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.     

VARIANCE ESTIMATION IN MULTI-STAGE SAMPLING

In multi-stage sampling with the first stage units (fsu) chosen without replacement (WOR) with varying probability schemes (VPS) unbiased estimators (UE) of variances of homogeneous linear (HL) functions of unbiased estimators (UE) T_i's of fsu totals Y_i's based on selection of subsequent stage units (SSU) from chosen fsu's are derived as homogeneous quadratic (HQ) functions of alternative less efficient UE's, say of T_i';'s of Y_i's. Specific strategies are illustrated.     

A New Proof of Murthy's Estimator which Applies to Sequential Sampling

Murthy's estimator has been used for constructing an unbiased estimator of a population total or mean from a sample of fixed size when there is unequal probability sampling without replacement. Traditionally, the estimator is derived by constructing an unordered version of Raj's ordered unbiased estimator. This paper presents an elementary proof of Murthy's estimator which applies the Rao–Blackwell theorem to a very simple estimator. This proof includes any sequential sampling scheme, thus extending the usefulness of Murthy's estimator. We demonstrate this extension by deriving unbiased estimators for inverse sampling.     

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.     

Comparison of Efficiency of Stratified and Unequal Probability Sampling

In this article, a choice of the optimum sampling design to study a finite population is studied. Three sampling schemes are compared, viz., Sunter's procedure of unequal probability sampling, stratified sampling under optimum stratification, and simple random sampling without replacement. The comparison is made against a background of various correlation between stratification and survey variables and various variability in the variables. Under weak correlation and large variability, stratification appeared to be more efficient than Sunter's procedure. Under strong correlation and/or low variability in the variables, the latter procedure was the most efficient. Simple random sampling was usually the least efficient.     

Examination of Influential Observations to Improve Discriminant Analysis

Three simple transformations are proposed in the context of ratio and product methods of estimation, based on any probability sampling design, and the usual unbiased estimation under varying probability sampling. These transformations may be effected

after the data are collected in a survey. The objective is to obtain improved estimators of the population total     

On optimality of the horvitz-thompson estimator under a markov process model

For any varying probability sampling design the Horvitz-Thompson (1952) estimator is shown to be optimal within the class of all unbiased estimators of a finite population total under a Markov process model     

Conditional and Restricted Pareto Sampling: Two New Methods for Unequal Probability Sampling

Abstract. Two new unequal probability sampling methods are introduced: conditional and restricted Pareto sampling. The advantage of conditional Pareto sampling compared with standard Pareto sampling, introduced by Rosén (J. Statist. Plann. Inference, 62, 1997, 135, 159), is that the factual inclusion probabilities better agree with the desired ones. Restricted Pareto sampling, preferably conditioned or adjusted, is able to handle cases where there are several restrictions on the sample and is an alternative to the recent cube method for balanced sampling introduced by Deville and Tillé (Biometrika, 91, 2004, 893). The new sampling designs have high entropy and the involved random numbers can be seen as permanent random numbers.     

Use of ranks in probability propoetional to size sampling

Use of ranks in unequal probability sampling is examined for sample selection, stratification as well as determining the strata boundaries. A few sampling schemes are proposed and investigated, For samples of size two, two sampling schemes and their 1PPS versions are discussed, An extension of these schemes to general sample sizes is outlined. Nonnegative unbiased variance estimators are proposed in each case, An empirical comparison is included.     

Accelerated life test sampling plans for the Weibull distribution under Type I progressive interval censoring with random removals

This paper considers the design of accelerated life test (ALT) sampling plans under Type I progressive interval censoring with random removals. We assume that the lifetime of products follows a Weibull distribution. Two levels of constant stress higher than the use condition are used. The sample size and the acceptability constant that satisfy given levels of producer's risk and consumer's risk are found. In particular, the optimal stress level and the allocation proportion are obtained by minimizing the generalized asymptotic variance of the maximum likelihood estimators of the model parameters. Furthermore, for validation purposes, a Monte Carlo simulation is conducted to assess the true probability of acceptance for the derived sampling plans.     

Ranked set sampling with unequal samples for skew distributions

A ranked set sampling procedure with unequal samples for positively skew distributions (RSSUS) is proposed and used to estimate the population mean. The estimators based on RSSUS are compared with the estimators based on ranked set sampling (RSS) and median ranked set sampling (MRSS) procedures. It is observed that the relative precisions of the estimators based on RSSUS are higher than those of the estimators based on RSS and MRSS procedures.     

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.     

Spatially correlated Poisson sampling

A new method for sampling from a finite population that is spread in one, two or more dimensions is presented. Weights are used to create strong negative correlations between the inclusion indicators of nearby units. The method can be used to produce unequal probability samples that are well spread over the population in every dimension, without any spatial stratification. Since the method is very general there are numerous possible applications, especially in sampling of natural resources where spatially balanced sampling has proven to be efficient. Two examples show that the method gives better estimates than other commonly used designs.     

Unbiased estimators for restricted adaptive cluster sampling

In adaptive cluster sampling the size of the final sample is random, thus creating design problems. To get round this, Brown (1994) and Brown & Manly (1998) proposed a modification of the method, placing a restriction on the size of the sample, and using standard but biased estimators for estimating the population mean. But in this paper a new unbiased estimator and an unbiased variance estimator are proposed, based on estimators proposed by Murthy (1957) and extended to sequential and adaptive sampling designs by Salehi & Seber (2001). The paper also considers a restricted version of the adaptive scheme of Salehi & Seber (1997a) in which the networks are selected without replacement, and obtains unbiased estimators. The method is demonstrated by a simple example. Using simulation from this example, the new estimators are shown to compare very favourably with the standard biased estimators.     

An efficient non-rejective implementation of the πps sampling designs

Poisson sampling is a method for unequal probabilities sampling with random sample size. There exist several implementations of the Poisson sampling design, with fixed sample size, which almost all are rejective methods, that is, the sample is not always accepted. Thus, the existing methods can be time-consuming or even infeasible in some situations. In this paper, a fast and non-rejective method, which is efficient even for large populations, is proposed and studied. The method is a new design for selecting a sample of fixed size with unequal inclusion probabilities. For the population of large size, the proposed design is very close to the strict πps sampling which is similar to the conditional Poisson (CP) sampling design, but the implementation of the design is much more efficient than the CP sampling. And the inclusion probabilities can be calculated recursively.     

Weighting for unequal selection probabilities in multilevel models

When multilevel models are estimated from survey data derived using multistage sampling, unequal selection probabilities at any stage of sampling may induce bias in standard estimators, unless the sources of the unequal probabilities are fully controlled for in the covariates. This paper proposes alternative ways of weighting the estimation of a two-level model by using the reciprocals of the selection probabilities at each stage of sampling. Consistent estimators are obtained when both the sample number of level 2 units and the sample number of level 1 units within sampled level 2 units increase. Scaling of the weights is proposed to improve the properties of the estimators and to simplify computation. Variance estimators are also proposed. In a limited simulation study the scaled weighted estimators are found to perform well, although non-negligible bias starts to arise for informative designs when the sample number of level 1 units becomes small. The variance estimators perform extremely well. The procedures are illustrated using data from the survey of psychiatric morbidity.     

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.     

Estimation of the parameters of Downton's bivariate exponential distribution using ranked set sampling scheme

The parameters of Downton's bivariate exponential distribution are estimated based on a ranked set sample. Parametric and nonparametric methods are considered. The suggested estimators are compared to the corresponding ones based on simple random sampling. It turns out that some of the suggested estimators are significantly more efficient than the ones based on simple random sampling.     

Poisson,modified poisson and collocated sampling

Poisson and collocated sampling are methods of selecting samples that allow for simple control as to which units are to be in sample and which not. They are particularly suitable for use when selecting more than one sample from the same framework. Sections 2 and 3 deal with Poisson sampling. Section 4 deals with modified Poisson sampling, a device to ensure that an empty sample is never selected. Sections 5, 6 and 7 deal with collocated sampling, another device for reducing the variance of sample size. In Section 8 a comparative study of variances and mean square errors is presented for a number of unequal probability sampling strategies.     

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.