Estimator Of Population Total Research Articles

Calibration Estimation of Population Total in Two-Stage Sampling Design under unavailability of Population Level Auxiliary Information for the selected PSUs

Calibration approach is a popular technique in sample surveys which incorporates auxiliary information in the estimation process assuming that population aggregates of auxiliary variable are available. Many often under two-stage sampling design, such population aggregates of auxiliary variable, i.e., population mean or total are unavailable and under such situations, estimation of population total has been limited to the use of two phase sampling. Therefore, in the present study, efficient estimator of population total is developed under two-stage sampling design when population aggregates of auxiliary variable are unavailable for the selected psu’s. The calibrated estimator is developed using the information on known population aggregates of additional auxiliary variable which is less linearly related to the study variable through two step calibration approach. The approximate variance and variance estimator of the proposed calibrated estimator has also been developed. Empirical evaluations using both real and simulated data shows the superior performance of the developed calibrated estimator in comparison to the existing estimators.

Rescaling bootstrap variance estimation technique under dual frame surveys with unknown domain sizes

Dual frame (DF) surveys are a special case of multiple frame (MF) surveys considering two frames covering the entire population. Dual frame surveys are applicable in those situations, where, one frame may cover the entire population but is very expensive to sample; so an alternate frame may be available that does not cover the entire population but is easily available. Unbiased variance estimation in dual frame surveys can be difficult and complicated than corresponding estimators under single frame surveys. Again, the variance of dual frame estimator involves population variances of the individual domains which are generally unknown. Due to this reason, obtaining an unbiased estimate of the variance of the dual frame estimator is quite complex in the case of dual frame surveys. In this article, we propose a Post-stratified Rescaling Bootstrap with Unknown Domain size (PstRBUD) method for variance estimation of the dual frame estimator of population total. The proposed rescaled bootstrap method was compared to that of standard bootstrap methods in simulation analysis. The proposed PstRBUD method provides an unbiased estimation of the variance of the dual frame estimator of population total, according to simulation results.

Improving Estimates of Natural Resources Using Model-Based Estimators: Impacts of Sample Design, Estimation Technique, and Strengths of Association

Natural resource managers need accurate depictions of existing resources to make informed decisions. The classical approach to describing resources for a given area in a quantitative manner uses probabilistic sampling and design-based inference to estimate population parameters. While probabilistic designs are accepted as being necessary for design-based inference, many recent studies have adopted non-probabilistic designs that do not include elements of random selection or balance and have relied on models to justify inferences. While common, model-based inference alone assumes that a given model accurately depicts the relationship between response and predictors across all populations. Within complex systems, this assumption can be difficult to justify. Alternatively, models can be trained to a given population by adopting design-based principles such as balance and spread. Through simulation, we compare estimates of population totals and pixel-level values using linear and nonlinear model-based estimators for multiple sample designs that balance and spread sample units. The findings indicate that model-based estimators derived from samples spread and balanced across predictor variable space reduce the variability of population and unit-level estimators. Moreover, if samples achieve approximate balance over feature space, then model-based estimates of population totals approached simple expansion-based estimates of totals. Finally, in all comparisons made, improvements in estimation were achieved using model-based estimation over design-based estimation alone. Our simulations suggest that samples drawn from a probabilistic design, that are spread and balanced across predictor variable space, improve estimation accuracy.

Benchmarked Estimators for a Small Area Mean Under a Onefold Nested Regression Model

SummaryIn this paper, we modify small area estimators, based on the unit‐level model, so that they add up to reliable higher‐level estimates of population totals. These modifications result in benchmarked small area estimators. We consider two benchmarking procedures. One is based on augmenting the unit‐level model with a suitable variable. The other one uses the calibrated weights of the direct estimators that are reliable at the higher levels. These weights are used in estimators that are based on the aggregation of the unit‐level model for each small area. The mean squared error estimators of the proposed benchmarked estimators are obtained by suitably modifying those associated with the corresponding non benchmarked estimators. The properties of the estimators are evaluated via simulation.

Finite Population Model-Assisted Estimation Using Combined Parametric and Nonparametric Regression Smoothers

This paper considers estimating finite population totals from complex sample surveys in the presence of auxiliary information. Model-assisted estimators which assume a working regression model relating the study variable with the auxiliary data are common in this context. Both parametric and nonparametric working models have been utilized individually in constructing several model-assisted estimators. Model-assisted estimators with parametric working models are known to be efficient when the assumed working model is correctly specified, while using nonparametric smoothers gives more robust estimates but requires relatively large sample sizes. In this paper, we consider the situation where the researcher has an idea of which parametric model can describe the relationship between the study variable and the auxiliary data, but this model may not be adequate in some areas of the data range. Using combined parametric and nonparametric regression smoothers for the working model, we introduce a new class of model-assisted estimators for finite population totals. The proposed estimators are shown to have the desirable asymptotic properties of traditional model-assisted estimators of population totals. The finite sample performance of the new estimators is studied via Monte Carlo simulations from both artificial and real populations. The empirical results suggest that our proposed estimators perform well relative to other model-based and model-assisted estimators as well as the customary Horvitz–Thompson estimator under different levels of misspecification in the working model. We also discuss the problem of variance estimation for the proposed estimators.

Comparison of Various Methods for Estimating Finite Population Total in Survey Sampling When Study Variable and Auxiliary Variable are Inversely Related

In the present paper, a model based calibration estimator of population total has been developed when study variable y and auxiliary variable x are inversely related. The relative performance of the proposed model based calibration estimator in comparison to model based estimator, the usual regression estimator and calibration based regression estimator have been examined by conducting a limited simulation study. In view of the results of the simulation study, it has been found that model based calibration estimator has outperformed the other estimators. However, calibration based regression estimator was found to be close to the model based calibration estimator.

Halton iterative partitioning: spatially balanced sampling via partitioning

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

Effects of uncertainty in upper-stem diameter information on tree volume estimates

Almost all relevant data in forestry databases arise from either field measurement or model prediction. In either case, these values have some amount of uncertainty that is often overlooked when doing analyses. In this study, the uncertainty associated with both measured and predicted data was quantified for upper-stem diameter at 5.27 m. This uncertainty was propagated through a tree taper model into predictions of individual-tree volume. The effects of uncertainty on individual-tree volume predictions and population estimates of total volume were assessed. Generally, when little or no systematic measurement deviation was present, less uncertainty was associated with field-measured diameters compared to model predictions. However, diameters predicted from a model were preferred when systematic deviations in field measurement exceeded approximately 0.2 cm. Comparisons of results obtained from an alternative taper model showed that more precise estimates of population totals might be obtained without upper-stem diameter information. Upper-stem diameter information increases the prediction accuracy of individual-tree volume, and thus, models using this information may be preferable in applications such as timber sales containing high-value trees. Due to the various factors that influence measurement and modeling uncertainty, foresters are encouraged to make similar evaluations in the context of their specific activities.

A Varying Coefficients Model For Estimating Finite Population Totals: A Hierarchical Bayesian Approach

In some finite sampling situations, there is a primary variable that is sampled, and there are measurements on covariates for the entire population. A Bayesian hierarchical model for estimating totals for finite populations is proposed. A nonparametric linear model is assumed to explain the relationship between the dependent variable of interest and covariates. The regression coefficients in the linear model are allowed to vary as a function of a subset of covariates nonparametrically based on B-splines. The generality of this approach makes it robust and applicable to data collected using a variety of sampling techniques, provided the sample is representative of the finite population. A simulation study is carried out to evaluate the performance of the proposed model for the estimation of the population total. Results indicate accurate estimation of population totals using the approach. The modeling approach is used to estimate the total production of avocado for a large group of groves in Mexico.

A balanced sampling and calibration

A balanced sampling design is a design in which Horvitz-Thompson estimators of population totals for a set of auxiliary variables equal the known totals of these variables. On the other hand, calibration is a technique where the modification of design weights occurs in such a way that the new weights, when applied to auxiliary variables, reproduce, i.e. estimate withouterror, the known totals for these variables. The general idea behind balanced sampling and calibration is thus the same — both techniques tend to reproduce known totals of the auxiliary variables. The purpose of the paper is to describe and compare both techniques, considering them as alternatives in achieving the same goal. More attention was devoted to balanced sampling. The algorithm for selecting a sample was illustrated with two numerical examples. The comparison between balanced sampling and calibration, as alternatives, indicates calibration, but the best strategy is to use both methods simultaneously.

Calibration Approach-based Product Estimator of Finite Population Total with Subsampling of Nonrespondents under Single and Two-phase Sampling

Hansen and Hurwitz (1946) technique–based estimator of population total is proposed using the calibration approach under the assumption that the auxiliary variable is negatively correlated with the study variable. The variance estimation is also considered. The two-phase sampling case is also explored. The theoretical results are demonstrated through empirical studies using both generated and real population data. The proposed estimator of population total outperforms the existing estimators in terms of the criteria of relative bias and relative root mean square error.

Calibration Approach-Based Regression-Type Estimator for Inverse Relationship Between Study and Auxiliary Variable

A regression-type estimator of population total is developed using the calibration approach under the assumption that the auxiliary variable is negatively correlated with the study variable. An estimator of the variance of the proposed estimator is developed. In addition, a higher order calibration approach has been described for the estimator of the variance of the proposed estimator. When the auxiliary information is not available on all the population units, then a two-phase sampling approach has been suggested. Theoretical results obtained are demonstrated through simulation studies and also with real data. Empirical results show that the proposed estimator outperforms the usual regression estimator in terms of the criteria of bias and mean square error.

Calibration-Based Product Estimator in Single- and Two-Phase Sampling

An estimator of population total is developed using the calibration approach under the assumption that the auxiliary variable is negatively correlated with the study variable. The proposed estimator outperforms the usual product estimator in terms of the criteria of bias and mean square error. An improved estimator of the variance of the proposed estimator is developed using the calibration approach a second time. The two-phase sampling case has also been dealt with. The theoretical results are demonstrated through a simulation study.

The effects of spatial population dataset choice on estimates of population at risk of disease

BackgroundThe spatial modeling of infectious disease distributions and dynamics is increasingly being undertaken for health services planning and disease control monitoring, implementation, and evaluation. Where risks are heterogeneous in space or dependent on person-to-person transmission, spatial data on human population distributions are required to estimate infectious disease risks, burdens, and dynamics. Several different modeled human population distribution datasets are available and widely used, but the disparities among them and the implications for enumerating disease burdens and populations at risk have not been considered systematically. Here, we quantify some of these effects using global estimates of populations at risk (PAR) of P. falciparum malaria as an example.MethodsThe recent construction of a global map of P. falciparum malaria endemicity enabled the testing of different gridded population datasets for providing estimates of PAR by endemicity class. The estimated population numbers within each class were calculated for each country using four different global gridded human population datasets: GRUMP (~1 km spatial resolution), LandScan (~1 km), UNEP Global Population Databases (~5 km), and GPW3 (~5 km). More detailed assessments of PAR variation and accuracy were conducted for three African countries where census data were available at a higher administrative-unit level than used by any of the four gridded population datasets.ResultsThe estimates of PAR based on the datasets varied by more than 10 million people for some countries, even accounting for the fact that estimates of population totals made by different agencies are used to correct national totals in these datasets and can vary by more than 5% for many low-income countries. In many cases, these variations in PAR estimates comprised more than 10% of the total national population. The detailed country-level assessments suggested that none of the datasets was consistently more accurate than the others in estimating PAR. The sizes of such differences among modeled human populations were related to variations in the methods, input resolution, and date of the census data underlying each dataset. Data quality varied from country to country within the spatial population datasets.ConclusionsDetailed, highly spatially resolved human population data are an essential resource for planning health service delivery for disease control, for the spatial modeling of epidemics, and for decision-making processes related to public health. However, our results highlight that for the low-income regions of the world where disease burden is greatest, existing datasets display substantial variations in estimated population distributions, resulting in uncertainty in disease assessments that utilize them. Increased efforts are required to gather contemporary and spatially detailed demographic data to reduce this uncertainty, particularly in Africa, and to develop population distribution modeling methods that match the rigor, sophistication, and ability to handle uncertainty of contemporary disease mapping and spread modeling. In the meantime, studies that utilize a particular spatial population dataset need to acknowledge the uncertainties inherent within them and consider how the methods and data that comprise each will affect conclusions.

A resampling variance estimator for the k nearest neighbours technique

Current estimators of variance for the k nearest neighbours (kNN) technique are designed for estimates of population totals. Their efficiency in small-area estimation problems can be poor. In this study, we propose a modified balanced repeated replication estimator of variance (BRR) of a kNN total that performs well in small-area estimation problems and under both simple random and cluster sampling. The BRR estimate of variance is the sum of variances and covariances of unit-level kNN estimates in the area of interest. In Monte Carlo simulations of simple random and cluster sampling from seven artificial populations with real and simulated forest inventory data, the agreement between averages of BRR estimates of variance and Monte Carlo sampling variances was good both for population and for small-area totals. The modified BRR estimator is currently limited to sample sizes no larger than 1984. An accurate approximation to the proposed BRR estimator allows significant savings in computing time.

When does an imperfect sampling frame produce more efficient estimators than a perfect frame?

The issue of when imperfect sampling frames can result in more efficient estimators of population totals than perfect frames is explored. Our analysis is based on an expression we call the difference score. We show how, when properly expanded it provides an illuminating basis for comparing a weighted estimator under an imperfect frame with that of a conventional estimator assuming the frame has been corrected. Specifically, the circumstances (i.e., population and frame characteristics) under which an imperfect frame results in estimates of population totals that are more precise than those from a perfect frame can in many cases be discerned by analytically examining the terms in the expansion of this difference score. In addition, a classification tree methodology was used to further explore circumstances under which imperfect frames result in more precise estimators. The results of this analytical study complement, strengthen, and in many cases explain those discovered in an earlier empirical investigation that lead to recommendations as to when to correct a frame or when to adjust for imperfection using a weighting methodology called the arc weight estimator.

A new approach to weighting and inference in sample surveys

The validity of design-based inference is not dependent on any model assumption. However, it is well known that estimators derived through design-based theory may be inefficient for the estimation of population totals when the design weights are weakly related to the variables of interest and have widely dispersed values. We propose estimators that have the potential to improve the efficiency of any estimator derived under the design-based theory. Our main focus is limited to the improvement of the Horvitz-Thompson estimator, but we also discuss the extension to calibration estimators. The new estimators are obtained by smoothing design or calibration weights using an appropriate model. Our approach to inference requires the modelling of only one variable, the weight, and it leads to a single set of smoothed weights in multipurpose surveys. This is to be contrasted with other model-based approaches, such as the prediction approach, in which it is necessary to postulate and validate a model for each variable of interest leading potentially to variable-specific sets of weights. Our proposed approach is first justified theoretically and then evaluated through a simulation study.

IPPS Sampling Plans Excluding Adjacent Units

The concept of inclusion probability proportional to size sampling plans excluding adjacent units separated by at most a distance of m (≥ 1) units {IPPSEA plans} is introduced. IPPSEA plans ensure that the first-order inclusion probabilities of units are proportional to size measures of the units, while the second-order inclusion probabilities are zero for pairs of adjacent units separated by a distance of m units or less. IPPSEA plans have been obtained by making use of binary, proper, and unequireplicated block designs and linear programing approach. The performance of IPPSEA plans using Horvitz–Thompson estimator of population total has been compared with existing sampling plans such as simple random sampling without replacement (SRSWOR), balanced sampling plans excluding adjacent units {BSA (m) plans}, probability proportional to size with replacement, Hartley and Rao's plan (1962), Rao et al.'s strategy (1962), and Sampford's IPPS plan (1967) using a real life population. Unbiased estimation of Horvitz–Thompson estimator of population total is not possible in these types of plans because some of the second-order inclusion probabilities are zero. To resolve this problem, one approximate variance estimation technique has been suggested.

Sampling Within Households in Household Surveys

SummaryThe number of people to select within selected households has significant consequences for the conduct and output of household surveys. The operational and data quality implications of this choice are carefully considered in many surveys, but the effect on statistical efficiency is not well understood. The usual approach is to select all people in each selected household, where operational and data quality concerns make this feasible. If not, one person is usually selected from each selected household. We find that this strategy is not always justified, and we develop intermediate designs between these two extremes. Current practices were developed when household survey field procedures needed to be simple and robust; however, more complex designs are now feasible owing to the increasing use of computer-assisted interviewing. We develop more flexible designs by optimizing survey cost, based on a simple cost model, subject to a required variance for an estimator of population total. The innovation lies in the fact that household sample sizes are small integers, which creates challenges in both design and estimation. The new methods are evaluated empirically by using census and health survey data, showing considerable improvement over existing methods in some cases.

Planning a sample survey with a many-to-many frame structure: an empirical investigation of the effects of frame imperfections on estimates of population totals

When faced with a sampling frame that has an imperfect correspondence with the target population, the survey researcher must decide whether to use the frame as is or to expend the resources to correct its imperfections. One important consideration is the effects of the frame's imperfections on the precision of the estimate. Using simulation techniques, this paper explores the effects of frame imperfections on the precision of an estimate of a population total in simple and stratified random sampling. Some factors studied were the amount of imperfection in the frame, the nature of the correspondence between the frame units and population elements, the population coefficient of variation, and the correlation of the weights used to correct for the frame's imperfections with the measure of interest. Recommendations for planning the survey design are presented based upon our findings.