Estimation of a population mean with two-way stratification using a systematic allocation scheme

A technique of systematically allocating a sample to the strata formed by double stratification is presented. The method can proportionally allocate the sample along each variable of stratification. If there are R strata and C strata for the first and second variable of stratification respectively, the technique requires that the total sample size be at least as large as max(R, C). An unbiased estimator of the population mean is given and its variance is obtained. The technique is compared with a random allocation procedure given by Bryant, Hartley, and Jessen (1960). Numerical examples are given suggesting when one technique is superior to the other.     

Pairwise comparisons for proportions estimated by pooled testing

When estimating the prevalence of a rare trait, pooled testing can confer substantial benefits when compared to individual testing. In addition to screening experiments for infectious diseases in humans, pooled testing has also been exploited in other applications such as drug testing, epidemiological studies involving animal disease, plant disease assessment, and screening for rare genetic mutations. Within a pooled-testing context, we consider situations wherein different strata or treatments are to be compared with the goals of assessing significant and practical differences between strata and ranking strata in terms of prevalence. To achieve these goals, we first present two simultaneous pairwise interval estimation procedures for use with pooled data. Our procedures rely on asymptotic results, so we investigate small-sample behavior and compare the two procedures in terms of simultaneous coverage probability and mean interval length. We then present a unified approach to determine pool sizes which deliver desired coverage properties while taking testing costs and interval precision into account. We illustrate our methods using data from an observational HIV study involving heterosexual males who use intravenous drugs.     

辅以排序集样本的分层抽样方法研究

分层抽样中,样本在各层中的不同获取方式会对估计量的精度和试验费用产生一定的影响,而已有的理论方法大多不能在提高精度的同时降低调查费用。为此,将排序抽样与分层抽样方法相结合,提出了辅以排序集样本的分层抽样方案,并得到了总体均值的估计量以及这一估计量的良好性质。这些结果表明,与单一的分层随机抽样相比,这种抽样设计的估计量具有更高的精度,同时也节约了各层抽样调查的费用。     

An exact small sample theory for post-stratification

A genuine small sample theory for post-stratification is developed in this paper. This includes the definition of a ratio estimator of the population mean ?, the derivation of its bias and its exact variance and a discussion of variance estimation. The estimator has both a within strata component of variance which is comparable with that obtained in proportional allocation stratified sampling and a between strata component of variance which will tend to zero as the overall sample size becomes large. Certain optimality properties of the estimator are obtained. The generalization of post-stratification from the simple random sampling to post-stratification used in conjunction with stratification and multi-stage designs is discussed.     

Combining multi‐observer information in partially rank‐ordered judgment post‐stratified and ranked set samples

This paper develops two sampling designs to create artificially stratified samples. These designs use a small set of experimental units to determine their relative ranks without measurement. In each set, the units are ranked by all available observers (rankers), with ties whenever the units cannot be ranked with high confidence. The rankings from all the observers are then combined in a meaningful way to create a single weight measure. This weight measure is used to create judgment strata in both designs. The first design constructs the strata through judgment post‐stratification after the data has been collected. The second design creates the strata before any measurements are made on the experimental units. The paper constructs estimators and confidence intervals, and develops testing procedures for the mean and median of the underlying distribution based on these sampling designs. We show that the proposed sampling designs provide a substantial improvement over their competitor designs in the literature. The Canadian Journal of Statistics 41: 304–324; 2013 © 2013 Statistical Society of Canada     

分层抽样估计方法新探

The paper puts forward two ways of thinking on stratified sampling estimation, the longitudinal auto-egression and multi-compound estimation method. The longitudinal auto-regression method probes into estimate of population mean based on the linear regression relationship between the means of the strata under investigation and ancillary characteristics. The multi-compound estimation method estimates the means of the strata separately by different methods and the population parameter is taken from these estimates. The paper further analyses conditions for the application of these methods; estimates and their error term; advantages and disadvantages and other problems of the two problems.     

Parametric inference based on judgment post stratified samples

In this paper, we consider a judgment post stratified (JPS) sample of set size $H$ from a location and scale family of distributions. In a JPS sample, ranks of measured units are random variables. By conditioning on these ranks, we derive the maximum likelihood (MLEs) and best linear unbiased estimators (BLUEs) of the location and scale parameters. Since ranks are random variables, by considering the conditional distributions of ranks given the measured observations we construct Rao-Blackwellized version of MLEs and BLUEs. We show that Rao-Blackwellized estimators always have smaller mean squared errors than MLEs and BLUEs in a JPS sample. In addition, the paper provides empirical evidence for the efficiency of the proposed estimators through a series of Monte Carlo simulations.     

A domain outlier robust design and smooth estimation approach

Outliers that commonly occur in business sample surveys can have large impacts on domain estimates. The authors consider an outlier‐robust design and smooth estimation approach, which can be related to the so‐called “Surprise stratum” technique [Kish, “Survey Sampling,” Wiley, New York (1965)]. The sampling design utilizes a threshold sample consisting of previously observed outliers that are selected with probability one, together with stratified simple random sampling from the rest of the population. The domain predictor is an extension of the Winsorization‐based estimator proposed by Rivest and Hidiroglou [Rivest and Hidiroglou, “Outlier Treatment for Disaggregated Estimates,” in “Proceedings of the Section on Survey Research Methods,” American Statistical Association (2004), pp. 4248–4256], and is similar to the estimator for skewed populations suggested by Fuller [Fuller, Statistica Sinica 1991;1:137–158]. It makes use of a domain Winsorized sample mean plus a domain‐specific adjustment of the estimated overall mean of the excess values on top of that. The methods are studied in theory from a design‐based perspective and by simulations based on the Norwegian Research and Development Survey data. Guidelines for choosing the threshold values are provided. The Canadian Journal of Statistics 39: 147–164; 2011 © 2010 Statistical Society of Canada     

Optimum multivariate stratified double sampling design: Chebyshev's Goal Programming approach

In stratified sampling when strata weights are unknown a double sampling technique may be used to estimate them. A large simple random sample from the unstratified population is drawn and units falling in each stratum are recorded. A stratified random sample is then selected and simple random subsamples are obtained out of the previously selected units of the strata. This procedure is called double sampling for stratification. If the problem of non-response is there, then subsamples are divided into classes of respondents and non-respondents. A second subsample is then obtained out of the non-respondents and an attempt is made to obtain the information by increasing efforts, persuasion and call backs. In this paper, the problem of obtaining a compromise allocation in multivariate stratified random sampling is discussed when strata weights are unknown and non-response is present. The problem turns out to be a multiobjective non-linear integer programming problem. An approximation of the problem to an integer linear programming problem by linearizing the non-linear objective functions at their individual optima is worked out. Chebyshev's goal programming technique is then used to solve the approximated problem. A numerical example is also presented to exhibit the practical application of the developed procedure.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

Rao and Wu's re-scaling bootstrap modified to achieve extended coverages

Horvitz and Thompson's (HT) [1952. A generalization of sampling without replacement from a finite universe. J. Amer. Statist. Assoc. 47, 663–685] well-known unbiased estimator for a finite population total admits an unbiased estimator for its variance as given by [Yates and Grundy, 1953. Selection without replacement from within strata with probability proportional to size. J. Roy. Statist. Soc. B 15, 253–261], provided the parent sampling design involves a constant number of distinct units in every sample to be chosen. If the design, in addition, ensures uniform non-negativity of this variance estimator, Rao and Wu [1988. Resampling inference with complex survey data. J. Amer. Statist. Assoc. 83, 231–241] have given their re-scaling bootstrap technique to construct confidence interval and to estimate mean square error for non-linear functions of finite population totals of several real variables. Horvitz and Thompson's estimators (HTE) are used to estimate the finite population totals. Since they need to equate the bootstrap variance of the bootstrap estimator to the Yates and Grundy's estimator (YGE) for the variance of the HTE in case of a single variable, i.e., in the linear case the YG variance estimator is required to be positive for the sample usually drawn.     

Estimation of a rare sensitive attribute for two-stage randomized response model in probability proportional to size sampling using Poisson probability distribution

This paper addresses the estimation of the mean number of individuals in the population who possess a rare sensitive attribute using Poisson distribution for the situations of (i) clustered population and (ii) stratified population with clusters are strata units. Properties of the proposed estimation procedures have been discussed when the proportion of a rare unrelated non-sensitive attribute is assumed to be known as well as unknown. Empirical studies are carried out to support the theoretical results which showed dominance over Lee et al. [Estimation of a rare sensitive attribute in probability proportional to size measures using Poisson distribution. Statistics (Ber). 2014;48(3):685–709] estimation procedures.     

Mean and sensitivity estimation in optional randomized response models

Optional randomized response models were introduced by Gupta et al. (2002). These models are based on the basic premise that a question may be sensitive for one respondent but may not be sensitive for another. In an optional RRT (randomized response technique) model, a respondent is asked to provide a scrambled response only if the respondent considers the question sensitive. Otherwise, the respondent provides a truthful response. The researcher does not know which type of response is provided. The proportion of respondents who provide a scrambled response is known as the sensitivity level of the question. In this paper, we estimate simultaneously the mean and the sensitivity level of a quantitative-response sensitive question using a two stage optional RRT model. The estimators are unbiased and asymptotically normally distributed. We discuss the advantages and disadvantages of using additive and multiplicative scrambling.     

Small area estimation of poverty indicators

The authors propose to estimate nonlinear small area population parameters by using the empirical Bayes (best) method, based on a nested error model. They focus on poverty indicators as particular nonlinear parameters of interest, but the proposed methodology is applicable to general nonlinear parameters. They use a parametric bootstrap method to estimate the mean squared error of the empirical best estimators. They also study small sample properties of these estimators by model‐based and design‐based simulation studies. Results show large reductions in mean squared error relative to direct area‐specific estimators and other estimators obtained by “simulated” censuses. The authors also apply the proposed method to estimate poverty incidences and poverty gaps in Spanish provinces by gender with mean squared errors estimated by the mentioned parametric bootstrap method. For the Spanish data, results show a significant reduction in coefficient of variation of the proposed empirical best estimators over direct estimators for practically all domains. The Canadian Journal of Statistics 38: 369–385; 2010 © 2010 Statistical Society of Canada     

A generalized class of estimators under two-phase stratified sampling for non response

In this paper, we propose a generalized class of estimators for finite population mean using two auxiliary variables in two-phase stratified sampling for non response. We identify 17 estimators as special cases of the proposed class of estimators. Expressions for the bias and mean squared error (MSE) of estimators are obtained up to first order of approximation. A data set is used for efficiency comparisons.     

Ratio estimators based on a ranked set sample in a finite population setting

This paper considers the ratio estimator in a finite population setting in a ranked set sampling (RSS) design, where the sample is constructed either with or without replacement policies. It is shown that the proposed ratio estimator is slightly biased, but the amount of bias is smaller than the amount of bias of a simple random sample (SRS) ratio estimator. For the proposed ratio estimator, the paper provides explicit expressions for its mean square error and precision relative to the other competing estimators. It is shown that the new estimator has a substantial amount of improvement in efficiency with respect to SRS estimator. The proposed estimator is applied to two different finite population settings to estimate population mean.     

Ranked set sampling theory with selective probability vector

In this paper an estimator of the population mean is introduced by using the idea of selective probability vector and the optimization algorithm of linear programming to find the optimal solution of the selective probability vector under the condition of unbiasedness.     

Optimum stratification for allocation proportional to strata totals for simple random sampling scheme

The paper considers the problem of finding optimum strata boundaries when sample sizes to different strata are allocated in proportion to the strata totals of the auxiliary variable. This variable is also treated as the stratification variable. Minimal equations, solutions to which provide the optimum boundaries, have been obtained. Because of the implicit nature of these equations their exact solutions cannot be obtained. Therefore, methods of obtaining their approximate solutions have been presented. A lim¬iting expression for the variance of the estimate of population mean, as the number of strata tend to become large, has also been obtained.     

Lower bounds on expectations of positive L-statistics from without-replacement models

We consider samples drawn without replacement from finite populations. We establish optimal lower non-negative and upper non-positive bounds on the expectations of linear combinations of order statistics centered about the population mean in units generated by the population central absolute moments of various orders. We also specify the general results for important examples of sample extremes, Gini mean differences and sample range. The paper completes the results of Papadatos and Rychlik [2004. Bounds on expectations of L-statistics from without replacement samples. J. Statist. Plann. Inference 124, 317–336], where sharp negative lower and positive upper bounds on the expectations of the combinations were presented for the without-replacement samples.