Entropy of unequal probability sampling designs

There exist many designs for unequal probability sampling. In this paper entropy, which is a measure of randomness, is used to compare eight designs. Both old and commonly used designs and more recent designs are included. Several different and general estimates of entropy are presented. In the quest of finding entropy, expressions for the probability function are derived for different designs. One of them is a recent general design called correlated Poisson sampling. Several designs are close to having maximum entropy, which means that the designs are robust. A few designs yield low entropy and should therefore in general be avoided.     

Properties of designs for sampling continuous spatial resources from a triangular grid

Precision of systematic designs for sampling continuous response variables defined over a continuous spatial region have been extensively investigated. Estimation of variance, particularly for triangular grids, has been given less attention. Two of the designs proposed for sampling environmental resources in the Environmental Protection Agency's Environmental Monitoring and Assessment Program (EMAP) are based on a triangular grid. Implementation of either design requires derivation and evaluation of a variance estimator adequate for the purposes of EMAP. The performance of the proposed variance estimator was assessed under various surface model representations of the continuous response variable. Of the designs examined, a tessellation-stratified design permitted better estimation of variance, and was generally more precise, than a strict systematic design.     

Weighted analyses for cohort sampling designs

Weighted analysis methods are considered for cohort sampling designs that allow subsampling of both cases and non-cases, but with cases generally sampled more intensively. The methods fit into the general framework for the analysis of survey sampling designs considered by Lin (Biometrika 87:37–47, 2000). Details are given for applying the general methodology in this setting. In addition to considering proportional hazards regression, methods for evaluating the representativeness of the sample and for estimating event-free probabilities are given. In a small simulation study, the one-sample cumulative hazard estimator and its variance estimator were found to be nearly unbiased, but the true coverage probabilities of confidence intervals computed from these sometimes deviated significantly from the nominal levels. Methods for cross-validation and for bootstrap resampling, which take into account the dependencies in the sample, are also considered. An erratum to this article can be found at     

One-cluster sampling designs for an autocorrelated population

The variance of the sampling distribution of the sample mean is derived for two sampling designs in which a single cluster is randomly drawn from an autocorrelated population. The derivations are motivated by potential applications to statistical quality control, where a "one-cluster" sampling design may often be used because of ease of implementation, and where it is likely that process output is autocorrelated Scenarios in statistical process control for which either non-overlapping or overlapping clusters are appropriate are described The sampling design variance under non-overlapping clusters is related to the sampling design variance under overlapping clusters through the use of a circular population.     

Performances of some goodness-of-fit tests for sampling designs in ranked set sampling

In this study, we consider different sampling designs of ranked set sampling (RSS) and give empirical distribution function (EDF) estimators for each sampling designs. We provide comparative graphs for the EDFs. Using these EDFs, power of five goodness-of-fit tests are obtained by Monte Carlo simulations for Tukey's g–h distributions under RSS and simple random sampling (SRS). Performances of these tests are compared with the tests based on the SRS. Also, critical values belong to these tests are obtained for different set and cycle sizes.     

Exact confidence limits for the probability of response in two-stage designs

In addition to point estimate for the probability of response in a two-stage design (e.g. Simon's two-stage design for binary endpoints), confidence limits should be computed and reported. The current method of inverting the p-value function to compute the confidence interval does not guarantee coverage probability in a two-stage setting. The existing exact approach to calculate one-sided limits is based on the overall number of responses to order the sample space. This approach could be conservative because many sample points have the same limits. We propose a new exact one-sided interval based on p-value for the sample space ordering. Exact intervals are computed by using binomial distributions directly, instead of a normal approximation. Both exact intervals preserve the nominal confidence level. The proposed exact interval based on the p-value generally performs better than the other exact interval with regard to expected length and simple average length of confidence intervals.     

Factorial hypercube designs for spatial correlation regression

SUMMARY The problem of generating a good experimental design for spatial correlation regression is studied in this paper. The quality of fit generated by random designs, Latin hypercube designs and factorial designs is studied for a particular response surface that arises in inkjet printhead design. These studies indicate that the quality of fit generated by spatial correlation models is highly dependent on the choice of design. A design strategy that we call 'factorial hypercubes' is introduced as a new method. This method can be thought of as an example of a more general class of hybrid designs. The quality of fit generated by these designs is compared with those of other methods. These comparisons indicate a better fit and less numerical problems with factorial hypercubes.     

Exchange algorithms for constructing large spatial designs

Exchange algorithms are widely used for optimizing design criteria and have proven to be highly effective. However, application of these algorithms can be computationally prohibitive for very large design problems, and in situations for which the design criterion is sufficiently complex so as to prevent efficient evaluation strategies. This paper reports on several modifications to exchange algorithms that lead to large reductions in the computational burden associated with optimizing any given design criterion. A small study indicates that these modifications do not significantly impact the quality of designs, while reductions in effort of over 90% can be achieved.     

A jackknife variance estimator for unequal probability sampling

Summary.  The jackknife method is often used for variance estimation in sample surveys but has only been developed for a limited class of sampling designs. We propose a jackknife variance estimator which is defined for any without-replacement unequal probability sampling design. We demonstrate design consistency of this estimator for a broad class of point estimators. A Monte Carlo study shows how the proposed estimator may improve on existing estimators.     

Chao's list sequential scheme for unequal probability sampling

The problem of unequal probability sampling is reviewed and discussed in the light of the list sequential scheme proposed by Chao. Chao's scheme is described fully, and its statistical properties are compared with systematic piPS sampling by simulation using standard populations.     

Variance estimation for measures of change in probability sampling

Variance estimation of changes requires estimates of variances and covariances that would be relatively straightforward to make if the sample remained the same from one wave to the next, but this is rarely the case in practice as successive waves are usually different overlapping samples. The author proposes a design‐based estimator for covariance matrices that is adapted to this situation. Under certain conditions, he shows that his approach yields non‐negative definite estimates for covariance matrices and therefore positive variance estimates for a large class of measures of change.     

Reconciliation of asymptotics for unequal probability sampling without replacement

In simple random sampling without replacement (SRSWOR), certain reverse martingale structures render simple asymptotics for the conventional linear statistics. In unequal probability sampling (UPS) WOR, such martingale-based methodology may not be generally adoptable. General asymptotics for UPSWOR sampling schemes, developed by Hartley and Rao (Ann. Math. Statist. 33 (1962) 350), and Hájek (Ann. Math. Statist. 35 (1964) 1491), rest on different sets of regularity assumptions, and they differ in their treatise too. Some anomalies in this context are eliminated here with a reconciliation of both the approaches, and estimation of the asymptotic variance of linear estimators is considered in the same vein. Applications to small area sampling are also stressed.     

Efficient circular neighbour designs for spatial interference model

We consider circular block designs for field-trials when there are two-sided spatial interference between neighbouring plots of the same blocks. The parameter of interest is total effects that is the sum of direct effect of treatment and neighbour effects, which correspond to the use of a single treatment in the whole field. We determine universally optimal approximate designs. When the number of blocks may be large, we propose efficient exact designs generated by a single sequence of treatment. We also give efficiency factors of the usual binary block neighbour balanced designs which can be used when the number of blocks is small.     

Closed inverse sampling procedure for selecting the largest multinomial cell probability

A class of closed inverse sampling procedures R(n,m) for selecting the multinomial cell with the largest probability is considered; here n is the maximum sample size that an experimenter can take and m is the maximum frequency that a multinomial cell can have. The proposed procedures R(n,m) achieve the same probability of a correct selection as do the corresponding fixed sample size procedures and the curtailed sequential procedures when m is at least n/2. A monotonicity property on the probability of a correct selection is proved and it is used to find the least favorable configurations and to tabulate the necessary probabilities of a correct selection and corresponding expected sample sizes     

Optimality of equidistant sampling designs for the Brownian motion with a quadratic drift

In the paper we show that the equidistant sampling designs are optimal for the model of Brownian motion with a quadratic drift and for any of its submodels. This result holds for all Loewner isotonic criteria of parametric optimality continuous on the set of regular information matrices, as well as for the mean squared error of the best linear unbiased predictor.     

Optimal sampling for spatial prediction of functional data

This paper combines optimal spatial sampling designs with geostatistical analysis of functional data. We propose a methodology and design criteria to find the set of spatial locations that minimizes the variance of the spatial functional prediction at unsampled sites for three functional predictors: ordinary kriging, simple kriging and simple cokriging. The last one is a modification of an existing predictor that uses ordinary cokriging based on the basis coefficients. Instead, we propose to use a simple cokriging predictor with the scores resulting from a representation of the functional data with the empirical functional principal components, allowing to remove restrictions and complexity of the covariance models and constraints on the estimation procedure. The methodology is applied to a network of air quality in Bogotá city, Colombia.     

A review of optimal designs in survey sampling

Results in five areas of survey sampling dealing with the choice of the sampling design are reviewed. In Section 2, the results and discussions surrounding the purposive selection methods suggested by linear regression superpopulation models are reviewed. In Section 3, similar models to those in the previous section are considered; however, random sampling designs are considered and attention is focused on the optimal choice of π_j. Then in Section 4, systematic sampling methods obtained under autocorrelated superpopulation models are reviewed. The next section examines minimax sampling designs. The work in the final section is based solely on the randomization. In Section 6 methods of sample selection which yield inclusion probabilities π_j = n/N and π_ij = n(n - 1)/N(N - 1), but for which there are fewer than _NC_n possible samples, are mentioned briefly.     

On the estimation of the Lorenz curve under complex sampling designs

Statistical Methods & Applications - This paper focuses on the estimation of the concentration curve of a finite population, when data are collected according to a complex sampling design with...