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.     

Comparison of distribution function estimators for forestry applications following variable probability sampling

Practical considerations often motivate employing variable probability sampling designs when estimating characteristics of forest populations. Three distribution function estimators, the Horvitz-Thompson estimator, a difference estimator, and a ratio estimator, are compared following variable probability sampling in which the inclusion probabilities are proportional to an auxiliary variable, X. Relative performance of the estimators is affected by several factors, including the distribution of the inclusion probabilities, the correlation () between X and the response Y, and the position along the distribution function being estimated. Both the ratio and difference estimators are superior to the Horvitz-Thompson estimator. The difference estimator gains better precision than the ratio estimator toward the upper portion of the distribution function, but the ratio estimator is superior toward the lower end of the distribution function. The point along the distribution function at which the difference estimator becomes more precise than the ratio estimator depends on the sampling design, as well as the coefficient of variation of X and . A simple confidence interval procedure provides close to nominal coverage for intervals constructed from both the difference and ratio estimators, with the exception that coverage may be poor for the lower tail of the distribution function when using the ratio estimator.     

Leveraging the Rao–Blackwell theorem to improve ratio estimators in adaptive cluster sampling

Rao-Blackwellization is used to improve the unbiased Hansen–Hurwitz and Horvitz–Thompson unbiased estimators in Adaptive Cluster Sampling by finding the conditional expected value of the original unbiased estimators given the sufficient or minimal sufficient statistic. In principle, the same idea can be used to find better ratio estimators, however, the calculation of taking all the possible combinations into account can be extremely tedious in practice. The simplified analytical forms of such ratio estimators are not currently available. For practical interest, several improved ratio estimators in Adaptive Cluster Sampling are proposed in this article. The proposed ratio estimators are not the real Rao-Blackwellized versions of the original ones but make use of the Rao-Blackwellized univariate estimators. How to calculate the proposed estimators is illustrated, and their performance are evaluated by both of the Bivariate Poisson clustered process and a real data. The simulation result indicates that the proposed improved ratio estimators are able to provide considerably advantageous estimation results over the original ones.     

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.     

Network inclusion probabilities and Horvitz-Thompson estimation for adaptive simple Latin square sampling

Consider a survey of a plant or animal species in which abundance or presence/absence will be recorded. Further assume that the presence of the plant or animal is rare and tends to cluster. A sampling design will be implemented to determine which units to sample within the study region. Adaptive cluster sampling designs Thompson (1990) are sampling designs that are implemented by first selecting a sample of units according to some conventional probability sampling design. Then, whenever a specified criterion is satisfied upon measuring the variable of interest, additional units are adaptively sampled in neighborhoods of those units satisfying the criterion. The success of these adaptive designs depends on the probabilities of finding the rare clustered events, called networks. This research uses combinatorial generating functions to calculate network inclusion probabilities associated with a simple Latin square sample. It will be shown that, in general, adaptive simple Latin square sampling when compared to adaptive simple random sampling will (i) yield higher network inclusion probabilities and (ii) provide Horvitz-Thompson estimators with smaller variability.     

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.     

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.

     

Empirical likelihood confidence intervals for adaptive cluster sampling

Adaptive cluster sampling (ACS) is an efficient sampling design for estimating parameters of rare and clustered populations. It is widely used in ecological research. The modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators based on small samples under ACS have often highly skewed distributions. In such situations, confidence intervals based on traditional normal approximation can lead to unsatisfactory results, with poor coverage properties. Christman and Pontius (Biometrics 56:503–510, 2000) showed that bootstrap percentile methods are appropriate for constructing confidence intervals from the HH estimator. But Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) showed that bootstrap confidence intervals from the HT estimator are even worse than the normal approximation confidence intervals. In this article, we consider two pseudo empirical likelihood functions under the ACS design. One leads to the HH estimator and the other leads to a HT type estimator known as the Hájek estimator. Based on these two empirical likelihood functions, we derive confidence intervals for the population mean. Using a simulation study, we show that the confidence intervals obtained from the first EL function perform as good as the bootstrap confidence intervals from the HH estimator but the confidence intervals obtained from the second EL function perform much better than the bootstrap confidence intervals from the HT estimator, in terms of coverage rate.     

An estimating function approach to the inference of catch-effort models

A class of catch-effort models, which allows for heterogeneous removal probabilities, is proposed for closed populations. The model includes three types of removal probabilities: multiplicative, Poisson and logistic. The usual removal and generalized removal models then become special cases. The equivalence of the proposed model and a special type of capture-recapture model is discussed. A unified estimating function approach is used to estimate the initial population size. For the homogeneous model, the resulting population size estimator based on optimal estimating functions is asymptotically equivalent to the maximum likelihood estimator. One advantage for our approach is that it can be extended to handle the heterogeneous populations in which the maximum likelihood estimators do not exist. The bootstrap method is applied to construct variance estimators and confidence intervals. We illustrate the method by two real data examples. Results of a simulation study investigating the performance of the proposed estimation procedure are presented.     

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.     

Adaptive two-stage one-per-stratum sampling

We briefly describe adaptive cluster sampling designs in which the initial sample is taken according to a Markov chain one-per-stratum design (Breidt, 1995) and one or more secondary samples are taken within strata if units in the initial sample satisfy a given condition C. An empirical study of the behavior of the estimation procedure is conducted for three small artificial populations for which adaptive sampling is appropriate. The specific sampling strategy used in the empirical study was a single random-start systematic sample with predefined systematic samples within strata when the initially sampled unit in that stratum satisfies C. The bias of the Horvitz-Thompson estimator for this design is usually very small when adaptive sampling is conducted in a population for which it is suited. In addition, we compare the behavior of several alternative estimators of the standard error of the Horvitz-Thompson estimator of the population total. The best estimator of the standard error is population-dependent but it is not unreasonable to use the Horvitz-Thompson estimator of the variance. Unfortunately, the distribution of the estimator is highly skewed hence the usual approach of constructing confidence intervals assuming normality cannot be used here.     

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.     

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.     

Regression-type estimators for adaptive two-stage sequential sampling

Adaptive two-stage sequential sampling (ATSSS) design was developed to observe more rare units and gain higher efficiency, in the sense of having a smaller variance estimator, than conventional sampling designs with equal effort for rare and spatially cluster populations. For certain rare populations, incorporating auxiliary variables into a sampling design can further improve the observation of rare units and increase efficiency. In this article, we develop regression-type estimators for ATSSS so that auxiliary variables can be incorporated into the ATSSS design when warranted. Simulation studies on two populations show that the regression-type estimators can significantly increase the efficiency of ATSSS and the detection of more rare units as compared to conventional sampling counterparts. Simulation of sampling of desert shrubs in Inner Mongolia (one of the two populations studied) showed that by incorporating a GIS auxiliary variable into ATSSS with the regression estimators resulted in a gain in efficiency over ATSSS without the auxiliary variable. Further, we found that the use of the GIS auxiliary variable in a conventional two-stage design with a regression estimator did not show a gain in efficiency.     

Randomized graph sampling

Randomized graph sampling (RGS) is an approach for sampling populations associated with or describable as graphs, when the structure of the graph is known and the parameter of interest is the total weight of the graph. RGS is related to, but distinct from, other graph-based approaches such as snowball and network sampling. Graph elements are clustered into walks that reflect the structure of the graph, as well as operational constraints on sampling. The basic estimator in RGS can be constructed as a Horvitz-Thompson estimator. I prove it to be design-unbiased, and also show design-unbiasedness of an estimator of the sample variance when walks are sampled with replacement. Covariates can be employed for variance reduction either through improved assignment of selection probabilities to walks in the design step, or through the use of alternative estimators during analysis. The approach is illustrated with a trail maintenance example, which demonstrates that complicated approaches to assignment of selection probabilities can be counterproductive. I describe conditions under which RGS may be efficient in practice, and suggest possible applications.     

Two-stage estimation of ungulate abundance in Mediterranean areas using pellet group count

A design-based strategy for estimating wildlife ungulate abundance in a Mediterranean protected area (Maremma Regional Park) is considered. The estimation is based on pellet group count (clearance count technique) in a set of plots, whose size and number is established on the basis of practical considerations and available resources. The sampling scheme involves a preliminary stratification and subsequent two-stage sampling. In the first stage, large strata (defined through habitat features) are partitioned into spatial units and a sample of units is selected by means of a sampling scheme ensuring inclusion probabilities proportional to unit size, but avoiding the selection of contiguous units. Then, the abundances of the selected units are estimated in a second stage, in which plots are located using a random scheme ensuring an even coverage of the units. In small strata, only the second stage is performed. Unbiased estimators of abundance and conservative estimators of their variances are derived for each strata and for the whole study area. The proposed strategy has been applied since the Summer of 2006 and the estimation results reveal substantial improvement with respect to the previous results obtained by means of an alternative strategy.     

Continuous-time capture-recapture population estimation when capture probabilities vary over time

Closed capture-recapture (CR) estimators have been used extensively to estimate population size. Most closed CR approaches have been developed and evaluated for discrete-time models, but there has been little effort to evaluate their continuous-time counterparts. Continuous-time estimators — developed using maximum likelihood theory by Craig (1953) and Darroch (1958), and martingale theory by Becker (1984) — that allow capture probabilities to vary over time were evaluated using Monte Carlo simulation. Overall, the ML estimators had a smaller MSE. The estimators performed well when model assumptions were upheld, and were somewhat robust to heterogeneity in capture probabilities. However, the estimators were not robust to behavioural effects in the capture probabilities. Time lag effects (periods when animals might be unavailable for immediate recapture) on continuous-time estimates were also investigated and results indicated a positive bias which was greater for smaller populations. There was no gain in performance when using a continuous-time estimator versus a discrete-time estimator on the same simulated data. Usefulness of the continuous-time approach may be limited to study designs where animals are easier to sample using continuous-time methodology.     

Comparison between Hansen-Hurwitz and Horvitz-Thompson estimators for adaptive cluster sampling

Thompson (1990) introduced the adaptive cluster sampling design. This sampling design has been shown to be a useful sampling method for parameter estimation of a clustered and scattered population (Roesch, 1993; Smith et al., 1995; Thompson and Seber, 1996). Two estimators, the modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators, are available to estimate the mean or total of a population. Empirical results from previous researches indicate that the modified HT estimator has smaller variance than the modified HH estimator. We analytically compare the properties of these two estimators. Some results are obtained in favor of the modified HT estimator so that practitioners are strongly recommended to use the HT estimator despite easiness of computations for the HH estimator.     

A framework for adapting survey design through time for wildlife population assessment

Sampling strategies for monitoring the status and trends in wildlife populations are often determined before the first survey is undertaken. However, there may be little information about the distribution of the population and so the sample design may be inefficient. Through time, as data are collected, more information about the distribution of animals in the survey region is obtained but it can be difficult to incorporate this information in the survey design. This paper introduces a framework for monitoring motile wildlife populations within which the design of future surveys can be adapted using data from past surveys whilst ensuring consistency in design-based estimates of status and trends through time. In each survey, part of the sample is selected from the previous survey sample using simple random sampling. The rest is selected with inclusion probability proportional to predicted abundance. Abundance is predicted using a model constructed from previous survey data and covariates for the whole survey region. Unbiased design-based estimators of status and trends and their variances are derived from two-phase sampling theory. Simulations over the short and long-term indicate that in general more precise estimates of status and trends are obtained using this mixed strategy than a strategy in which all of the sample is retained or all selected with probability proportional to predicted abundance. Furthermore the mixed strategy is robust to poor predictions of abundance. Estimates of status are more precise than those obtained from a rotating panel design.