Ranked set sampling allocation models for multiple skewed variables: an application to agricultural data

The mean of a balanced ranked set sample is more efficient than the mean of a simple random sample of equal size and the precision of ranked set sampling may be increased by using an unbalanced allocation when the population distribution is highly skewed. The aim of this paper is to show the practical benefits of the unequal allocation in estimating simultaneously the means of more skewed variables through real data. In particular, the allocation rule suggested in the literature for a single skewed distribution may be easily applied when more than one skewed variable are of interest and an auxiliary variable correlated with them is available. This method can lead to substantial gains in precision for all the study variables with respect to the simple random sampling, and to the balanced ranked set sampling too.     

Ranked set sample design for environmental investigations

Ranked set sampling can provide an efficient basis for estimating parameters of environmental variables, particularly when sampling costs are intrinsically high. Various ranked set estimators are considered for the population mean and contrasted in terms of their efficiencies and useful- ness, with special concern for sample design considerations. Specifically, we consider the effects of the form of the underlying random variable, optimisation of efficiency and how to allocate sampling effort for best effect (e.g. one large sample or several smaller ones of the same total size). The various prospects are explored for two important positively skew random variables (lognormal and extreme value) and explicit results are given for these cases. Whilst it turns out that the best approach is to use the largest possible single sample and the optimal ranked set best linear estimator (ranked set BLUE), we find some interesting qualitatively different conclusions for the two skew distributions     

Ranked set sampling: its essence and some new applications

Ranked set sampling is a simple idea of great use. It was proposed half a century ago. The last 15 years or so have witnessed considerable development in the research and applications of ranked set sampling. In this paper, we give an overview on ranked set sampling. We review several variants of ranked set sampling developed since the original idea was proposed. We discuss the essence and the theoretical foundation of ranked set sampling. We present some novel applications of ranked set sampling in areas such as clinical trials, genetic quantitative trait loci mappings and others. By doing so, we wish to provide the reader with a philosophical view on ranked set sampling and shed some lights on a broader range of its applications.     

On ranked set sampling for multiple characteristics

We consider the selection of samples in ranked set sampling when several attributes of each sample are of interest. We describe approaches that have appeared previously in the literature and present a novel method that seeks to achieve samples that are nearly balanced with respect to the ranks of all attributes. This method is shown to result in very little loss of precision compared to problems in which only a single sample attribute is of interest.     

Sampling from partially rank-ordered sets

In this paper we introduce a new sampling design. The proposed design is similar to a ranked set sampling (RSS) design with a clear difference that rankers are allowed to declare any two or more units are tied in ranks whenever the units can not be ranked with high confidence. These units are replaced in judgment subsets. The fully measured units are then selected from these partially ordered judgment subsets. Based on this sampling scheme, we develop unbiased estimators for the population mean and variance. We show that the proposed sampling procedure has some advantages over standard ranked set sampling.     

Best linear unbiased estimators for the simple linear regression model using ranked set sampling

When sample observations are expensive or difficult to obtain, ranked set sampling is known to be an efficient method for estimating the population mean, and in particular to improve on the sample mean estimator. Using best linear unbiased estimators, this paper considers the simple linear regression model with replicated observations. Use of a form of ranked set sampling is shown to be markedly more efficient for normal data when compared with the traditional simple linear regression estimators.     

The ‘Bayesics’ of Ranked Set Sampling

Ranked set sampling can be useful when measurements are expensive but units from the popu- lation can be easily ranked. In this situation one may draw k units from the population, rank them, select one on which to make the expensive measurement, draw another k units, rank them, select one, and so on. The method was originally suggested by McIntyre (1952) in connection with pasture yields and is obviously applicable in other situations as well. Dell and Clutter (1972) and Patil et al. (1994) explain the basics from a classical point of view. Our aim is to examine the procedure from a Bayesian point of view, determine whether ranked set sampling provides advantages over simple random sampling and explore some optimality questions     

Estimation of Normal Variance Based on Balanced and Unbalanced Ranked Set Samples

In this paper we address the problem of estimation of the variance of a normal population based on a balanced as well as an unbalanced ranked set sample (RSS), which is a modification of the original RSS of McIntyre (1952).We have proposed several methods of estimation of variance by combining different unbiased between and within estimators, and compared their performances     

Design-based ranked set sampling using auxiliary variables

A ranked set sampling protocol is proposed when an auxiliary variable is available in addition to the target variable in sample surveys. The protocol may be practically carried out without additional sampling effort or costs. Under the suggested sampling scheme, the estimators usually adopted in surveys with auxiliary information - such as the ratio estimator or the regression estimator - display surprising theoretical properties as well as high performance in practice.     

Ranked set sampling: an annotated bibliography

The paper provides an up-to-date annotated bibliography of the literature on ranked set sampling. The bibliography includes all pertinent papers known to the authors, and is intended to cover applications as well as theoretical developments. The annotations are arranged in chronological order and are intended to be sufficiently complete and detailed that a reading from beginning to end would provide a statistically mature reader with a state-of-the-art survey of ranked set sampling, including historical development, current status, and future research directions and applications. A final section of the paper gives a listing of all annotated papers, arranged in alphabetical order by author.This paper was prepared with partial support from the United States Environmental Protection Agency under a Cooperative Agreement Number CR-821531. The contents have not been subject to Agency review and therefore do not necessarily reflect the views or policies of the Agency and no official endorsement should be inferred.     

Optimal allocation procedure in ranked set sampling for unimodal and multi-modal distributions

This paper presents a ranked set sample allocation procedure that is optimal for a number of nonparametric test procedures. We define a function that measures the amount of information provided by each observation given the actual joint ranking of all the units in a set. The optimal ranked set sample allocates order statistics by maximizing this information function. This paper shows that the optimal allocation of order statistics in a ranked set sample is determined by the location of the mode(s) of the underlying distribution. For unimodal, symmetric distributions, optimal allocation always quantifies the middle observation(s). If the underlying distribution with cdf F is a multi-modal distribution with modes , then the optimal allocation procedure quantifies observations at in a set of size m. We provide similar results for unimodal, asymmetric distributions. We also propose a new sign test which considers the relative positions of the quantified observations from the same cycle in a ranked set sample. The proposed sign test provides improvement in the Pitman efficiency over the ranked set sample sign test of Hettmansperger (1995). It is shown that the information optimal allocation procedure induced by Pitman efficiency is equivalent to the optimal allocation procedure induced by the information criteria. We show that the finite sample distribution of the proposed test based on this optimal design is binomial.     

Trophic food web and ecosystem attributes of a water reservoir of the Ria Formosa (south Portugal)

Using a top-down modelling approach, a balanced trophic food web model of a water reservoir of the Ria Formosa lagoonal system was constructed. Few adjustments of input data were necessary to run the model since most of the parameters were estimated at the study site and the characteristics of the water reservoir allowed a high degree of control and precision in terms of sampling and data collection. Trophic levels of the 14 compartments included in the model varied between 1.0 for primary producers and detritus to 3.4 for carnivorous fish and the 14 groups were aggregated in a food chain with six trophic levels. The water reservoir has a detritus based food chain, with the majority of the biomass concentrated in the first two levels, the producers level and the herbivore/detritivore level (97.6%). The transfer efficiencies were low, and decreased with increasing level number, varying between 6.0 and 0.2%. The degree of “ecosystem maturity” was difficult to establish, but several parameters indicate that it could be high. The water reservoir studied, which has similar environmental and ecological characteristics as the Ria Formosa lagoon, is near its carrying capacity.     

Design based estimation for ranked set sampling in finite populations

In this paper, we consider design-based estimation using ranked set sampling (RSS) in finite populations. We first derive the first and second-order inclusion probabilities for an RSS design and present two Horvitz–Thompson type estimators using these inclusion probabilities. We also develop an alternate Hansen–Hurwitz type estimator and investigate its properties. In particular, we show that this alternate estimator always outperforms the usual Hansen–Hurwitz type estimator in the simple random sampling with replacement design with comparable sample size. We also develop formulae for ratio estimator for all three developed estimators. The theoretical results are augmented by numerical and simulation studies as well as a case study using a well known data set. These show that RSS design can yield a substantial improvement in efficiency over the usual simple random sampling design in finite populations.     

Randomly selected order statistics in ranked set sampling: A less expensive comparable alternative to simple random sampling

Rank-based sampling designs are powerful alternatives to simple random sampling (SRS) and often provide large improvements in the precision of estimators. In many environmental, ecological, agricultural, industrial and/or medical applications the interest lies in sampling designs that are cheaper than SRS and provide comparable estimates. In this paper, we propose a new variation of ranked set sampling (RSS) for estimating the population mean based on the random selection technique to measure a smaller number of observations than RSS design. We study the properties of the population mean estimator using the proposed design and provide conditions under which the mean estimator performs better than SRS and some existing rank-based sampling designs. Theoretical results are augmented with some numerical studies and a real-life example, where we also study the performance of our proposed design under perfect and imperfect ranking situations.     

不同海岸梯度上短枝木麻黄小枝金属元素含量及其再吸收率动态

对福建东山赤山林场不同海岸梯度上短枝木麻黄小枝中的钾(K)、钙(Ca)、镁(Mg)、钠(Na)、铁(Fe)、锰(Mn)和锌(Zn)元素含量及其再吸收率动态进行了研究.结果表明,海岸梯度对短枝木麻黄小枝中各元素含量及其再吸收率均具有显著影响.在各海岸梯度上,K再吸收率为正值,且基干林带的再吸收率显著低于其它样地,Fe和Zn为负值,Ca、Mg和Na再吸收率在防护林带前沿为负值,而在后沿林为正值,Mn则相反.这表明,基干林带严重的环境胁迫显著降低了短枝木麻黄小枝的养分再吸收率.短枝木麻黄成熟小枝中的K、Mg含量与其再吸收率存在显著正相关,而Na和Fe与其再吸收率存在显著负相关,其它元素与其再吸收率之间没有显著相关性.除K外,各元素在衰老小枝中的含量与相应元素再吸收率之间均存在显著负相关,表明元素再吸收程度越高,其再吸收率就越高.Ca、Mg、Na和Fe再吸收率之间具有显著的正相关关系(P<0.01),Zn除与Mn再吸收率呈显著正相关外,与其它元素均表现为负相关,Mn则与其它元素再吸收率之间没有显著相关性.因而短枝木麻黄小枝中不同类型的养分状况对其再吸收率具有不同的影响.表5参42     

Halton iterative partitioning: spatially balanced sampling via partitioning

A new spatially balanced sampling design for environmental surveys is introduced, called Halton iterative partitioning (HIP). The design draws sample locations that are well spread over the study area. Spatially balanced designs are known to be efficient when surveying natural resources because nearby locations tend to be similar. The HIP design uses structural properties of the Halton sequence to partition a resource into nested boxes. Sample locations are then drawn from specific boxes in the partition to ensure spatial diversity. The method is conceptually simple and computationally efficient, draws spatially balanced samples in two or more dimensions and uses standard design-based estimators. Furthermore, HIP samples have an implicit ordering that can be used to define spatially balanced over-samples. This feature is particularly useful when sampling natural resources because we can dynamically add spatially balanced units from the over-sample to the sample as non-target or inaccessible units are discovered. We use several populations to show that HIP sampling draws spatially balanced samples and gives precise estimates of population totals.     

Combining ranking information in judgment post stratified and ranked set sampling designs

Judgment post stratified (JPS) and ranked set sampling (RSS) designs rely on the ability of a ranker to assign ranks to potential observations on available experimental units. In many settings, there are often more than one rankers available and each of these rankers provide judgment ranks. This paper proposes two sampling schemes, one for JPS and the other for RSS, to combine the judgment ranks of these rankers to produce a strength of agreement measure for each fully measured unit. This strength measure is used to draw inference for the population mean and cumulative distribution function. The paper shows that the estimators constructed based on this strength measure provide a substantial improvement over the same estimators based on judgment ranking information of a single best ranker.     

环境因素对密云水库上游流域土壤有机碳和全氮含量影响的通径分析

土壤是全球重要的碳库和氮库,在全球碳氮循环中具有重要地位。密云水库是华北地区最大的水库和北京市最重要的地表水水源地,其上游流域山地广布,地形复杂,气候变化明显,土壤类型和植被类型多样,影响土壤碳氮库的环境因素具有较强的变异性。为揭示环境因素对密云水库上游流域土壤有机碳（soil organic carbon,SOC）和全氮（total nitrogen,TN）含量的作用效应,采用野外采样、实验室分析与逐步回归分析和通径分析相结合的方法,研究了气候（温度和降水）、地形（海拔和坡度）、土壤理化性质（土壤容重、含水量、pH值和粘粒含量）等环境因素对流域SOC和TN含量的影响。结果表明：温度、土壤容重、含水量、pH值和粘粒含量对SOC含量的影响显著（P＆lt;0.001）,其中各因子的直接通径系数依次为：土壤含水量（0.439）＞土壤容重（-0.324）＞pH值（-0.238）＞温度（-0.209）＞土壤粘粒含量（0.092）,间接通径系数依次为：土壤容重（-0.425）＞土壤粘粒含量（0.305）＞土壤含水量（0.287）＞pH值（-0.179）＞温度（-0.043）,因此,土壤含水量、pH值和温度主要通过直接作用影响SOC含量,而土壤容重和粘粒含量则主要通过其它因子的作用间接影响SOC含量。海拔、土壤容重、含水量、pH 值和粘粒含量对 TN 含量的影响显著（P＆lt;0.001）,其中各因子的直接通径系数依次为：土壤含水量（0.456）＞海拔（0.234）＞土壤容重（-0.228）＞pH 值（-0.190）＞土壤粘粒含量（0.094）,间接通径系数依次为：土壤容重（-0.484）＞土壤粘粒含量（0.301）＞海拔（0.247）＞土壤含水量（0.257）＞pH 值（-0.202）,因此,土壤含水量主要通过直接作用影响TN含量,而土壤容重和粘粒含量主要通过土壤含水量的作用间接影响TN含量,海拔和土壤pH值的直接作用与间接作用     

Estimator bias and efficiency for adaptive cluster sampling with order statistics and a stopping rule

Practical problems facing adaptive cluster sampling with order statistics (acsord) are explored using Monte Carlo simulation for three simulated fish populations and two known waterfowl populations. First, properties of an unbiased Hansen-Hurwitz (HH) estimator and a biased alternative Horvitz-Thompson (HT) estimator are evaluated. An increase in the level of population aggregation or the initial sample size increases the efficiencies of the two acsord estimators. For less aggregated fish populations, the efficiencies decrease as the order statistic parameter r (the number of units about which adaptive sampling is carried out) increases; for the highly aggregated fish and waterfowl populations, they increase with r. Acsord is almost always more efficient than simple random sampling for the highly aggregated populations. Positive bias is observed for the HT estimator, with the maximum bias usually occurring at small values of r. Secondly, a stopping rule at the Sth iteration of adaptive sampling beyond the initial sampling unit was applied to the acsord design to limit the otherwise open-ended sampling effort. The stopping rule induces relatively high positive bias to the HH estimator if the level of the population aggregation is high, the stopping level S is small, and r is large. The bias of HT is not very sensitive to the stopping rule and its bias is often reduced by the stopping rule at smaller values of r. For more aggregated populations, the stopping rule often reduces the efficiencies of the estimators compared to the non-stopping-rule scheme, but acsord still remains more efficient than simple random sampling. Despite its bias and lack of theoretical grounding, the HT estimator is usually more efficient than the HH estimator. In the stopping rule case, the HT estimator is preferable, because its bias is less sensitive to the stopping level.     

Spatially-balanced sampling versus unbalanced stratified sampling for assessing forest change: evidences in favour of spatial balance

Large-scale remote sensing-based inventories of forest cover are usually carried out by combining unsupervised classifications of satellite pixels into forest/non forest classes (map data) with subsequent time-consuming visual on-screen imagery classification of a probabilistic sample of pixels taken as the ground truth (reference data). In this paper the estimation of forest change from a sample of reference data is approached by: (i) exploiting map data to construct strata in which changes are occurred, and then adopting the stratified sampling joined with the HT estimator with most sampling effort devoted to strata where changes are occurred irrespective of their size, as suggested in most remote sensing literature regarding land change assessments; (ii) adopting a spatial scheme ensuring spatially balanced samples, as suggested in most recent statistical literature regarding spatial surveys, and exploiting the map data in the difference estimator. The results of a comparison performed on an artificial population of reference data generated from a real population of map data recorded in Sardinia (Italy) discourage the use of unbalanced stratified samples that achieve the worst precision. The best results are obtained by means of spatially balanced samples or stratification with nearly proportional allocation to strata.