An Extension of Sampford's Method for Unequal Probability Sampling

Abstract. Sampford's unequal probability sampling method is extended to the case that the inclusion probabilities do not sum to an integer. In this case, the sampling outcome is left open for exactly one randomly chosen unit and that unit gets a new inclusion probability. Three applications are presented. Two of them challenge traditional sampling routines. The simple Pareto sampling design, which was introduced by Rosén in 1997, is also extended. The extended Pareto design is shown to be close to the extended Sampford design.     

How to Select Representative Samples

We give a formal definition of a representative sample, but roughly speaking, it is a scaled‐down version of the population, capturing its characteristics. New methods for selecting representative probability samples in the presence of auxiliary variables are introduced. Representative samples are needed for multipurpose surveys, when several target variables are of interest. Such samples also enable estimation of parameters in subspaces and improved estimation of target variable distributions. We describe how two recently proposed sampling designs can be used to produce representative samples. Both designs use distance between population units when producing a sample. We propose a distance function that can calculate distances between units in general auxiliary spaces. We also propose a variance estimator for the commonly used Horvitz–Thompson estimator. Real data as well as illustrative examples show that representative samples are obtained and that the variance of the Horvitz–Thompson estimator is reduced compared with simple random sampling.     

Robust sampling strategies for regression estimation

For probability linear regression estimation, conditions are derived where sampling will be robust against violations of the commonly assumed heterogeneous variance model. Stratified pps (spps) and stratified random sampling (spscx) are shown to satisfy these conditions approximately and are more efficient generally than restricted simple random sampling (RSRS) for some real populations and for artificial populations with weights of k = 0, 0.5, 1.0, 1.5 and 2.0. The criteria needs some additional refinement to better predict relative efficiency of spps and spscx.     

使用普查数据模拟MPPS抽样方法的研究

MPPS抽样即多变量与规模成比例的概率抽样,是20世纪90年代才提出来的一种抽样设计。近年来,中国有关部门与美国农业部国家农业署合作,进行了MPPS抽样设计的试点,来解决多目标调查问题。但是MPPS抽样在中国的应用非常有限。对MPPS抽样进行简单的回顾,介绍了它的基本估计,并对其应用进行了数据模拟研究。模拟中采用了系统抽样和泊松抽样的方法,根据实际调查数据得到了明确的结果。还对泊松抽样的一种变形永久随机数抽样的方法进行了模拟研究,并对它的一种误用情况进行了模拟比较,得到了具有说服力的结果。     

Estimation of a rare sensitive attribute in a stratified two-stage unrelated randomized response model by using Poisson probability distribution

The present article deals with the estimation of mean number of respondents who possess a rare sensitive character in presence of known and unknown proportion of a rare unrelated non-sensitive attribute by using the Poisson probability distribution in stratified random sampling as well as in stratified random double sampling. The variance of rare sensitive character is also derived under proportional and optimal allocation methods in stratified random sampling when stratum sizes are known and unknown. The properties of the suggested estimation procedures have been deeply examined. The proposed model is found to be dominant over Lee et al. [Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution. Statistics. 2013;47:575–589] model. Numerical illustrations are presented to support the theoretical results. Results are analysed and suitable recommendations are put forward to the survey practitioners.     

Random variate transformations in the Gibbs sampler: issues of efficiency and convergence

In the non-conjugate Gibbs sampler, the required sampling from the full conditional densities needs the adoption of black-box sampling methods. Recent suggestions include rejection sampling, adaptive rejection sampling, generalized ratio of uniforms, and the Griddy-Gibbs sampler. This paper describes a general idea based on variate transformations which can be tailored in all the above methods and increase the Gibbs sampler efficiency. Moreover, a simple technique to assess convergence is suggested and illustrative examples are presented.     

Variance Estimation under Two‐Phase Sampling

We consider the variance estimation of the weighted likelihood estimator (WLE) under two‐phase stratified sampling without replacement. Asymptotic variance of the WLE in many semiparametric models contains unknown functions or does not have a closed form. The standard method of the inverse probability weighted (IPW) sample variances of an estimated influence function is then not available in these models. To address this issue, we develop the variance estimation procedure for the WLE in a general semiparametric model. The phase I variance is estimated by taking a numerical derivative of the IPW log likelihood. The phase II variance is estimated based on the bootstrap for a stratified sample in a finite population. Despite a theoretical difficulty of dependent observations due to sampling without replacement, we establish the (bootstrap) consistency of our estimators. Finite sample properties of our method are illustrated in a simulation study.     

Probability proportional to size (πps) sampling using ranks

There can be gains in estimation efficiency over equal probability samplin methods when one makes use of auxiliary information for probability proporti onal to size with replacement (πpswr) sampling methods. The usual method is simple to execute, but might lead to more than one appearance in the sampl e for any particular unit. When a suitable variable x is not available, one may know how to rank units reasonably well relative to the unknown y values before sample selection. When such ranking is possible, we introduce a simple and efficient sampling plan using the ranks as the unknown x measures of size. The proposed sampling plan is similar to, has the simplicity of, and has no greater sampling variance than with replacement sampling, but is without replacement.     

Analysis of bivariate dichotomous data from a stratified two-stage cluster sample

In the health and social sciences, researchers often encounter categorical data for which complexities come from a nested hierarchy and/or cross-classification for the sampling structure. A common feature of these studies is a non-standard data structure with repeated measurements which may have some degree of clustering. In this paper, methodology is presented for the joint estimation of quantities of interest in the context of a stratified two-stage sample with bivariate dichotomous data. These quantities are the mean value π of an observed dichotomous response for a certain condition or time-point and a set of correlation coefficients for intra-cluster association for each condition or time period and for inter-condition correlation within and among clusters. The methodology uses the cluster means and pairwise joint probability parameters from each cluster. They together provide appropriate information across clusters for the estimation of the correlation coefficients.     

The Horvitz-Thompson Theorem as a Unifying Perspective for Probability Sampling: With Examples from Natural Resource Sampling

Courses in sampling often lack a coherent structure because many related sampling designs, estimators, variances, and variance estimators are presented as separate cases. The Horvitz-Thompson theorem offers a needed integrating perspective for teaching the methods and fundamental concepts of probability sampling. Development of basic concepts in sampling via this approach provides the student with tools to solve more complicated problems, and helps to avoid some common stumbling blocks of beginning students. Examples from natural resource sampling are provided to illustrate applications and insight gained from this approach.     

Nonresponse weighting adjustment using estimated response probability

To reduce nonresponse bias in sample surveys, a method of nonresponse weighting adjustment is often used which consists of multiplying the sampling weight of the respondent by the inverse of the estimated response probability. The authors examine the asymptotic properties of this estimator. They prove that it is generally more efficient than an estimator which uses the true response probability, provided that the parameters which govern this probability are estimated by maximum likelihood. The authors discuss variance estimation methods that account for the effect of using the estimated response probability; they compare their performances in a small simulation study. They also discuss extensions to the regression estimator.     

On a sequential subset selection procedure for the least probable multinomial cell

This paper studies a sequential procedure R for selecting a random size subset that contains the multinomial cell which has the smallest cell probability. The stopping rule of the proposed procedure R is the composite of the stopping rules of curtailed sampling, inverse sampling, and the Ramey-Alam sampling. A reslut on the worst configuration is shown and it is employed in computing the procedure parameters that guarantee certain probability requirements. Tables of these procedure parameters, the corresponding probability of correct selection, the expected sample size, and the expected subset size are given for comparison purpose.     

Double bootstrap estimation of variance under systematic sampling with probability proportional to size

Variance estimation under systematic sampling with probability proportional to size is known to be a difficult problem. We attempt to tackle this problem by the bootstrap resampling method. It is shown that the usual way to bootstrap fails to give satisfactory variance estimates. As a remedy, we propose a double bootstrap method which is based on certain working models and involves two levels of resampling. Unlike existing methods which deal exclusively with the Horvitz–Thompson estimator, the double bootstrap method can be used to estimate the variance of any statistic. We illustrate this within the context of both mean and median estimation. Empirical results based on five natural populations are encouraging.     

Bayesian multiple change-point estimation for exponential distribution with truncated and censored data

This paper considers the multiple change-point estimation for exponential distribution with truncated and censored data by Gibbs sampling. After all the missing data of interest is filled in by some sampling methods such as rejection sampling method, the complete-data likelihood function is obtained. The full conditional distributions of all parameters are discussed. The means of Gibbs samples are taken as Bayesian estimations of the parameters. The implementation steps of Gibbs sampling are introduced in detail. Finally random simulation test is developed, and the results show that Bayesian estimations are fairly accurate.     

Semiparametric Analysis of Truncated Data

Randomly truncated data are frequently encountered in many studies where truncation arises as a result of the sampling design. In the literature, nonparametric and semiparametric methods have been proposed to estimate parameters in one-sample models. This paper considers a semiparametric model and develops an efficient method for the estimation of unknown parameters. The model assumes that K populations have a common probability distribution but the populations are observed subject to different truncation mechanisms. Semiparametric likelihood estimation is studied and the corresponding inferences are derived for both parametric and nonparametric components in the model. The method can also be applied to two-sample problems to test the difference of lifetime distributions. Simulation results and a real data analysis are presented to illustrate the methods.     

On kernel density estimation based on different stratified sampling with optimal allocation

Kernel density estimation is probably the most widely used non parametric statistical method for estimating probability densities. In this paper, we investigate the performance of kernel density estimator based on stratified simple and ranked set sampling. Some asymptotic properties of kernel estimator are established under both sampling schemes. Simulation studies are designed to examine the performance of the proposed estimators under varying distributional assumptions. These findings are also illustrated with the help of a dataset on bilirubin levels in babies in a neonatal intensive care unit.     

Comparing three bootstrap methods for survey data

Various bootstrap methods for variance estimation and confidence intervals in complex survey data, where sampling is done without replacement, have been proposed in the literature. The oldest, and perhaps the most intuitively appealing, is the without-replacement bootstrap (BWO) method proposed by Gross (1980). Unfortunately, the BWO method is only applicable to very simple sampling situations. We first introduce extensions of the BWO method to more complex sampling designs. The performance of the BWO and two other bootstrap methods, the rescaling bootstrap (Rao and Wu 1988) and the mirror-match bootstrap (Sitter 1992), are then compared through a simulation study. Together these three methods encompass the various bootstrap proposals.     

Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution

This study proposes the estimators for the mean and its variance of the number of respondents who possessed a rare sensitive attribute based on stratified sampling schemes (stratified sampling and stratified double sampling). This study deals with the extension of the estimation reported in Land et al. [Estimation of a rare sensitive attribute using Poisson distribution, Statistics (2011), in press. DOI: 10.1080/02331888.2010.524300] using a Poisson distribution and an unrelated question randomized response model reported in Greenberg et al. [The unrelated question randomized response model: Theoretical framework, J. Amer. Statist. Assoc. 64 (1969), 520–539]. In the stratified sampling, the estimators are proposed when the parameter of the rare unrelated attribute is known and unknown. The variances of estimators using a proportional and optimum allocation are also suggested. The proposed estimators are evaluated using a relative efficiency comparing variances of the estimators reported in Land et al. depending on the parameters and the probability of selecting a question. We showed that our proposed methods have better efficiencies than Land et al.’s randomized response model in some conditions. When the sizes of stratified populations are not given, other estimators are suggested using a stratified double sampling. For the proportional allocation, the difference between two variances in the stratified sampling and the stratified double sampling is given with the known rare unrelated attribute.     

Systematic and nonrandom sampling in the presence of linear trends

In the case of a finite population with a linear trend, improvements in estimation procedures are suggested for several well-known systematic samplingschemes. The use of the least squares regression estimator is examined and a sampling scheme which reduces its expected mean squared error is proposed.     

PPS抽样设计的比估计及其模拟分析

本文在介绍简单随机抽样比估计的基础上,对PPS抽样设计的比估计进行了系统的研究,并使用了第一次农业普查数据对方法进行了模拟。从模拟结果可以看到,比估计方法可以解决一些调查精度和多目标调查的问题。