Horvitz-Thompson Estimator Research Articles

Generalized ratio-type and ratio-exponential-type estimators for population mean under modified Horvitz-Thompson estimator in adaptive cluster sampling

ABSTRACTIn the present article, we propose the generalized ratio-type and generalized ratio-exponential-type estimators for population mean in adaptive cluster sampling (ACS) under modified Horvitz-Thompson estimator. The proposed estimators utilize the auxiliary information in combination of conventional measures (coefficient of skewness, coefficient of variation, correlation coefficient, covariance, coefficient of kurtosis) and robust measures (tri-mean, Hodges-Lehmann, mid-range) to increase the efficiency of the estimators. Properties of the proposed estimators are discussed using the first order of approximation. The simulation study is conducted to evaluate the performances of the estimators. The results reveal that the proposed estimators are more efficient than competing estimators for population mean in ACS under both modified Hansen-Hurwitz and Horvitz-Thompson estimators.

Impact of non-farm income on welfare in rural Bangladesh: Multilevel mixed-effects regression approach

This study uses a two-stage econometric framework based on multilevel mixed-effects linear regression model with nationally representative Household Income Expenditure Survey (HIES) data in 2010 to investigate the effects of non-farm income on asset ownership of rural Bangladeshi households. Result reveals that non-farm income has a significant positive effect on household’s asset ownership. This study also uses Horvitz-Thompson (HT) estimator of the Foster, Greer and Thorbecke (FGT) indices and Gini, Theil’s and Atkinson income inequality measures to examine the impact of non-farm income on poverty and income distribution. The poverty measures reveal that the inclusion of non-farm income reduces the level, depth and severity of poverty as well as poverty risk at division level and in rural Bangladesh. However, non-farm income increased income inequality among rural households. As the agriculture of Bangladesh is subsistence and semi-commercial in nature, it is essential to encourage non-farm income activities to accelerate well-being among rural agricultural households.

Design-based estimation of mark variograms in forest ecosystem surveys

Mark variograms are widely adopted in forest ecosystem studies for analyzing spatial interaction among trees. Inference on mark variograms can be performed under a model-dependent perspective (ergodic variograms) or under a deterministic perspective (non-ergodic variograms). A simple and workable definition of non-ergodic mark variogram is introduced based on distances between pairs of trees distinguished by distance classes. Design-based estimators of mark variogram values are proposed as the ratio of two Horvitz–Thompson estimators using replicated plots randomly located on the study area and making use of the inclusion probabilities of the pairs of trees. Jackknife estimators of their variances are considered. A simulation study is performed on a real forest stand to check the statistical performance of the proposed strategy.

Horvitz-Thompson estimator under partial information with an application to network degree distribution

We present an extension of the Horvitz-Thompson estimator for estimating the total number of sets of a given size within a population. We study the limitations of the Horvitz-Thompson under partial information where we have a sample of population elements rather than of sets. The developed estimator is the chained Horvitz-Thompson. We apply this estimator to the estimate the network degree distribution to be used under probability sampling designs, in particular, induced subgraph and incident subgraph. Finally, we present Monte Carlo simulations to assess the accuracy of the proposed estimator.

A Monte Carlo appraisal of tree abundance and stand basal area estimation in forest inventories based on terrestrial laser scanning

Non-detection of trees is an important issue when using single-scan TLS in forest inventories. A hybrid inference approach is adopted. Quoting from distance sampling, a detection function is assumed, so that the inclusion probability of each tree included within each plot can be determined. A simulation study is performed to compare the TLS-based estimators corrected and uncorrected for non-detection with the Horvitz–Thompson estimator based on conventional plot sampling, in which all the trees within plots are recorded. Results show that single-scan TLS provides more efficient estimators with respect to those provided by the conventional plot sampling in the case of low-density forests when no distance sampling correction is performed. In low-density forests, uncorrected estimators lead to a small bias (1%–6%), increasing with plot size. Therefore, care must be taken in enlarging the plot radius too much. The bias increases in forests with clustered spatial structures and in dense forests, where the bias levels (30%–50%) deteriorate the performance of uncorrected estimators. Even if the bias-corrected estimators prove to be effective in reducing the bias (below 15%), these reductions are not sufficient to outperform conventional plot sampling. Therefore, there is no convenience in using TLS-based estimation in high-density forests.

Developing an area-based sampling system in the urban district of Bobo Dioulasso, Burkina Faso

Nous avons propose une approche simple pour construire un systeme d’echantillonnage dans des situations a revenus limites. L’etude a ete menee a Bobo-Dioulasso, Burkina Faso. La methode a consiste a selectionner des lieux-dits bien connus dans la zone d’etude. Des informations cles etaient collectees sur ces lieux et la densite de la population etait estimee. Nous avons ensuite predit la densite de la population sur tout le territoire par krigeage, en controlant la variance. Nous pouvons ensuite proposer, pour un point designe dans la zone d’etude, un groupe virtuel avec une densite de population estimee. Nous avons ensuite decrit une methode de sondage a deux degres a probabilites egales et l’estimateur sans biais de Horvitz-Thompson associe.

Quantifying the trade-off between cost and precision in estimating area of forest loss and degradation using probability sampling in Guyana

Guyana is a high forest cover, low deforestation country. Since 2011–2014 the Guyana Forestry Commission (GFC) has used visual interpretation of 5 m resolution RapidEye imagery to map forest loss and nearby degradation for the entire country. According to the GFC produced national map, 58% of all forest loss events cover less than 1 ha, so forest loss is very rare and occurs at a scale that can be difficult to detect and monitor. Nearly all (~97%) of the forest loss and nearby degradation is due to small-scale mining and its associated infrastructure. For any country wanting to accurately map and monitor forest loss, sample-based area estimation can be considerably less expensive than wall-to-wall mapping. To quantify the tradeoff between precision and cost, we evaluated the standard errors of area estimators for several sample-based strategies using Guyana as a case study. We partitioned Guyana into 374 blocks, with each block 24 km × 24 km to align with the image tiles provided by RapidEye's 3A product. The area of forest loss and area of degradation for each of the 374 blocks were determined from the GFC map to create the population data used to evaluate the precision of different sample-based strategies. We compared the standard errors of estimators of area of forest loss and area of degradation obtained from simple random, stratified random and systematic sampling as applied to these population data. To construct strata for the stratified design, we evaluated two forest loss maps produced from Landsat data, a 30 m global forest loss product and a 30 m national forest loss map produced specifically for Guyana. For both of these maps, several options for defining stratum boundaries and allocating sample size to strata were evaluated. All stratified design options reduced the standard error of the area estimators relative to simple random sampling. The Dalenius-Hodges and Jenks methods for choosing stratum boundaries yielded greater improvements in precision than the Equal Area method. For this Guyana case study, optimal and equal allocation of sample size to strata led to substantially better precision than proportional allocation. Only small improvements in precision were attained by increasing the number of strata from three to five. For simple random sampling, incorporating the global and national forest loss map information in a regression estimator reduced standard errors of estimated area of forest loss and area of degradation relative to the standard error of the Horvitz-Thompson estimator that does not incorporate the forest loss map information. However, the best performing stratified sampling design options had better precision than the regression estimator. RapidEye data cost 800 USD per ‘3A’ tile so the annual imagery cost for mapping Guyana would be 274,000 USD. For an investment of 75,200 USD (the cost of 94 RapidEye tiles, the largest sample size evaluated), the best performing stratified random sample achieved a relative standard error (100% times the standard error divided by area) of 7% for estimating area of forest loss and 6% for estimating area of degradation. This study demonstrates the utility of Landsat-based forest loss maps to provide effective strata to improve precision of area estimators produced from a sample of RapidEye imagery in a country for which forest loss and degradation are very rare features of the landscape.

Geographically weighted regression model-assisted estimation in survey sampling

ABSTRACTA geographically weighted regression model-assisted method is proposed to estimate the finite population totals using survey data with the aid of spatial and other auxiliary information. The local linear method is used to the estimation of geographically weighted regression model. Our proposed GWR-assisted (geographically weighted regression model-assisted) estimators are more efficient than the well-known Horvitz–Thompson estimators. These estimators are calibrated and asymptotically design-unbiased. Some theoretical results have been established for GWR-assisted estimators. Simulation experiments show that the GWR-assisted estimators are more efficient than the LM-assisted (linear regression model-assisted) estimators and NP-assisted (nonparametric regression model-assisted) estimators. Finally, the Boston housing data are used in the simulation study to demonstrate the importance of location information in spatial modelling.

Register data in sample allocations for small-area estimation

ABSTRACTThe inadequate control of sample sizes in surveys using stratified sampling and area estimation may occur when the overall sample size is small or auxiliary information is insufficiently used. Very small sample sizes are possible for some areas. The proposed allocation based on multi-objective optimization uses a small-area model and estimation method and semi-collected empirical data annually collected empirical data. The assessment of its performance at the area and at the population levels is based on design-based sample simulations. Five previously developed allocations serve as references. The model-based estimator is more accurate than the design-based Horvitz–Thompson estimator and the model-assisted regression estimator. Two trade-off issues are between accuracy and bias and between the area- and the population-level qualities of estimates. If the survey uses model-based estimation, the sampling design should incorporate the underlying model and the estimation method.

Adaptive Design Strategies for Nonresponse Follow-Up in Economic Surveys

Abstract The U.S. Census Bureau is investigating nonrespondent subsampling strategies for use in the 2017 Economic Census. In previous research, we developed an optimized allocation procedure for subsampling nonrespondents that selects larger systematic samples in domains with lower initial response. This article expands on our previous research by exploring improvements to the optimal allocation method; we investigate refinements to the previous procedure that incorporate measures of respondent balance with respect to the original sample. The revised allocation procedures have simultaneous objectives of allocating high proportions of sample in domains that indicate potential nonresponse bias and of equalizing response rates across domains. We examine the effects of the alternative allocation approaches on Horvitz-Thompson estimates via a simulation study using data from the 2014 Annual Survey of Manufactures.

An 18-Year Mark–Recapture Study of Wood Turtles (Glyptemys insculpta) in Michigan

Wood Turtles (Glyptemys insculpta) occupy forested streams at midlatitudes in eastern North America and are listed as endangered by the International Union for Conservation of Nature, but few populations have been rigorously studied. We studied a population of Wood Turtles in Michigan for 18 yr, individually marking 260 different turtles (146 females, 88 males, and 26 unsexed juveniles), including 118 turtles that we followed for one or more years using radiotelemetry. We analyzed our encounter data using a Cormack–Jolly–Seber model in Program MARK; and we estimated total population size using a Bayesian integrated population model that combined Horvitz–Thompson estimates of annual population size, mark–recapture estimates of annual survival, and derived estimates of annual recruitment. Annual adult survival was 0.970 ± 0.016 SD and annual recruitment to age 15 (mean age of first capture) was 0.058 ± 0.019 SD. Over the 18-yr study, estimated population size grew from 770 (95% CI 631–928) to 1,196 (95% CI 977–1,444) individuals.

Spatially Balanced Sampling of Continuous Populations

AbstractWhen sampling from a continuous population (or distribution), we often want a rather small sample due to some cost attached to processing the sample or to collecting information in the field. Moreover, a probability sample that allows for design‐based statistical inference is often desired. Given these requirements, we want to reduce the sampling variance of the Horvitz–Thompson estimator as much as possible. To achieve this, we introduce different approaches to using the local pivotal method for selecting well‐spread samples from multidimensional continuous populations. The results of a simulation study clearly indicate that we succeed in selecting spatially balanced samples and improve the efficiency of the Horvitz–Thompson estimator.

Factors differentiating the level of crop insurance at Polish farms

PurposeThe purpose of this paper is to identify the factors that determine demand for crop insurance in Poland.Design/methodology/approachTo examine the determinants of decisions regarding crop insurance, the authors used logistic regression. The base source of data for the analysis was the 2013 FADN sample. The scale of yield losses, the indemnities received and the Arrow-Pratt risk aversion coefficient were examined in a representative sample of farms in consecutive years in the period 2004-2013.FindingsLosses are the major determinants of crop insurance uptake. Additionally, it was observed that the economic determinants are in line with the expected utility theory, while contrary to expectations, farmer’s characteristics such as education level, age or even risk aversion did not prove to have any influence on crop insurance uptake.Research limitations/implicationsThe FADN sample is representative as regards the type of farming, economic size of farm and location of the farm. Every farm in the sample represents a specific number of similar farms in the population. However, it must be emphasised that the representativeness of the sample with respect to other determinants, e.g., yield losses in previous years, using crop insurance or the farmers’ age and education has not been verified due to lack of data characterizing the general population with regard to these factors.Practical implicationsIt could be argued that the system of crop insurance subsidies should be targeted to encourage the farmers who previously had not used insurance to join the system.Originality/valueThe paper presents the analysis of crop insurance uptake in a country with a strongly polarised agriculture. The Polish farm sector consists of 1.4 million farms with sizes ranging from 1 ha to over a few thousands hectares. The research is based on a data set of 5,202 farms which contains data from ten years (2004-2013). The novelty of the methodological approach is that it includes information on the number of farms represented by every farm in the FADN sample in the Horvitz-Thompson estimator in order to achieve results which are valid for the general population of Polish farms.

Calibrated Bayesian shrinkage of finite population totals in survey sampling

In this article a Bayesian methodology (parametric and nonparametric) is proposed in order to estimate, by means of calibration, the population total from a sample with unequal inclusion probabilities. By means of some simulation studies, it was empirically determined that, through an adequate choice of an prior distribution with the proposed methodology, unbiased estimators are obtained. In addition to this, with an appropriate sample size, a smaller variance and confidence intervals with higher levels and minor length were obtained in comparison with the intervals induced by classic estimators (Horvitz-Thompson and calibration estimators). Finally, the implementation of the methodology to estimate the income is exemplified using real data from a labor force survey in Colombia.

Using Social Network Information for Survey Estimation

Abstract Model-based and model-assisted methods of survey estimation aim to improve the precision of estimators of the population total or mean relative to methods based on the nonparametric Horvitz-Thompson estimator. These methods often use a linear regression model defined in terms of auxiliary variables whose values are assumed known for all population units. Information on networks represents another form of auxiliary information that might increase the precision of these estimators, particularly if it is reasonable to assume that networked population units have similar values of the survey variable. Linear models that use networks as a source of auxiliary information include autocorrelation, disturbance, and contextual models. In this article we focus on social networks, and investigate how much of the population structure of the network needs to be known for estimation methods based on these models to be useful. In particular, we use simulation to compare the performance of the best linear unbiased predictor under a model that ignores the network with model-based estimators that incorporate network information. Our results show that incorporating network information via a contextual model seems to be the most appropriate approach. We also show that one does not need to know the full population network, but that knowledge of the partial network linking the sampled population units to the non-sampled population units is necessary. Finally, we also provide an estimator for the mean-squared error to make an informed decision about using the contextual information, as well as the results showing that this adaptive strategy leads to higher precision.

K-step adaptive cluster sampling with Horvitz–Thompson estimator

Adaptive cluster sampling (ACS) has been widely used for data collection of environment and natural resources. However, the randomness of its final sample size often impedes the use of this method. To control the final sample sizes, in this study, a [Formula: see text]-step ACS based on Horvitz–Thompson (HT) estimator was developed and an unbiased estimator was derived. The [Formula: see text]-step ACS-HT was assessed first using a simulated example and then using a real survey for numbers of plants for three species that were characterized by clustered and patchily spatial distributions. The effectiveness of this sampling design method was assessed in comparison with ACS Hansen–Hurwitz (ACS-HH) and ACS-HT estimators, and [Formula: see text]-step ACS-HT estimator. The effectiveness of using different [Formula: see text]-step sizes was also compared. The results showed that [Formula: see text]-step ACS-HT estimator was most effective and ACS-HH was the least. Moreover, stable sample mean and variance estimates could be obtained after a certain number of steps, but depending on plant species. [Formula: see text]-step ACS without replacement was slightly more effective than that with replacement. In [Formula: see text]-step ACS, the variance estimate of one-step ACS is much larger than other [Formula: see text]-step ACS ([Formula: see text]), but it is smaller than ACS. This implies that [Formula: see text]-step ACS is more effective than traditional ACS, besides, the final sample size can be controlled easily in population with big clusters.

A proposal for a two-step sampling design to oversample units responding to prescribed characteristics

We introduce a novel method to extract a sample from a finite population where units with desired characteristics are over-represented. The approach is both sequential and adaptive and allows, via suitable compositions of predictive and objective functions, to target specific subsets of the population. We consider the problem of estimation and conjecture the validity of a modified Horvitz–Thompson estimator capable to account for the imbalance induced by the targeting procedure. After discussing how to apply the method to the sampling of geographically distributed units, we investigate its potential via simulations.

Time series sightability modeling of animal populations.

Logistic regression models—or “sightability models”—fit to detection/non-detection data from marked individuals are often used to adjust for visibility bias in later detection-only surveys, with population abundance estimated using a modified Horvitz-Thompson (mHT) estimator. More recently, a model-based alternative for analyzing combined detection/non-detection and detection-only data was developed. This approach seemed promising, since it resulted in similar estimates as the mHT when applied to data from moose (Alces alces) surveys in Minnesota. More importantly, it provided a framework for developing flexible models for analyzing multiyear detection-only survey data in combination with detection/non-detection data. During initial attempts to extend the model-based approach to multiple years of detection-only data, we found that estimates of detection probabilities and population abundance were sensitive to the amount of detection-only data included in the combined (detection/non-detection and detection-only) analysis. Subsequently, we developed a robust hierarchical modeling approach where sightability model parameters are informed only by the detection/non-detection data, and we used this approach to fit a fixed-effects model (FE model) with year-specific parameters and a temporally-smoothed model (TS model) that shares information across years via random effects and a temporal spline. The abundance estimates from the TS model were more precise, with decreased interannual variability relative to the FE model and mHT abundance estimates, illustrating the potential benefits from model-based approaches that allow information to be shared across years.

On Local Linear Regression Estimation of Finite Population Totals in Model Based Surveys

In this paper, nonparametric regression is employed which provides an estimation of unknown finite population totals. A robust estimator of finite population totals in model based inference is constructed using the procedure of local linear regression. In particular, robustness properties of the proposed estimator are derived and a brief comparison between the performances of the derived estimator and some existing estimators is made in terms of bias, MSE and relative efficiency. Results indicate that the local linear regression estimator is more efficient and performing better than the Horvitz-Thompson and Dorfman estimators, regardless of whether the model is specified or mispecified. The local linear regression estimator also outperforms the linear regression estimator in all the populations except when the population is linear. The confidence intervals generated by the model based local linear regression method are much tighter than those generated by the design based Horvitz-Thompson method. Generally the model based approach outperforms the design based approach regardless of whether the underlying model is correctly specified or not but that effect decreases as the model variance increases.

The continuous population approach to forest inventories and use of information in the design

An extended theoretical framework for the continuous population approach to forest inventories is derived. Here, we treat a simultaneous selection of sample points with any prescribed sampling intensity over a continuous population. Different ways to use available auxiliary information, for example, from remote sensing, by selection of approximately balanced or spatially balanced samples are considered. A large data set of spatially continuous individual tree‐level data is used to demonstrate the potential of these theoretical approaches. This study shows new ways to integrate remote sensing information in designs for forest inventory applications, which can significantly reduce the variance of the Horvitz–Thompson estimator for target variables related to the auxiliary information.