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.     

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.     

Density surface modelling with line transect sampling as a tool for abundance estimation of marine benthic species: the <Emphasis Type="Italic">Pinna nobilis</Emphasis> example in a marine lake

Density estimation of marine benthic fauna is most often conducted with fishery surveys using dredges or trawls. These estimates are often unreliable due to low and variable efficiency and are inappropriate when dealing with rare or endangered species. In the marine Lake Vouliagmeni, a density surface modelling (DSM) approach using survey data from line transects, integrated with a Geographic Information System (GIS), was used to estimate the population density of the endangered fan mussel Pinna nobilis. This is the first time that such an approach has been applied for a marine benthic species. DSM was beneficial in relation to traditional distance sampling. Apart from providing a more precise total abundance estimate, it related the density of the species to spatial covariates of interest, gave a depiction of the species dispersion in the study area, and provided abundance estimates in any sub-region of the study area. In Lake Vouliagmeni, a marked zonation of P. nobilis distribution was revealed, with the species being restricted in the shallow peripheral zone at depths <22 m. Two density peaks were observed, a major peak at depths between 12 and 13 m and a secondary peak at ∼4 m. A main hotspot of high density was also observed in the northeastern part of the lake. Total abundance of the species was estimated to be 6,770 individuals with a 95% confidence interval of 5,460–8,393 individuals.     

Sampling for weed spatial distribution mapping need not be adaptive

Weeds are species of interest for ecologists because they are competitors of the crop for resources but they also play an important role in maintaining biodiversity in agroecosystems. To study their spatial distribution at the field scale, only sampled observations are available due to the cost of sampling. Weeds sampling strategies are static. However, in the domain of spatial sampling, adaptive strategies have also been developed with, for some of them, an important on-line or off-line computational cost. In this article we provide answers to the following question: Are the current adaptive sampling methods efficient enough to motivate a wider use in practice when sampling a weed species at a field scale? We provide a comparison of the behaviour of 8 static strategies and 3 adaptive ones on four criteria: density class estimation, map restoration, spatial aggregation estimation, and sampling duration. From two weeds data sets, we estimated six contrasted Markov Random Field (MRF) models of weed density class spatial distribution and a model for sampling duration. The MRF models were then used to compare the strategies on a large set of simulated maps. Our main finding was that there is no clear gain in using adaptive sampling strategies rather than static ones for the three first criteria, and adaptive strategies were associated to longer sampling duration. This conclusion points out that for weed mapping, it is more important to build a good model of spatial distribution, than to propose complex adaptive sampling strategies.     

Bayes Estimation of a Distribution Function Using Ranked Set Samples

Aranked set sample (RSS), if not balanced, is simply a sample of independent order statistics gener- ated from the same underlying distribution F. Kvam and Samaniego (1994) derived maximum likelihood estimates of F for a general RSS. In many applications, including some in the environ- mental sciences, prior information about F is available to supplement the data-based inference. In such cases, Bayes estimators should be considered for improved estimation. Bayes estimation (using the squared error loss function) of the unknown distribution function F is investigated with such samples. Additionally, the Bayes generalized maximum likelihood estimator (GMLE) is derived. An iterative scheme based on the EM Algorithm is used to produce the GMLE of F. For the case of squared error loss, simple solutions are uncommon, and a procedure to find the solution to the Bayes estimate using the Gibbs sampler is illustrated. The methods are illustrated with data from the Natural Environmental Research Council of Great Britain (1975), representing water discharge of floods on the Nidd River in Yorkshire, England     

Propagation functions for monitoring distributional changes

In environmental assessment and monitoring, a primary objective of the investigator is to describe the changes occurring in the environmentally important variables over time. Propagation functions have been proposed to describe the distributional changes occurring in the variable of interest at two different times. McDonald et al. (1992, 1995) proposed an estimator of propagation function under the assumption of normality. We conduct a detailed sensitivity analysis of inference based on the normal model. It turns out that this model is appropriate only for small departures from normality whereas, for moderate to large departures, both estimation and testing of hypothesis break down. Non-parametric estimation of the propagation function based on kernel density estimation is also considered and the robustness of the choice of bandwidth for kernel density estimation is investigated. Bootstrapping is employed to obtain confidence intervals for the propagation function and also to determine the critical regions for testing the significance of distributional changes between two sampling epochs. Also studied briefly is the mathematical form and graphical shape of the propagation function for some parametric bivariate families of distributions. Finally, the proposed estimation techniques are illustrated on a data set of tree ring widths.     

Nonparametric mean estimation using partially ordered sets

In ranked-set sampling (RSS), the ranker must give a complete ranking of the units in each set. In this paper, we consider a modification of RSS that allows the ranker to declare ties. Our sampling method is simply to break the ties at random so that we obtain a standard ranked-set sample, but also to record the tie structure for use in estimation. We propose several different nonparametric mean estimators that incorporate the tie information, and we show that the best of these estimators is substantially more efficient than estimators that ignore the ties. As part of our comparison of estimators, we develop new results about models for ties in rankings. We also show that there are settings where, to achieve more efficient estimation, ties should be declared not just when the ranker is actually unsure about how units rank, but also when the ranker is sure about the ranking, but believes that the units are close.     

Estimation of rare and clustered population mean using stratified adaptive cluster sampling

For many clustered populations, the prior information on an initial stratification exists but the exact pattern of the population concentration may not be predicted. Under this situation, the stratified adaptive cluster sampling (SACS) may provide more efficient estimates than the other conventional sampling designs for the estimation of rare and clustered population parameters. For practical interest, we propose a generalized ratio estimator with the single auxiliary variable under the SACS design. The expressions of approximate bias and mean squared error (MSE) for the proposed estimator are derived. Numerical studies are carried out to compare the performances of the proposed generalized estimator over the usual mean and combined ratio estimators under the conventional stratified random sampling (StRS) using a real population of redwood trees in California and generating an artificial population by the Poisson cluster process. Simulation results show that the proposed class of estimators may provide more efficient results than the other estimators considered in this article for the estimation of highly clumped population.

     

Bayesian nonparametric inference for the overlap of daily animal activity patterns

The study of the interaction among species is an active area of research in Ecology. In particular, it is of interest to evaluate the overlap of their ecological niches. Temporal activity is one of the niche’s axes most commonly used to explore ecological segregation among animal species, and many contributions focus on the overlap of this variable. Once the information of the temporal activity is obtained in the wild, the data is treated as a random sample. There exist different methods to estimate the overlap. Specifically, in the case of two species, one possibility is to estimate the density of the temporal activity of each species and then evaluate the overlap between these density functions. This leads naturally to the analysis of circular data. Most of the procedures currently in use impose some rather restrictive assumptions on the probabilistic models used to describe the phenomena, and only provide approximate measures of the uncertainty involved in the process. In this article, we propose a Bayesian nonparametric approach which incorporates a well-defined noninformative prior. We take advantage of the data structure to define such a prior in terms of the predictive distribution. To the best of our knowledge, this is a novel approach. Our procedure is compared with a well-known method using simulated data, and applied to the analysis of real camera-trap data concerning two mammalian species from the El Triunfo biosphere reserve (Chiapas, Mexico).     

Spatially-balanced adaptive web sampling

The paper deals with sampling from a finite population that is distributed over space and has a highly uneven spatial distribution. It suggests a sampling design that allocates a portion of the sample units that are well spread over the population and sequentially selects the remaining units in sub-areas that appear to be of more interest according to the study variable values observed during the survey. In order to estimate the population mean while using this sampling design, a computationally intense estimator, obtained via the Rao–Blackwell approach, is proposed and a resampling method is used that makes the inference computationally feasible. The whole sampling strategy is evaluated through several Monte Carlo experiments.     

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.     

Remote sensing: Using the spatial domain

Objects in the terrestrial environment interact differentially with electromagnetic radiation according to their essential physical, chemical and biological properties. This differential interaction is manifest as variability in scattered radiation according to wavelength, location, time, geometries of illumination and observation and polarization. If the population of scattered radiation could be measured, then estimation of these essential properties would be straightforward. The only problem would be linking such estimates to environmental variables of interest. This review paper is divided into three parts. Part 1 is an overview of the attempts that have been made to sample the five domains of scattered radiation (spectral, spatial, temporal, geometrical, polarization) and then to use the results of this sampling to estimate environmental variables of interest. Part one highlights three issues: first, that relationships between remotely sensed data and environmental variables of interest are indirect; second, our ability to estimate these environmental variables is dependent upon our ability to capture a sound representation of variability in scattered radiation and third, a considerable portion of the useful information in remotely sensed images resides in the spatial domain (within the relations between the pixels in the image). This final point is developed in Part 2 that explores ways in which the spatial domain is utilized to describe spatial variation in remotely sensed and ground data; to design optimum sampling schemes for image data and ground data and to increase the accuracy with which remotely sensed data can be used to estimate both discontinuous and continuous variables. Part 3 outlines two specific uses of information in the spatial domain; first, to select an optimum spatial resolution and second, to inform an image classification.     

The application of adaptive cluster sampling for rare subtidal macroalgae

Adaptive cluster sampling (ACS) is a targeting sampling method that provides unbiased abundance estimators for populations of rare species that may be inadequately sampled with simple random sampling (SRS). ACS has been used successfully to estimate abundances of rockfish and sardine larvae from shipboard surveys. In this study, we describe the application of ACS for subtidal macroalgae. Using SCUBA, we measured abundances of Codium mamillosum, C. pomoides, and Halimeda cuneata at three islands and two levels of wave exposure. The three species were relatively patchy and could be sampled with ACS at one site per dive. Their distributions differed among islands and with exposure to wave energy, with H. cuneata found at only one island. ACS is a useful tool for understanding the spatial distribution and abundance of populations of rare benthic species, but, as was the case in this study, may not be as efficient as sampling with SRS with comparable replication.     

Toward accurate and precise estimates of lion density

Reliable estimates of animal density are fundamental to understanding ecological processes and population dynamics. Furthermore, their accuracy is vital to conservation because wildlife authorities rely on estimates to make decisions. However, it is notoriously difficult to accurately estimate density for wide‐ranging carnivores that occur at low densities. In recent years, significant progress has been made in density estimation of Asian carnivores, but the methods have not been widely adapted to African carnivores, such as lions (Panthera leo). Although abundance indices for lions may produce poor inferences, they continue to be used to estimate density and inform management and policy. We used sighting data from a 3‐month survey and adapted a Bayesian spatially explicit capture‐recapture (SECR) model to estimate spatial lion density in the Maasai Mara National Reserve and surrounding conservancies in Kenya. Our unstructured spatial capture‐recapture sampling design incorporated search effort to explicitly estimate detection probability and density on a fine spatial scale, making our approach robust in the context of varying detection probabilities. Overall posterior mean lion density was estimated to be 17.08 (posterior SD 1.310) lions >1 year old/100 km², and the sex ratio was estimated at 2.2 females to 1 male. Our modeling framework and narrow posterior SD demonstrate that SECR methods can produce statistically rigorous and precise estimates of population parameters, and we argue that they should be favored over less reliable abundance indices. Furthermore, our approach is flexible enough to incorporate different data types, which enables robust population estimates over relatively short survey periods in a variety of systems. Trend analyses are essential to guide conservation decisions but are frequently based on surveys of differing reliability. We therefore call for a unified framework to assess lion numbers in key populations to improve management and policy decisions.     

Characterizing Spatial Uncertainty when Integrating Social Data in Conservation Planning

Recent conservation planning studies have presented approaches for integrating spatially referenced social (SRS) data with a view to improving the feasibility of conservation action. We reviewed the growing conservation literature on SRS data, focusing on elicited or stated preferences derived through social survey methods such as choice experiments and public participation geographic information systems. Elicited SRS data includes the spatial distribution of willingness to sell, willingness to pay, willingness to act, and assessments of social and cultural values. We developed a typology for assessing elicited SRS data uncertainty which describes how social survey uncertainty propagates when projected spatially and the importance of accounting for spatial uncertainty such as scale effects and data quality. These uncertainties will propagate when elicited SRS data is integrated with biophysical data for conservation planning and may have important consequences for assessing the feasibility of conservation actions. To explore this issue further, we conducted a systematic review of the elicited SRS data literature. We found that social survey uncertainty was commonly tested for, but that these uncertainties were ignored when projected spatially. Based on these results we developed a framework which will help researchers and practitioners estimate social survey uncertainty and use these quantitative estimates to systematically address uncertainty within an analysis. This is important when using SRS data in conservation applications because decisions need to be made irrespective of data quality and well characterized uncertainty can be incorporated into decision theoretic approaches.     

Density estimation in tiger populations: combining information for strong inference

A productive way forward in studies of animal populations is to efficiently make use of all the information available, either as raw data or as published sources, on critical parameters of interest. In this study, we demonstrate two approaches to the use of multiple sources of information on a parameter of fundamental interest to ecologists: animal density. The first approach produces estimates simultaneously from two different sources of data. The second approach was developed for situations in which initial data collection and analysis are followed up by subsequent data collection and prior knowledge is updated with new data using a stepwise process. Both approaches are used to estimate density of a rare and elusive predator, the tiger, by combining photographic and fecal DNA spatial capture-recapture data. The model, which combined information, provided the most precise estimate of density (8.5 +/- 1.95 tigers/100 km2 [posterior mean +/- SD]) relative to a model that utilized only one data source (photographic, 12.02 +/- 3.02 tigers/100 km2 and fecal DNA, 6.65 +/- 2.37 tigers/100 km2). Our study demonstrates that, by accounting for multiple sources of available information, estimates of animal density can be significantly improved.     

Design-based estimation in environmental surveys with positional errors

The recent increased availability of information about the micro-geographic positions of population units in environmental surveys has led to important developments in spatial sampling methodologies and, as a result, has improved the estimation accuracy. In real data, however, information about the location of units is often affected by inaccuracy about their exact spatial positions, and these non-sampling errors can affect the estimation procedure. This paper aims to investigate the effects of positional errors on total estimation through a Monte-Carlo simulation study based on real populations of trees. Starting from perfect positioning, we examine two typical types of coarsening that frequently impact two different species of trees. The simulation results show that the exploitation of spatial information to estimate population totals continues to be relevant in the context of environmental surveys, even in the presence of inaccuracies.     

A state-space model approach to optimum spatial sampling design based on entropy

We consider the spatial sampling design problem for a random field X. This random field is in general assumed not to be directly observable, but sample information from a related variable Y is available. Our purpose in this paper is to present a state-space model approach to network design based on Shannon's definition of entropy, and describe its main points with regard to some of the most common practical problems in spatial sampling design. For applications, an adaptation of Ko et al.'s (1995) algorithm for maximum entropy sampling in this context is provided. We illustrate the methodology using piezometric data from the Velez aquifer (Malaga, Spain). © Rapid Science 1998     

Sampling design trade-offs in occupancy studies with imperfect detection: examples and software.

Researchers have used occupancy, or probability of occupancy, as a response or state variable in a variety of studies (e.g., habitat modeling), and occupancy is increasingly favored by numerous state, federal, and international agencies engaged in monitoring programs. Recent advances in estimation methods have emphasized that reliable inferences can be made from these types of studies if detection and occupancy probabilities are simultaneously estimated. The need for temporal replication at sampled sites to estimate detection probability creates a trade-off between spatial replication (number of sample sites distributed within the area of interest/inference) and temporal replication (number of repeated surveys at each site). Here, we discuss a suite of questions commonly encountered during the design phase of occupancy studies, and we describe software (program GENPRES) developed to allow investigators to easily explore design trade-offs focused on particularities of their study system and sampling limitations. We illustrate the utility of program GENPRES using an amphibian example from Greater Yellowstone National Park, U.S.A.     

Moment and maximum likelihood estimators for Weibull distributions under length- and area-biased sampling

Many of the most popular sampling schemes used in forestry are probability proportional to size methods. These methods are also referred to as size-biased because sampling is actually from a weighted form of the underlying population distribution. Length- and area-biased sampling are special cases of size-biased sampling where the probability weighting comes from a lineal or areal function of the random variable of interest, respectively. Often, interest is in estimating a parametric probability density of the data. In forestry, the Weibull function has been used extensively for such purposes. Estimating equations for method of moments and maximum likelihood for two- and three-parameter Weibull distributions are presented. Fitting is illustrated with an example from an area-biased angle-gauge sample of standing trees in a woodlot. Finally, some specific points concerning the form of the size-biased densities are reported.