Hansen-Hurwitz Estimator Research Articles

Optimum second call imputation in PPS sampling.

The current study deals with imputation of item non-response in probability proportional to size (PPS) sampling. A new imputation procedure is proposed by using the known co-variance between the study variable and the auxiliary variable in the case of quantitative sensitive study variable by considering the non-response in a randomization mechanism on the second call. An empirical study is conducted at the optimum values of kog and nog for the relative comparisons of ratio, difference, and proposed estimators, respectively, with the Hansen-Hurwitz estimator.

Optimal subsampling for linear quantile regression models

AbstractSubsampling techniques are efficient methods for handling big data. Quite a few optimal sampling methods have been developed for parametric models in which the loss functions are differentiable with respect to parameters. However, they do not apply to quantile regression (QR) models as the involved check function is not differentiable. To circumvent the non‐differentiability problem, we consider directly estimating the linear QR coefficient by minimizing the Hansen–Hurwitz estimator of the usual loss function for QR. We establish the asymptotic normality of the resulting estimator under a generic sampling method, and then develop optimal subsampling methods for linear QR. In particular, we propose a one‐stage subsampling method, which depends only on the lengths of covariates, and a two‐stage subsampling method, which is a combination of the one‐stage sampling and the ideal optimal subsampling methods. Our simulation and real data based simulation studies show that the two recommended sampling methods always outperform simple random sampling in terms of mean square error, whether the linear QR model is valid or not.

Exponential type estimator for the population mean in the presence of non-response

In this article, we adapt the estimator based on the exponential function for the estimation of the population mean in the presence of non-response on both the study and the auxiliary variables. The expressions for the Bias and Mean Square Error (MSE) are derived to the first degree of approximation and theoretical comparisons are made with existing estimators in literature. We demonstrate that the adapted estimator is more efficient than compared estimators, such as classical ratio and regression estimators, Hansen-Hurwitz unbiased estimator, and the estimator of Singh et al. (2009) under the obtained conditions. Furthermore, these theoretical results are supported numerically by providing an empirical study on seven different data sets presenting the efficiency of the adapted estimator.

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.

Inferring the Population Mean with Second-Order Information in Online Social Networks

With the increasing use of online social networking platforms, online surveys are widely used in many fields, e.g., public health, business and sociology, to collect samples and to infer the population characteristics through self-reported data of respondents. Although the online surveys can protect the privacy of respondents, self-reporting is challenged by a low response rate and unreliable answers when the survey contains sensitive questions, such as drug use, sexual behaviors, abortion or criminal activity. To overcome this limitation, this paper develops an approach that collects the second-order information of the respondents, i.e., asking them about the characteristics of their friends, instead of asking the respondents’ own characteristics directly. Then, we generate the inference about the population variable with the Hansen-Hurwitz estimator for the two classic sampling strategies (simple random sampling or random walk-based sampling). The method is evaluated by simulations on both artificial and real-world networks. Results show that the method is able to generate population estimates with high accuracy without knowing the respondents’ own characteristics, and the biases of estimates under various settings are relatively small and are within acceptable limits. The new method offers an alternative way for implementing surveys online and is expected to be able to collect more reliable data with improved population inference on sensitive variables.

Adaptive cluster sampling for negatively correlated data

Adaptive cluster sampling is a design specifically developed for rare and clustered populations. Using this sampling design, we consider the case when an auxiliary variable is available in addition to the variable of interest. The use of auxiliary information has been shown to improve the efficiency of estimators although this results in asymptotically design‐unbiased estimators. Consider wildlife population in a protected area. Its distribution and abundance can partly be influenced by such factors as disease and pollution where the presence of wildlife diseases or higher environmental pollution decreases population totals and the distribution of wildlife. This paper proposes two product estimators and their associated variance estimators for the adaptive cluster sampling design to be used when the study and auxiliary variables are negatively correlated. The exact expression of the bias together with the mean square error to the first degree of approximation has been obtained. We derived the conditions under which the proposed estimators provided a more accurate estimation than the Horvitz–Thompson and Hansen–Hurwitz estimators with adaptive cluster sampling and the product estimator with simple random sampling. A simulation study was carried out to show the performance of the proposed estimators. Moreover, theoretical findings were supported by a numerical example using real data. Copyright © 2016 John Wiley & Sons, Ltd.

Methods for Estimating Private Forest Ownership Statistics: Revised Methods for the USDA Forest Service's National Woodland Owner Survey

The USDA Forest Service's National Woodland Owner Survey (NWOS) is conducted to better understand the attitudes and behaviors of private forest ownerships, which control more than half of US forestland. Inferences about the populations of interest should be based on theoretically sound estimation procedures. A recent review of the procedures disclosed an error in the application of estimation used in the NWOS. These issues are addressed and resolved in this article. The NWOS is administered to ownerships that contain a sample point determined to be forested and privately owned. The random selection of the sample points results in selection of ownerships with probability proportional to area of forestland owned. Inferences from survey results, including numbers of ownerships, area of forestland and associated variances, can be made using appropriate techniques, such as the Hansen-Hurwitz Estimator.

A Horvitz-Thompson-type estimator of species richness

A Horvitz–Thompson-type estimator of species richness for plot (cluster) sampling is constructed by considering species sampling as sampling with an unequal probability. Inclusion probabilities are estimated from sample-based estimates of relative species incidence. Bias is addressed by adding, to each observed species, the expected number of unseen species with the same relative incidence. A Hansen–Hurwitz estimator of variance is adopted and augmented by the anticipated variance from sample-based inclusion probabilities and the number of observed species. In Monte Carlo simulation of simple random plot (cluster) sampling from 11 large finite populations of forest trees and three sample sizes, the proposed estimator achieved the best overall ranking in terms of relative root mean square error efficiency when compared to 12 alternative estimators. The proposed estimator ranked third in terms of bias. The augmented Hansen–Hurwitz estimator of variance was liberal (median −13%). No richness estimator was uniformly best across populations and sample sizes. Across all settings, the performance of the best four estimators was similar, both in terms of bias and relative root mean square error efficiency. Copyright © 2011 John Wiley & Sons, Ltd.

Improving the Hansen­Hurwitz Estimator in PPSWR Sampling

An attempt has been made to improve upon the Hansen-Hurwitz (1943) estimator based on PPSWR sampling scheme through Rao-Blackwellisation. In order to derive the sampling variance of the improved estimator obtained by Rao-Blackwellisation, it is of interest to derive the probability distribution of the number of distinct units in the sample drawn according to PPSWR sampling scheme, as the improved estimator is based solely on the distinct units in the sample. It has been possible to write down the exact distribution of the number of distinct units in the sample drawn by PPSWR sampling scheme in a closed form. Assuming a super-population model, the model-expected design-variance of the improved estimator is worked out and it is compared with that of the Hansen-Hurwitz estimator under the same super-population model. The percentage gain in efficiency in using the improved estimator over the Hansen-Hurwitz estimator is worked out for certain selected values of the model parameters. AMS (2000) Subject Classification : 62D05.

On The Use Of Network Sampling In Diabetic Surveys

Diabetic patients' surveys have to deal with the lack of proper formal sampling frames. For survey researches at the community level, often some partial sampling frames such as medical centres or households to which a person is linked are available. These frames can be used to draw a network sample. At a selected household, the adult occupants are asked to report on the occurrence of the characteristics not only in them but also in their siblings. Using this network sampling design, the total number of people with diabetics can be estimated with lower variance than conventional procedures. The design is illustrated by an analysis of the network data of Nafiu (2007) in a survey to estimate the population of diabetic patients in Niger State, Nigeria. Two estimators: Hansen-Hurwitz estimator and Horvitz-Thompson estimator were considered; and the results were obtained using a program written in Microsoft Visual C++ programming language. Keywords: Graph, Sampling Frame, Households, Hansen-Hurwitz estimator and Horvitz-Thompson estimatorJORIND Vol. 5 (2) 2007: pp. 2-2

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 Hajek 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.

A comparison of conditional Poisson sampling versus unequal probability sampling with replacement

The variance of the Horvitz–Thompson estimator for a fixed size Conditional Poisson sampling scheme without replacement and with unequal inclusion probabilities is compared to the variance of the Hansen–Hurwitz estimator for a sampling scheme with replacement. We show, using a theorem by Gabler, that the sampling design without replacement is more efficient than the sampling design with replacement.

Inventory of sparse forest populations using adaptive cluster sampling

In many studies, adaptive cluster sampling (ACS) proved to be a powerful tool for assessing rare clustered populations that are difficult to estimate by means of conventional sampling methods. During 2002 and 2003, severe drought-caused damage was observed in the park forests of the City of Helsinki, Finland, especially in barren site pine and spruce stands. The aim of the present study was to examine sampling and measurement methods for assessing drought damage by analysing the effectiveness of ACS compared with simple random sampling (SRS). Horvitz-Thompson and Hansen-Hurwitz estimators of the ACS method were used for estimating the population mean and variance of the variable of interest. ACS was considerably more effective than SRS in assessing rare clustered populations such as those resulting from drought damage. The variances in the ACS methods were significantly smaller and the inventory efficiency in the field better than in SRS.

A Simple Variance Estimator for Unequal Probability Sampling without Replacement

Survey sampling textbooks often refer to the Sen–Yates–Grundy variance estimator for use with without-replacement unequal probability designs. This estimator is rarely implemented because of the complexity of determining joint inclusion probabilities. In practice, the variance is usually estimated by simpler variance estimators such as the Hansen–Hurwitz with replacement variance estimator; which often leads to overestimation of the variance for large sampling fractions that are common in business surveys. We will consider an alternative estimator: the Hájek (1964) variance estimator that depends on the first-order inclusion probabilities only and is usually more accurate than the Hansen–Hurwitz estimator. We review this estimator and show its practical value. We propose a simple alternative expression; which is as simple as the Hansen–Hurwitz estimator. We also show how the Hájek estimator can be easily implemented with standard statistical packages.

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.

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.

Optimal cluster selection probabilities to estimate the finite population distribution function under PPS cluster sampling

The estimation of the finite population distribution function under several sampling strategies based on a PPS cluster sampling, i.e., with cluster selection probabilities proportional to size, is studied. For the estimation of population means and totals, it is well-known that this type of strategies gives good results if the cluster selection probabilities are proportional to the total of the variable under study or to a related auxiliary variable over the cluster. It is proved that, for the estimation of the distribution function using cluster sampling, this solution is not good in general and, under an appropriate criteria, the optimal cluster selection probabilities that minimize the variance of the estimation, is obtained. This methodology is applied to two classical PPS sampling strategies: sampling with replacement, with the Hansen-Hurwitz estimator, and random groups sampling with the Rao-Hartley-Cochran estimator. Finally a small simulation to compare the efficiency of this approach with other methods is presented.

Regression analysis and random sampling

Abstract Consider a linear regression model. We wish to estimate a regression parameter or predict the sum of all N values of an endogeneous variable based on a sample of n (n observations. If the variances of the residual variables are known up to a common factor the sample may be selected purposively. In case of incomplete knowledge random selection is preferable and an optimal random design exists. To demonstrate this we consider a model with only one exogeneous variable, generalizations being obvious. We interpret the situation as a linear game and derive an equilibrium point determining an optimal estimator and an optimal sampling design. Under weak additional conditions we obtain the Hansen–Hurwitz estimator and a design with inclusion probabilities proportionate to the values of the exogeneous variable.

Treatment of non-response in successive sampling

Estimators of the population mean have been presented for sampling on two successive occasions when there is non-response in the sample. These estimators have been derived using the Hansen and Hurwitz (1946) procedure for treating non-response. The estimators were found to be more efficient than the Hansen-Hurwitz estimator.

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 \(y > 0\), one can expect some Rao-Blackwell improvement in the HH estimator but not in the HT estimator. Two examples are given.