extension of its traditional counterpart for non-normal populations. Finally, we will use a real data set to illustrate the computation of the proposed estimators.     

Paired double-ranked set sampling

Abstract

In environmental monitoring and assessment, the main focus is to achieve observational economy and to collect data with unbiased, efficient and cost-effective sampling methods. Ranked set sampling (RSS) is one traditional method that is mostly used for accomplishing observational economy. In this article, we propose an unbiased sampling scheme, named paired double RSS (PDRSS) for estimating the population mean. We study the performance of the mean estimators under PDRSS based on perfect and imperfect rankings. It is shown that, for perfect ranking, the variance of the mean estimator under PDRSS is always less than the variance of mean estimator based on simple random sampling, paired RSS and RSS. The mean estimators under RSS, median RSS, PDRSS, and double RSS are also compared with the regression estimator of population mean based on SRS. The procedure is also illustrated with a case study using a real data set.     

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.     

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.     

The global minimum variance unbiased estimator of the parameter for a truncated parameter family under the optimal ranked set sampling

The minimum variance unbiased estimators (MVUEs) of the parameters for various distributions are extensively studied under ranked set sampling (RSS). However, the results in existing literatures are only locally MVUEs, i.e. the MVUE in a class of some unbiased estimators is obtained. In this paper, the global MVUE of the parameter in a truncated parameter family is obtained, that is to say, it is the MVUE in the class of all unbiased estimators. Firstly we find the optimal RSS according to the character of a truncated parameter family, i.e. arrange RSS based on complete and sufficient statistics of independent and identically distributed samples. Then under this RSS, the global MVUE of the parameter in a truncated parameter family is found. Numerical simulations for some usual distributions in this family fully support the result from the above two-step optimizations. A real data set is used for illustration.     

Reliability estimation based on ranked set sampling

In this paper we consider the problem of estimating the reliability of an exponential component based on a Ranked Set Sample (RSS) of size n. Given the first r observations of that sample, 1≤r≤n, we construct an unbiased estimator for this reliability and we show that these n unbiased estimators are the only ones in a certain class of estimators. The variances of some of these estimators are compared. By viewing the observations of the RSS of size n as the lifetimes of n independent k-out-of-n systems, 1≤k≤n, we are able to utilize known properties of these systems in conjunction with the powerful tools of majorization and Schur functions to derive our results.     

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.     

A Remark on the Zhang Omnibus Test for Normality

Zhang (1999) proposed a novel test statistic Q for testing normality based on the ratio of two unbiased standard deviation estimators, q₁ and q₂, for the true population standard deviation σ. Mingoti & Neves (2003) discussed some properties of q₁ and q₂ and showed that the variance of q₁ increases as the true population variance increases. In this paper, we show that the distribution of q₁ is not normal. As a result, normality percentage points for Q are not appropriate. In this paper, percentage points of Q are obtained using simulations. Monte Carlo simulations are provided to evaluate the performance of the new method and Zhang's method.     

Nonnegative Unbiased Estimation of Scale Parameters and Associated Quantiles Based on a Ranked Set Sample

In this article, we are interested in estimating the scale parameter in location and scale families. It is well known that the best linear unbiased estimator (BLUE) of scale parameter based on a simple random sample (SRS) is nonnegative. However, the BLUE of scale parameter based on a ranked set sample (RSS) can assume negative values. We suggest various modifications of BLUE of scale parameter based on RSS so that the resulting estimators are unbiased as well as nonnegative. Their performances in terms of relative efficiencies are compared and some recommendations are made for normal, logistic, double exponential, two-parameter exponential and Weibull distributions. We also briefly discuss an application of the proposed nonnegative BLUE of scale parameter for quantile estimation for the above populations.     

Best linear unbiased estimators of parameters of a simple linear regression model based on ordered ranked set samples

As an alternative to an estimation based on a simple random sample (BLUE-SRS) for the simple linear regression model, Moussa-Hamouda and Leone [E. Moussa-Hamouda and F.C. Leone, The o-blue estimators for complete and censored samples in linear regression, Technometrics, 16 (3) (1974), pp. 441–446.] discussed the best linear unbiased estimators based on order statistics (BLUE-OS), and showed that BLUE-OS is more efficient than BLUE-SRS for normal data. Using the ranked set sampling, Barreto and Barnett [M.C.M. Barreto and V. Barnett, Best linear unbiased estimators for the simple linear regression model using ranked set sampling. Environ. Ecoll. Stat. 6 (1999), pp. 119–133.] derived the best linear unbiased estimators (BLUE-RSS) for simple linear regression model and showed that BLUE-RSS is more efficient for the estimation of the regression parameters (intercept and slope) than BLUE-SRS for normal data, but not so for the estimation of the residual standard deviation in the case of small sample size. As an alternative to RSS, this paper considers the best linear unbiased estimators based on order statistics from a ranked set sample (BLUE-ORSS) and shows that BLUE-ORSS is uniformly more efficient than BLUE-RSS and BLUE-OS for normal data.     

On quantiles estimation based on different stratified sampling with optimal allocation

This work considers the problem of estimating a quantile function based on different stratified sampling mechanism. First, we develop an estimate for population quantiles based on stratified simple random sampling (SSRS) and extend the discussion for stratified ranked set sampling (SRSS). Furthermore, the asymptotic behavior of the proposed estimators are presented. In addition, we derive an analytical expression for the optimal allocation under both sampling schemes. Simulation studies are designed to examine the performance of the proposed estimators under varying distributional assumptions. The efficiency of the proposed estimates is further illustrated by analyzing a real data set from CHNS.     

Testing equality of proportions with incomplete correlated data

Let (ψ_i,φ_i) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of ψ_i and φ_i within the pair. For n pairs, both variables are observed, but for m₁ additional pairs only ψ_i is observed and for m₂ others φ_i is observed. If π_1· = P{ψ_i = 1} and π_·1=P{φ_i, the problem is to test π_1·=π_·1. Maximum likelihood estimates of π_1· and π_·1 are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either m₁ or m₂ tend to infinity). If m₁ = m₂ = 0 the statistic reduces to that of McNemar's test; if n = 0, it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.     

Ordered partially ordered judgment subset sampling with applications to parametric inference

ABSTRACT

In this paper, we use the idea of order statistics from independent and non-identically distributed random variables to propose ordered partially ordered judgment subset sampling (OPOJSS) and then develop optimal linear parametric inferences. The best linear unbiased and invariant estimators of the location and scale parameters of a location-scale family are developed based on OPOJSS. It is shown that, despite the presence or absence of ranking errors, the proposed estimators with OPOJSS are uniformly better than the existing estimators with simple random sampling (SRS), ranked set sampling (RSS), ordered RSS (ORSS) and partially ordered judgment subset sampling (POJSS). Moreover, we also derive the best linear unbiased estimators (BLUEs) of the unknown parameters of the simple linear regression model with replicated observations using POJSS and OPOJSS. It is found that the BLUEs with OPOJSS are more precise than the BLUEs based on SRS, RSS, ORSS and POJSS.     

Quality Control Chart for the Mean using Double Ranked Set Sampling

In this paper, an attempt is made to develop Quality Control Charts for monitoring the process mean based on Double Ranked Set Sampling (DRSS) rather than the traditional Simple Random Sampling (SRS). Considering a normal population and several shift values, the performance of the Average Run Length (ARL) of these new charts was compared with the control charts based on Ranked Set Sampling (RSS) and SRS with the same number of observations. It is shown that the new charts do a better job of detecting changes in process mean compared with SRS and RSS.     

General procedure for estimating the population mean using ranked set sampling

In this paper, we suggest a class of estimators for estimating the population mean ? of the study variable Y using information on X?, the population mean of the auxiliary variable X using ranked set sampling envisaged by McIntyre [A method of unbiased selective sampling using ranked sets, Aust. J. Agric. Res. 3 (1952), pp. 385–390] and developed by Takahasi and Wakimoto [On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 20 (1968), pp. 1–31]. The estimator reported by Kadilar et al. [Ratio estimator for the population mean using ranked set sampling, Statist. Papers 50 (2009), pp. 301–309] is identified as a member of the proposed class of estimators. The bias and the mean-squared error (MSE) of the proposed class of estimators are obtained. An asymptotically optimum estimator in the class is identified with its MSE formulae. To judge the merits of the suggested class of estimators over others, an empirical study is carried out.     

Partial groups ranked set sampling and mean estimation

Ranked set sampling (RSS) is an advanced sampling method which is very effective for estimating mean of the population when exact measurement of observation is difficult and/or expensive. Balanced Groups RSS (BGRSS) is one of the modification of RSS where only the lowest, the median and the largest ranked units are taken into account. Although BGRSS is advantageous and useful for some specific cases, it has strict restrictions regarding the set size which could be problematic for sampling plans. In this study, we make an improvement on BGRSS and propose a new design called Partial Groups RSS which offers a more flexible sampling plan providing the independence of the set size and sample size. Partial Groups RSS also has a cost advantage over BGRSS. We construct a Monte Carlo simulation study comparing the performance of the mean estimators of the proposed sampling design and BGRSS according to their sampling costs and mean squared errors for various type of distributions. In addition, we give a biometric data application for investigating the efficiency of Partial Groups RSS in real life applications.