On Sampling with Prescribed Second‐order Inclusion Probabilities

Abstract. Methods to perform fixed size sampling with prescribed second‐order inclusion probabilities are presented. The focus is on a conditional Poisson design of order 2, a CP(2) design. It is an exponential design of quadratic type and it is carefully studied. In particular, methods to find the suitable values of the parameters and methods to sample are described. Small examples illustrate.     

Probabilities of Failure in Mixed Acceptance Sampling Schemes

Mixed acceptance sampling schemes are commonly used for consumer protection. In a typical application, a sample is taken from a product lot and tested to check that the average value of the sample is not less than the labelled net content, and that there is no 'unreasonable' deficiency in any individual item. Exact and approximate expressions are obtained for the probabilities of failure or success under such schemes and under a variety of conditions.     

Inverse Sampling with Unequal Selection Probabilities

In some real situations the population of interest is divided into two groups, of which one contains only a few units. In other cases, the population may be considered as subdivided into two group', for example, if only a few units display a value of the variable of interest which is highly different from zero, while all the other units show a value equal to or near zero. In both cases, inverse sampling is more efficient than classical fixed sample-size designs to obtain the parameter estimators for the whole population as well as for its groups (e.g., Salehi and Seber, 2004 Salehi , M. M. , Seber , G. A. F. ( 2004 ). A general inverse sampling scheme and its application to adaptive cluster sampling . Austral. NZ J. Statist. 46 : 483 – 494 .[Crossref], [Web of Science ®] , [Google Scholar]). In fact, in this design the procedure selection continues until a prefixed number of units with the characteristic of interest is sampled. Since it is not known a priori to which group the population units belong, the sample size is a random variable. Christman and Lan (2001 Christman , M. C. , Lan , F. ( 2001 ). Inverse adaptive cluster sampling . Biometrics 57 : 1096 – 1105 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Salehi and Seber (2001 Salehi , M. M. , Seber , G. A. F. ( 2001 ). A new proof of Murthy's estimator which applies to sequential sampling . Austral. NZ J. Statist. 43 : 281 – 286 . [Google Scholar] 2004 Salehi , M. M. , Seber , G. A. F. ( 2004 ). A general inverse sampling scheme and its application to adaptive cluster sampling . Austral. NZ J. Statist. 46 : 483 – 494 .[Crossref], [Web of Science ®] , [Google Scholar]) considered inverse sampling designs when all the population units have equal selection probabilities. In this article, we consider the general case in which the units may have unequal probabilities of being included in the sample. In fact, in many real situations different units may have different selection probabilities because of some inherent features of the sampling procedure, or in order to obtain better estimates. We derive unbiased estimators of the totals of the two groups, their variance and the corresponding unbiased variance estimators in inverse sampling with replacement. Finally, we derive similar results for more complex designs, where the selection procedure stops before observing the prefixed number of units from the rare group.     

Pareto Sampling versus Sampford and Conditional Poisson Sampling

Abstract.  Pareto sampling was introduced by Rosén in the late 1990s. It is a simple method to get a fixed size π ps sample though with inclusion probabilities only approximately as desired. Sampford sampling, introduced by Sampford in 1967, gives the desired inclusion probabilities but it may take time to generate a sample. Using probability functions and Laplace approximations, we show that from a probabilistic point of view these two designs are very close to each other and asymptotically identical. A Sampford sample can rapidly be generated in all situations by letting a Pareto sample pass an acceptance–rejection filter. A new very efficient method to generate conditional Poisson ( CP ) samples appears as a byproduct. Further, it is shown how the inclusion probabilities of all orders for the Pareto design can be calculated from those of the CP design. A new explicit very accurate approximation of the second-order inclusion probabilities, valid for several designs, is presented and applied to get single sum type variance estimates of the Horvitz–Thompson estimator.     

Negative Dependence in Sampling

Abstract. The strong Rayleigh property is a new and robust negative dependence property that implies negative association; in fact it implies conditional negative association closed under external fields (CNA+). Suppose that and are two families of 0‐1 random variables that satisfy the strong Rayleigh property and let . We show that {Z_i} conditioned on is also strongly Rayleigh; this turns out to be an easy consequence of the results on preservation of stability of polynomials of Borcea & Brändén (Invent. Math., 177, 2009, 521–569). This entails that a number of important π ps sampling algorithms, including Sampford sampling and Pareto sampling, are CNA+. As a consequence, statistics based on such samples automatically satisfy a version of the Central Limit Theorem for triangular arrays.     

A New Sampling Design for Systematic Sampling

A sampling design called “Modified Systematic Sampling (MSS)” is proposed. In this design each unit has an equal probability of selection. Moreover, it works for both situations: N = nk or N ≠ nk. Consequently, the Linear Systematic Sampling (LSS) and Circular Systematic Sampling (CSS) become special cases of the proposed MSS design.     

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.     

A List Sequential Sampling Method Suitable for Real-Time Sampling

Abstract.  A flexible list sequential π ps sampling method is introduced and studied. It can reproduce any given sampling design without replacement, of fixed or random sample size. The method is a splitting method and uses successive updating of inclusion probabilities. The main advantage of the method is in real-time sampling situations where it can be used as a powerful alternative to Bernoulli and Poisson sampling and can give any desired second-order inclusion probabilities and thus considerably reduce the variability of the sample size.     

Matching Performance of Continuous Sampling Plans with Single Sampling Plans

Single sampling plans are widely used for appraising incoming product quality. However, for situations where a continuous product flow exists, lot-by-lot demarcations may not exist, and it may be necessary to use alternate procedures, such as CSP-1, for continuous processes. In this case, one would like to be able to understand how average performance of the continuous sampling procedures compares to the more commonly used single sampling plans.

In this study, a model is devised which can be used to relate plan performance between single sample lot acceptance procedures and Dodge's(1943) CSP-1 continuous sampling plan. It is shown that it is generally not possible to match up performance based upon operating characteristic curve expressions for the two plans. Instead, the plans are matched by equating expressions for π(p), the long run proportion of product which is accepted, under both procedures. This is shown to be equivalent to matching up properties on an average outgoing quality basis. The methodology may be extended for any derivative plan under MIL-STD-1235B (1982), the military standard for continuous acceptance sampling.     

Two-Stage Cluster Sampling with Ranked Set Sampling in the Secondary Sampling Frame

Motivated by a real-life problem, we develop a Two-Stage Cluster Sampling with Ranked Set Sampling (TSCRSS) design in the second stage for which we derive an unbiased estimator of population mean and its variance. An unbiased estimator of the variance of mean estimator is also derived. It is proved that the TSCRSS is more efficient—in the sense of having smaller variance—than the conventional two-stage cluster simple random sampling in which the second-stage sampling is with replacement. Using a simulation study on a real-life population, we show that the TSCRSS is more efficient than the conventional two-stage cluster sampling when simple random sampling without replacement is used in both stages.     

Statistical Education for Engineers: An Initial Task Force Report

This article describes a method for producing size-biased probability samples as originally proposed by Hanurav (1967) and Vijayan (1968). The complexity of the procedure has led to the development of microcomputer software that greatly facilitates the production of sampling plans as well as the computation of population estimates.     

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.     

Estimation of Finite Population Mean in Stratified Random Sampling with Two Auxiliary Variables under Double Sampling Design

In this article, a chain ratio-product type exponential estimator is proposed for estimating finite population mean in stratified random sampling with two auxiliary variables under double sampling design. Theoretical and empirical results show that the proposed estimator is more efficient than the existing estimators, i.e., usual stratified random sample mean estimator, Chand (1975) chain ratio estimator, Choudhary and Singh (2012) estimator, chain ratio-product-type estimator, Sahoo et al. (1993) difference type estimator, and Kiregyera (1984) regression-type estimator. Two data sets are used to illustrate the performances of different estimators.     

Estimation of a Parameter of Bivariate Pareto Distribution by Ranked Set Sampling

Ranked set sampling is applicable whenever ranking of a set of sampling units can be done easily by a judgement method or based on the measurement of an auxiliary variable on the units selected. In this work, we derive different estimators of a parameter associated with the distribution of the study variate Y, based on a ranked-set sample obtained by using an auxiliary variable X correlated with Y for ranking the sample units, when (X, Y) follows a bivariate Pareto distribution. Efficiency comparisons among these estimators are also made. Real-life data have been used to illustrate the application of the results obtained.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

基于双重抽样框的二阶段抽样调查方法研究

由于被调查对象的频繁变动,单一抽样框很难覆盖所有的目标单位。为了克服单一抽样框覆盖不完全的缺陷,在各阶段抽样调查采用双重（或多重）抽样框是一种有效的办法。对双重抽样框下的二阶段抽样估计方法进行了研究,得出简单随机抽样下的总体总值估计及其估计量方差,并利用拉格朗日函数求出双重抽样框重叠部分的最优抽样权重系数及各抽样阶段不同子域的样本容量,分析结果可为实际部门在双重抽样框下进行二阶段抽样调查提供相关的理论基础。     

A New Design on Attributes Single Sampling Plans

In this article, two new approaches are introduced to design attributes single plans, and the corresponding models are constructed separately. For Approach I, an algorithm is proposed to design sampling plans by setting a goal function to fulfill the two-point conditions on the operating characteristic curve. For Approach II, the plan parameters are solved by a nonlinear optimization model which minimizes the integration of the probability of acceptance in the interval from the producer's risk quality to the consumer's risk quality. Then numerical examples and discussions based on numerical computation results are given to illustrate the approaches, and tables of the designed plans under various conditions are provided. Moreover, a fact is given to be proved that there is a relation between the conventional design and the new approaches.     

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.     

A Comparison Of Two Adaptive Sampling Designs

Stratified sampling is a technique commonly used for ecological surveys. In this study there appears to be little gain in using a stratified design with adaptive cluster sampling. Two-phase adaptive sampling is preferable to adaptive cluster sampling. Even though two-phase adaptive sampling can give biased estimates, it is found that two-phase adaptive sampling has a lower MSE than adaptive cluster sampling for most populations.