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.

     

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.     

Use of post-stratification in composite sampling for estimating mean

We show with the results of a study conducted in the Hamadan Province, Iran as to how the use of composite sampling for estimating mean zinc concentration in the soil can help reduce analytical costs by reducing the number of analysis required. We also introduce post-stratification methodology in the composition step to take advantage of possible spatial dispersion. We speculate that the zinc concentration value depends on the sample location, we first stratified the sample set and then composite units from different strata randomly. The results of a simulation study show that the use of this approach not only reduces the total costs but also increases the precision of the estimator.     

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.     

Spatially stratified sampling using auxiliary information for geostatistical mapping

This paper presents a method of spatial sampling based on stratification by Local Moran’s I _i calculated using auxiliary information. The sampling technique is compared to other design-based approaches including simple random sampling, systematic sampling on a regular grid, conditional Latin Hypercube sampling and stratified sampling based on auxiliary information, and is illustrated using two different spatial data sets. Each of the samples for the two data sets is interpolated using regression kriging to form a geostatistical map for their respective areas. The proposed technique is shown to be competitive in reproducing specific areas of interest with high accuracy.     

Rao-Blackwell versions of the Horvitz-Thompson and Hansen-Hurwitz in adaptive cluster sampling

Thompson (1990) introduced the adaptive cluster sampling design and developed two unbiased estimators, the modified Horvitz-Thompson (HT) and Hansen-Hurwitz (HH) estimators, for this sampling design and noticed that these estimators are not a function of the minimal sufficient statistics. He applied the Rao-Blackwell theorem to improve them. Despite having smaller variances, these latter estimators have not received attention because a suitable method or algorithm for computing them was not available. In this paper we obtain closed forms of the Rao-Blackwell versions which can easily be computed. We also show that the variance reduction for the HH estimator is greater than that for the HT estimator using Rao-Blackwell versions. When the condition for extra samples is 0$$ " align="middle" border="0"> , one can expect some Rao-Blackwell improvement in the HH estimator but not in the HT estimator. Two examples are given.     

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.     

Bootstrap confidence intervals for adaptive cluster sampling design based on Horvitz–Thompson type estimators

Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) introduced several bootstrap methods under adaptive cluster sampling using a Horvitz–Thompson type estimator. Using a simulation study, they showed that their proposed methods provide confidence intervals with highly understated coverage rates. In this article, we first show that their bootstrap methods provide biased bootstrap estimates. We then define two bootstrap methods, based on the method of Gross (Proceeding of the survey research methods section. American Statistical Association, Alexandria, VA, pp 181–184, 1980) and Bootstrap With Replacement, that provide unbiased bootstrap estimates of the population mean with bootstrap variances matching the corresponding unbiased variance estimator. Using a simulation study, we show that the bootstrap confidence intervals based on our proposed methods have better performance than those based on available bootstrap methods, in the sense of having coverage proportion closer to the nominal coverage level. We also compare the proposed intervals to empirical likelihood based intervals in small samples.     

Using the bootstrap with two-phase adaptive stratified samples from multiple populations at multiple locations

Recently the two-phase adaptive stratified sampling design proposed by Francis (1984) has been extended by Manly et al. (2002) for situations where several biological populations are sampled simultaneously, and where this is done at several different geographical locations in order to estimate population totals or means. The method uses the results from a first phase sample to decide how best to allocate a second phase sample to locations and strata, in order to maximise a criterion (based on estimated coefficients of variation) that measures the accuracy of estimation for population totals, for all variables at all locations. One potential problem with this method is bias in the estimators of the population totals and means. In this paper bootstrapping is considered as a means of overcoming these biases. It is shown using model populations of Pacific walrus and shellfish, based on real data, that bootstrapping is a useful tool for removing about half of the bias. This is also confirmed from some simulations using artificial data.     

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.     

Markov Chain Monte Carlo on optimal adaptive sampling selections

Under a Bayesian population model with a given prior distribution, the optimal sampling strategy with a fixed sample size n is an n-phase adaptive one. That is, the selection of the next sampling units should sequentially depend on the information obtained from the previously selected units, including the observed values of interest. Such an optimal strategy is in general not executable in practice due to its intensive computation. In many survey sampling situations, an important problem is that one would like to select a set of units in addition to a certain number of sampling units which have been observed. If the optimal strategy is an adaptive one, the selection of the additional units should take both the labels and the observed values of the already selected units into account. Hence, a simpler optimal two-phase adaptive sampling strategy under a Bayesian population model is proposed in this article for practical interest. A Markov chain Monte Carlo method is used to approximate the posterior joint distribution of the unobserved population units after the first phase sampling, for the optimal selection of the second phase sample. This approximation method is found to be successful to select the optimal second-phase sample. Finally, this optimal strategy is applied to a set of data from a study of geothermal CO₂ emissions in Yellowstone National Park as a practical illustrative example.     

Variances and variance estimators of the improved ratio estimators under adaptive cluster sampling

Although not design-unbiased, the ratio estimator is recognized as more efficient when a certain degree of correlation exists between the variable of primary interest and the auxiliary variable. Meanwhile, the Rao–Blackwell method is another commonly used procedure to improve estimation efficiency. Various improved ratio estimators under adaptive cluster sampling (ACS) that make use of the auxiliary information together with the Rao–Blackwellized univariate estimators have been proposed in past research studies. In this article, the variances and the associated variance estimators of these improved ratio estimators are proposed for a thorough framework of statistical inference under ACS. Performance of the proposed variance estimators is evaluated in terms of the absolute relative percentage bias and the empirical mean-squared error. As expected, results show that both the absolute relative percentage bias and the empirical mean-squared error decrease as the initial sample size increases for all the variance estimators. To evaluate the confidence intervals based on these variance estimators and the finite-population Central Limit Theorem, the coverage rate and the interval width are used. These confidence intervals suffer a disadvantage similar to that of the conventional ratio estimator. Hence, alternative confidence intervals based on a certain type of adjusted variance estimators are constructed and assessed in this article.     

A new design for sampling with adaptive sample plots

Adaptive cluster sampling (ACS) is a sampling technique for sampling rare and geographically clustered populations. Aiming to enhance the practicability of ACS while maintaining some of its major characteristics, an adaptive sample plot design is introduced in this study which facilitates field work compared to “standard” ACS. The plot design is based on a conditional plot expansion: a larger plot (by a pre-defined plot size factor) is installed at a sample point instead of the smaller initial plot if a pre-defined condition is fulfilled. This study provides insight to the statistical performance of the proposed adaptive plot design. A design-unbiased estimator is presented and used on six artificial and one real tree position maps to estimate density (number of objects per ha). The performance in terms of coefficient of variation is compared to the non-adaptive alternative without a conditional expansion of plot size. The adaptive plot design was superior in all cases but the improvement depends on (1) the structure of the sampled population, (2) the plot size factor and (3) the critical value (the minimum number of objects triggering an expansion). For some spatial arrangements the improvement is relatively small. The adaptive design may be particularly attractive for sampling in rare and compactly clustered populations with an appropriately chosen plot size factor.     

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.     

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.     

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.     

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.     

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.     

A robust weighted EM algorithm for use-availability data

Wildlife sampling for habitat selection often combines a random background sample with a random sample of used sites, because the background sample could contain too few used sites to be informative for rare species. This approach is referred to as use-availability sampling. Two variants are considered where there is: (1) a random background sample including used and unused sites augmented with a sample of used sites, and (2) a sample of used sites augmented with a contaminated background sample, i.e. use is not recorded. A weighted estimator first proposed by Manski and Lerman (Econometrica 45(8):1977?C1988, 1977) forms the basis for our suggested approach. The weighted estimator has been shown to perform better than the usual unweighted approach with uncontaminated data and mis-specified logit models (Xie and Manski in Sociol Methods Res 17(3):283?C302, 1989). A weighted EM algorithm is developed for use with contaminated background data. We show that the weighted estimator continues to perform well with contaminated data and maintains its robustness to model mis-specification. The weighted estimator has not been previously used for use-availability sampling due to reliance on the assumption that only the intercept is biased, which is valid for a correct logit model. We show that adjusting the intercept may not eliminate the bias with an incorrect logit model. In this case, the weighted estimator is a relatively simple and effective alternative.     

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.