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.     

Stratified Double Sampling with Continuous Outcomes: Design and Analysis

We provide a method for estimating the sample mean of a continuous outcome in a stratified population using a double sampling scheme. The stratified sample mean is a weighted average of stratum specific means. It is assumed that the fallible and true outcome data are related by a simple linear regression model in each stratum. The optimal stratified double sampling plan, i.e. , the double sampling plan that minimizes the cost of sampling for fixed variances, or alternatively, minimizes the variance for fixed costs, is found and compared to a standard sampling plan. The design parameters are the total sample size and the number of doubly sampled units in each stratum. We show that the optimal double sampling plan is a function of the between-strata and within-strata cost and variance ratios. The efficiency gains, relative to standard sampling plans, under broad set of conditions, are considerable.     

中国人口变动调查的自加权抽样设计

自加权分层多阶段抽样设计具有三大特征：一为除第一阶抽样外其余各阶抽样的样本量均为常数，二为样本量按照各层的最终单元数量在各层比例分配，三为前几阶采用抽样而最后一阶采用放回或不放回的简单随机抽样。根据上述三个特征设计了中国人口变动调查的自加权抽样设计。     

Adaptive sampling without replacement of clusters

In a common form of adaptive cluster sampling, an initial sample of units is selected by random sampling without replacement and, whenever the observed value of the unit is sufficiently high, its neighboring units are added to the sample, with the process of adding neighbors repeated if any of the added units are also high valued. In this way, an initial selection of a high-valued unit results in the addition of the entire network of surrounding high-valued units and some low-valued “edge” units where sampling stops. Repeat selections can occur when more than one initially selected unit is in the same network or when an edge unit is shared by more than one added network. Adaptive sampling without replacement of networks avoids some of this repeat selection by sequentially selecting initial sample units only from the part of the population not already in any selected network. The design proposed in this paper carries this step further by selecting initial units only from the population, exclusive of any previously selected networks or edge units.     

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.     

A note on stratification and gain in precision in estimating diversity from large samples

Several indices of entropy have been suggested in the literature as weighted diversity measures of a population with respect to a classification process. Among them, Shannon's entropy and Havrda -Charvát's non-additive entropies of order a, have been exhaustively used.

When the population is finite but too large to be censused, the diversity with respect to a given classification process must be estimated from a sample.

In this note, on the basis of an asymptotic study of the sample indices in the stratified random sampling, we are going to confirm that when we deal with large samples one can guarantee a gain in precision from stratified random over simple random sampling. This gain becomes considerable when the ‘inaccuracy" (as intended by Kerridge and Rathie and Kannapan) between the frequency vector in each stratum and that in the whole population, varies greatly from stratum to stratum.     

A MODEL OF SYSTEMATIC SAMPLING WITH UNEQUAL PROBABILITIES

A new method is described of drawing, without replacement, two sample units per stratum from any population. The method is developed from a consideration of the asymptotic properties of systematic sampling with unequal probabilities, as the sizes of the population units tend to zero. The essential properties of this method are very easily analysed. They also converge, over a large number of strata, to those of systematic sampling from the same strata with their population units arranged in random order. In proving this, the assumption is made that the underlying population is of the type to which it is appropriate to apply ratio estimation. The sampling method described is, however, simple enough to commend itself as an alternative to systematic sampling when the underlying population is not of this type. Consideration is given to the case where the sizes of some of the population units exceed the skip interval.     

Edgeworth expansion and the bootstrap for stratified sampling without replacement from a finite population

An Edgeworth expansion for a linear combination of stratum means in stratified sampling without replacement from a finite population is derived. The expansion is applied to a bootstrap proposed for this context to show that the bootstrap captures the second-order term of the expansion.     

Gap-based inverse sampling

We present a new inverse sampling design for surveys of rare events, Gap-Based Inverse Sampling. In the design, sampling stops if after a predetermined interval, or gap, no new rare events are found. The length of the gap that follows after finding a rare event is used as a way of limiting sample effort. We present stopping rules using decisions based on the gap length, the total number of rare events found, and a fixed upper limit of survey effort. We illustrate the use of the design with stratified sampling of two biological populations. The design uses the intuitive behavior of a field biologist in stratified sampling, where if in a stratum nothing is found after a long search, the field surveyor would like to consider the stratum is empty and stop searching. Our design has appeal for surveying rare events (for example, a rare species) with stratified sampling where there are likely to be some completely empty strata.     

INVERSE SAMPLING FOR DOMAIN ESTIMATION IN A STRATIFIED POPULATION

For a stratified population under inverse sampling, we propose and study an unbiased estimator for the mean of units belonging to a domain with specific features. An alternative, simpler, ratio-type estimator is also considered. Empirical studies show that strategies based on inverse sampling can be superior to a more traditional strategy based on stratified simple random sampling with a fixed number of draws in each stratum.     

An Optimal Multivariate Stratified Sampling Design Using Dynamic Programming

Numerous optimization problems arise in survey designs. The problem of obtaining an optimal (or near optimal) sampling design can be formulated and solved as a mathematical programming problem. In multivariate stratified sample surveys usually it is not possible to use the individual optimum allocations for sample sizes to various strata for one reason or another. In such situations some criterion is needed to work out an allocation which is optimum for all characteristics in some sense. Such an allocation may be called an optimum compromise allocation. This paper examines the problem of determining an optimum compromise allocation in multivariate stratified random sampling, when the population means of several characteristics are to be estimated. Formulating the problem of allocation as an all integer nonlinear programming problem, the paper develops a solution procedure using a dynamic programming technique. The compromise allocation discussed is optimal in the sense that it minimizes a weighted sum of the sampling variances of the estimates of the population means of various characteristics under study. A numerical example illustrates the solution procedure and shows how it compares with Cochran's average allocation and proportional allocation.     

SOME CONTRIBUTIONS TO TWO-PHASE SAMPLING

A new two-phase sampling procedure is suggested in this paper. The information on the ancillary variate in the preliminary sample is used for selecting the units for sub-sample with unequal probabilities. Unbiased estimators for the population mean and their variance expressions are obtained for sub-sampling procedures, both with and without replacement.     

Design effect of randomized systematic sampling

In statistical practice, systematic sampling (SYS) is used in many modifications due to its simple handling. In addition, SYS may provide efficiency gains if it is well adjusted to the structure of the population under study. However, if SYS is based on an inappropriate picture of the population a high decrease of efficiency, i.e. a high increase in variance may result by changing from simple random sampling to SYS. In the context of two-stage designs SYS so far seems often in use for subsampling within the primary units. As an alternative to this practice, we propose to randomize the order of the primary units, then to select systematically a number of primary units and, thereafter, to draw secondary units by simple random sampling without replacement within the primary units selected. This procedure is more efficient than simple random sampling with replacement from the whole population of all secondary units, i.e. the variance of an adequate estimator for a total is never increased by changing from simple random sampling to randomized SYS whatever be the values associated by a characteristic with the secondary units, while there are values for which the variance decreases for the change mentioned. This result should hold generally, even if our proof, so far, is not complete for general sample sizes.     

Anderson's classification statistic based on a post-stratified training sample

The performance of Anderson's classification statistic based on a post-stratified random sample is examined. It is assumed that the training sample is a random sample from a stratified population consisting of two strata with unknown stratum weights. The sample is first segregated into the two strata by post-stratification. The unknown parameters for each of the two populations are then estimated and used in the construction of the plug-in discriminant. Under this procedure, it is shown that additional estimation of the stratum weight will not seriously affect the performance of Anderson's classification statistic. Furthermore, our discriminant enjoys a much higher efficiency than the procedure based on an unclassified sample from a mixture of normals investigated by Ganesalingam and McLachlan (1978).     

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.     

存在测量误差时分层抽样层均值方差的估计

利用模型的方法研究出现测量误差时多变量间的关系是目前国际上的流行方法,但这不利于对单指标的估计。因此,通过在估计量的设计中纳入测量误差信息,推导测量误差方差的定量测度方法,实现了存在测量误差时分层抽样各层均值方差的估计。采用2007年广东省三个市（县）城镇住户调查中的人均消费性支出数据进行实证分析,定量测度了测量误差在层均值方差估计中的大小及其影响,并对不考虑测量误差的估计结果进行了修正。     

A Non-Typical Least-Squares Solution to an Engineering Problem

In preposterior analysis, Bayesians use an Expected-Net-Gain chart to identify the optimal sample size. This kind of chart, it turns out, is also an excellent educational vehicle for illustrating many of the reasons given for preferring sampling from a population over taking a census, preferring one type of sampling over another (e.g., stratified sampling rather than simple random sampling), or allocating part of a fixed budget to reduce systematic error rather than using it all to reduce sampling errors. The use of such a chart in a basic statistics course is described.     

Empirical Bayes estimation of undercount in the decennial census

Empirical Bayes methods are used to estimate the extent of the undercount at the local level in the 1980 U.S. census. "Grouping of like subareas from areas such as states, counties, and so on into strata is a useful way of reducing the variance of undercount estimators. By modeling the subareas within a stratum to have a common mean and variances inversely proportional to their census counts, and by taking into account sampling of the areas (e.g., by dual-system estimation), empirical Bayes estimators that compromise between the (weighted) stratum average and the sample value can be constructed. The amount of compromise is shown to depend on the relative importance of stratum variance to sampling variance. These estimators are evaluated at the state level (51 states, including Washington, D.C.) and stratified on race/ethnicity (3 strata) using data from the 1980 postenumeration survey (PEP 3-8, for the noninstitutional population)."     

On a generalization of Poisson sampling

In real-time sampling, the units of a population pass a sampler one by one. Alternatively the sampler may successively visit the units of the population. Each unit passes only once and at that time it is decided whether or not it should be included in the sample. The goal is to take a sample and efficiently estimate a population parameter. The list sequential sampling method presented here is called correlated Poisson sampling. The method is an alternative to Poisson sampling, where the units are sampled independently with given inclusion probabilities. Correlated Poisson sampling uses weights to create correlations between the inclusion indicators. In that way it is possible to reduce the variation of the sample size and to make the samples more evenly spread over the population. Simulation shows that correlated Poisson sampling improves the efficiency in many cases.     

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.