Density estimation using ranked-set sampling data

The ranked-set sampling (RSS) is applicable in practical problems where the variable of interest for an observed item is costly or time-consuming but the ranking of a set of items according to the variable can be easily done without actual measurement. In the context of RSS, the need for density estimation arises in certain statistical procedures. The density estimation also has its own interest. In this article, we develop a method for the density estimation using RSS data. We derive the properties of the resulted density estimate and compare it with its counterpart in simple random sampling (SRS). It is shown that the density estimate using RSS data provides a better estimate of the density than the usual density estimate using SRS data. The density estimate developed in this article can well serve various purposes in the context of RSS.     

Distribution-free ranked-set sample procedures for umbrella alternatives in the m-sample setting

In settings where measurements are costly and/or difficult to obtain but ranking of the potential sample data is relatively easy and reliable, the use of statistical methods based on a ranked-set sampling approach can lead to substantial improvement over analogous methods associated with simple random samples. Previous nonparametric work in this area has been concentrated almost exclusively on the one- and two-sample location problems. In this paper we develop ranked-set sample procedures for the m-sample location setting where the treatment effect parameters follow a restricted umbrella pattern. Distribution-free testing procedures are developed for both the case where the peak of the umbrella is known and for the case where it is unknown. Small sample and asymptotic null distribution properties are provided for the peak-known test statistic.     

Conservation threats and the phylogenetic utility of IUCN Red List rankings in Incilius toads

Phylogenetic analysis of extinction threat is an emerging tool in the field of conservation. However, there are problems with the methods and data as commonly used. Phylogenetic sampling usually extends to the level of family or genus, but International Union for Conservation of Nature (IUCN) rankings are available only for individual species, and, although different species within a taxonomic group may have the same IUCN rank, the species may have been ranked as such for different reasons. Therefore, IUCN rank may not reflect evolutionary history and thus may not be appropriate for use in a phylogenetic context. To be used appropriately, threat‐risk data should reflect the cause of extinction threat rather than the IUCN threat ranking. In a case study of the toad genus Incilius, with phylogenetic sampling at the species level (so that the resolution of the phylogeny matches character data from the IUCN Red List), we analyzed causes of decline and IUCN threat rankings by calculating metrics of phylogenetic signal (such as Fritz and Purvis’ D). We also analyzed the extent to which cause of decline and threat ranking overlap by calculating phylogenetic correlation between these 2 types of character data. Incilius species varied greatly in both threat ranking and cause of decline; this variability would be lost at a coarser taxonomic resolution. We found far more phylogenetic signal, likely correlated with evolutionary history, for causes of decline than for IUCN threat ranking. Individual causes of decline and IUCN threat rankings were largely uncorrelated on the phylogeny. Our results demonstrate the importance of character selection and taxonomic resolution when extinction threat is analyzed in a phylogenetic context.     

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.     

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.     

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.     

The Relative Efficiency of Ranked Set Sampling in Ordinary Least Squares Regression

Ranked set sampling was developed for situations where measurement cost is expensive compared with unit acquisition. This paper presents results of simulations and theory examining the impact of balanced ranked set sampling on the relative efficiencies of the slope and intercept estimators of an ordinary least squares regression. Perfect ranking of either the independent or the dependent variable is assumed throughout. In contradistinction to most of the published ranked set sampling work, it is demonstrated that balanced ranked set sampling offers at most little improvement in the relative efficiencies of the slope estimator at any sample size.     

Variance estimation in the EMAP strategy for sampling discrete ecological resources

The United States Environmental Protection Agency's Environmental Monitoring and Assessment Program (EMAP) is designed to describe status, trends and spatial pattern of indicators of condition of the nation's ecological resources. The proposed sampling design for EMAP is based on a triangular systematic grid and employs both variable probability and double sampling. The Horvitz-Thompson estimator provides the foundation of the design-based estimation strategy used in EMAP. However, special features of EMAP designed to accommodate the complexity of sampling environmental resources on a national scale require modifications of standard variance estimation procedures as well as development of new techniques. An overview of variance estimation methods proposed for application to EMAP's sampling strategy for discrete resources is presented.     

A further contribution to the study of fish populations by capture-recapture experiments

In capture-recapture experiments, fish populations can be studied by two different sampling procedures. In both procedures, tagged fish are released on capture, but untagged fish are in one procedure released after tagging, in the second procedure they are retained. Using the two sampling techniques, Rafail (1971a,b) gave expressions for the estimation of an assumed constant (C) of proportionality between probabilities of capture of tagged to untagged fish which are simplified here to forms easier for calculation. The estimation of this constant (C) aids in estimation of abundance and mortality rates of untagged fish which are assumed to differ from those of tagged fish.     

Sequential sampling designs for catching the tail of dispersal kernels

Methods to design a sampling strategy should depend on the research question involved when conducting the experiment. The objective of this study is to design a seed trap configuration surrounding a parent plant when the long distance component of the seed dispersal kernel is of interest. In particular, as a population’s invasion speed depends mainly on the tail of the dispersal kernel, the sampling design in this study is based on calculating this quantity. The optimality criterion is to minimize the mean squared error (MSE) of the estimated invasion speed (using a limited number of traps) with respect to the “true” calculated invasion speed. Detailed procedures are given on how to calculate an invasion speed, both in a 1D and a 2D setting, with examples on how to implement the method to get a local optimal sampling strategy using Calluna vulgaris as a test system. Results show a trade-off between nearby sampling (many seeds, no long-distance dispersal measured) and distant sampling (few seeds, but long-distance dispersal measured).     

A proposed standard method for composite sampling of water chemistry and plankton in small lakes

In this paper a method for collection of vertically and horizontally integrated volume-weighted composite samples for analysis of water chemistry and plankton is presented. The method, which requires a proper knowledge of lake morphometry parameters, includes proposed standard procedures for determination of sampling interval thickness, maximum depth of sampling, selection of sampling stations, and distribution of discrete samples. An example of the outcome of the method in a lake with uncomplicated basin morphometry is given and the results are discussed against background of general lake basin morphometry data. The aim of the paper is to start a debate about optimization (statistical as well as ecological) of volume weighted composite sampling.     

A combination line transect and capture-recapture sampling model for multiple observers in aerial surveys

We present a robust sampling methodology to estimate population size using line transect and capture-recapture procedures for aerial surveys. Aerial surveys usually underestimate population density due to animals being missed. A combination of capture-recapture and line transect sampling methods with multiple observers allows violation of the assumption that all animals on the centreline are sighted from the air. We illustrate our method with an example of inanimate objects which shows evidence of failure of the assumption that all objects on the centreline have probability 1 of being detected. A simulation study is implemented to evaluate the performance of three variations of the Lincoln-Petersen estimator: the overall estimator, the stratified estimator, and the general stratified estimator based on the combined likelihood proposed in this paper. The stratified Lincoln-Petersen estimator based on the combined likelihood is found to be generally superior to the other estimators.     

Statistical inference using stratified judgment post-stratified samples from finite populations

This paper develops statistical inference for population mean and total using stratified judgment post-stratified (SJPS) samples. The SJPS design selects a judgment post-stratified sample from each stratum. Hence, in addition to stratum structure, it induces additional ranking structure within stratum samples. SJPS is constructed from a finite population using either a with or without replacement sampling design. Inference is constructed under both randomization theory and a super population model. In both approaches, the paper shows that the estimators of population mean and total are unbiased. The paper also constructs unbiased estimators for the variance (mean square prediction error) of the sample mean (predictor of population mean), and develops confidence and prediction intervals for the population mean. The empirical evidence shows that the proposed estimators perform better than their competitors in the literature.     

Semi-parametric Bayesian density estimation using ranked set sample in the presence of ranking error

In this paper, we propose a Bayesian method to estimate the underlying density function of a study variable Y using a ranked set sample in which an auxiliary variable X is used to rank the sampling units. The amount of association between X and Y is not known, resulting in an unknown degree of ranking error. We assume that (X, Y) follows a Morgenstern family of distributions. The study variable Y is assumed to have a parametric distribution, with the distribution of the parameters having a Dirichlet process prior. A Markov chain Monte Carlo procedure is developed to obtain a Bayesian estimator of the desired density function as well as of the ranking error. A simulation study is used to evaluate the performance of the proposed method. An example from forestry is used to illustrate a real-life application of the proposed methodology.     

Raumbezug in der ökosystemaren Umweltbeobachtung

Using the example of the transboundary Biosphere Reserve Rhone, experiments were performed and expounded upon with regard to the concept of ‘integrated monitoring’. The paper describes the components of a step-by-step harmonisation of data sampling and analysis procedures. Special emphasis is given to topics dealing with suitable methods for a sound selection of areas and plots to be monitored, as well as on rules for the spatial integration and generalisation of sampling results. As tools for this purpose the concept of ‘integrated monitoring’ uses the federal ‘Classification System of Ecoregions’ (Standortökologische Raumgliederung) and geostatistical methods for the spatial integration of existing monitoring programmes and sampling grids. Further, the paper outlines how to judge the development of water catchment areas using existing data from hydrological analyses and by means of an ecosystem-oriented water balance model.     

Effects of sampling error and temporal correlations in population growth on process variance estimators

Estimates of a population’s growth rate and process variance from time-series data are often used to calculate risk metrics such as the probability of quasi-extinction, but temporal correlations in the data from sampling error, intrinsic population factors, or environmental conditions can bias process variance estimators and detrimentally affect risk predictions. It has been claimed (McNamara and Harding, Ecol Lett 7:16–20, 2004) that estimates of the long-term variance that incorporate observed temporal correlations in population growth are unaffected by sampling error; however, no estimation procedures were proposed for time-series data. We develop a suite of such long-term variance estimators, and use simulated data with temporally autocorrelated population growth and sampling error to evaluate their performance. In some cases, we get nearly unbiased long-term variance estimates despite ignoring sampling error, but the utility of these estimators is questionable because of large estimation uncertainty and difficulties in estimating correlation structure in practice. Process variance estimators that ignored temporal correlations generally gave more precise estimates of the variability in population growth and of the probability of quasi-extinction. We also found that the estimation of probability of quasi-extinction was greatly improved when quasi-extinction thresholds were set relatively close to population levels. Because of precision concerns, we recommend using simple models for risk estimates despite potential biases, and limiting inference to quantifying relative risk; e.g., changes in risk over time for a single population or comparative risk among populations.     

Estimation bias under model selection for distance sampling detection functions

Many simulation studies have examined the properties of distance sampling estimators of wildlife population size. When assumptions hold, if distances are generated from a detection model and fitted using the same model, they are known to perform well. However, in practice, the true model is unknown. Therefore, standard practice includes model selection, typically using model comparison tools like Akaike Information Criterion. Here we examine the performance of standard distance sampling estimators under model selection. We compare line and point transect estimators with distances simulated from two detection functions, hazard-rate and exponential power series (EPS), over a range of sample sizes. To mimic the real-world context where the true model may not be part of the candidate set, EPS models were not included as candidates, except for the half-normal parameterization. We found median bias depended on sample size (being asymptotically unbiased) and on the form of the true detection function: negative bias (up to 15% for line transects and 30% for point transects) when the shoulder of maximum detectability was narrow, and positive bias (up to 10% for line transects and 15% for point transects) when it was wide. Generating unbiased simulations requires careful choice of detection function or very large datasets. Practitioners should collect data that result in detection functions with a shoulder similar to a half-normal and use the monotonicity constraint. Narrow-shouldered detection functions can be avoided through good field procedures and those with wide shoulder are unlikely to occur, due to heterogeneity in detectability.     

Detecting Rare Species with Random or Subjective Sampling: a Case Study of Red-Listed Saproxylic Beetles in Boreal Sweden

Abstract:  Efficient sampling design in field studies is important for economical and statistical reasons. We compared two ways to distribute sampling effort over an area, either randomly or subjectively. We searched for red-listed saproxylic (wood-living) beetles in 30 spruce stands in boreal Sweden by sifting wood from dead trees. We randomly selected positions within each stand with a geographic positioning system and sampled the nearest dead tree (random sample). In the same stand we also sampled dead trees that, based on literature, were likely to host such species (subjective sampling). The subjective sampling (two to five samples per stand, depending on stand size) was compared with the higher, random sampling effort (fixed level of 12 samples/stand). Subjective sampling was significantly more efficient. Red-listed species were found in 36% of the subjective samples and in 16% of the random samples. Nevertheless, the larger random effort resulted in a comparable number of red-listed species per stand and in 13 detected species in total (vs. 12 species with subjective sampling). Random sampling was less efficient, but provided an unbiased alternative more suitable for statistical purposes, as needed in, for example, monitoring programs. Moreover, new species-specific knowledge can be gained through random searches.     

Estimating the complexity of bird song by using capture-recapture approaches from community ecology

Repertoire size, the number of unique song or syllable types in the repertoire, is a widely used measure of song complexity in birds, but it is difficult to calculate this exactly in species with large repertoires. A new method of repertoire size estimation applies species richness estimation procedures from community ecology, but such capture-recapture approaches have not been much tested. Here, we establish standardized sampling schemes and estimation procedures using capture-recapture models for syllable repertoires from 18 bird species, and suggest how these may be used to tackle problems of repertoire estimation. Different models, with different assumptions regarding the heterogeneity of the use of syllable types, performed best for different species with different song organizations. For most species, models assuming heterogeneous probability of occurrence of syllables (so-called detection probability) were selected due to the presence of both rare and frequent syllables. Capture-recapture estimates of syllable repertoire size from our small sample did not differ significantly from previous estimates using larger samples of count data. However, the enumeration of syllables in 15 songs yielded significantly lower estimates than previous reports. Hence, heterogeneity in detection probability of syllables should be addressed when estimating repertoire size. This is neglected using simple enumeration procedures, but is taken into account when repertoire size is estimated by appropriate capture-recapture models adjusted for species-specific song organization characteristics. We suggest that such approaches, in combination with standardized sampling, should be applied in species with potentially large repertoire size. On the other hand, in species with small repertoire size and homogenous syllable usage, enumerations may be satisfactory. Although researchers often use repertoire size as a measure of song complexity, listeners to songs are unlikely to count entire repertoires and they may rely on other cues, such as syllable detection probability.Communicated by A. Cockburn