Asymptotic theory for order sampling

Sampling with varying probabilities, notably πps (= probability π proportional to size sampling, is one of many vehicles for utilization of auxiliary information. We introduce and study a novel general class of varying probabilities sampling schemes, called order sampling schemes. The main results concern asymptotic distributions of linear statistics. Even if the results lie on the theoretical side, they lay ground for applications of practical sampling interest. Rosén (1997), which is a follow-up to the present paper, shows that order sampling yields interesting contributions to the problem of finding simple and good πps schemes.     

Computing inclusion probabilities for order sampling

Rosèn [1997. J. Statist. Plann. Inference 62, 159–191] introduced order sampling schemes of fixed shape which have inclusion probabilities roughly proportional to given size measures ($\pi$ps schemes). Three particular cases where the fixed shape distributions are Pareto, exponential and uniform, respectively, are specially treated. In this paper, we give general algorithms for computing the first- and second-order inclusion probabilities for a general fixed shape order sampling scheme and explicit formulae for the three special cases. Identities are given that can be used to check the accuracy of the numerical results. Examples are included as well as some comments on improving the computational efficiency and accuracy of the algorithms.     

On inclusion probabilities and relative estimator bias for Pareto πps sampling

A means for utilizing auxiliary information in surveys is to sample with inclusion probabilities proportional to given size values, to use a πps design, preferably with fixed sample size. A novel candidate in that context is Pareto πps. This sampling scheme was derived by limit considerations and it works with a degree of approximation for finite samples. Desired and factual inclusion probabilities do not agree exactly, which in turn leads to some estimator bias. The central topic in this paper is to derive conditions for the bias to be negligible.Practically useful information on small sample behavior of Pareto πps can, to the best of our understanding, be gained only by numerical studies. Earlier investigations to that end have been too limited to allow general conclusions, while this paper reports on findings from an extensive numerical study. The chief conclusion is that the estimator bias is negligible in almost all situations met in survey practice.     

Order Sampling Design with Prescribed Inclusion Probabilities

Order sampling with fixed distribution shape is a class of sampling schemes with inclusion probabilities approximately proportional to given size measures. In a recent article, methods were provided to compute the exact first and second order inclusion probabilities numerically when the distribution shape is of the Pareto type. In the same article, procedures were also provided for this case to adjust the parameters to get predetermined inclusion probabilities. In this paper we prove the existence and uniqueness of a solution for the latter problem, in general for any order sampling of fixed distribution shape.     

An empirical bayes approach to a variables acceptance sampling plan problem

An empirical Bayes approach to a variables acceptance sampling plan problem is presented and an empirical Bayes rule is developed which is shown to be asymptotically optimal under general conditions. The problem considered is one in which the ratio of the costs of accepting defective items and rejecting non-defective items is specified. Sampling costs are not considered and the size of the sample taken from each lot is fixed and constant. The empirical Bayes estimation of the Bayes rule is shown to require the estimation of a conditional probability. An estimator for conditional probabilities of the form needed is derived and shown to have good asymptotic properties.     

On discrete distributions generated from drawing balls with bivariate labels

In this article, we consider urn models under three types of sampling schemes in terms of the probability-generating functions. The tools are developed for the evaluation of the distributions arising from the urn models along with some examples. Furthermore, the distributions are investigated by making use of the Bell polynomials. The results presented here provide a wide framework for developing the theory of urn models. As examples, we propose new class of probability models, which are called multiple-player problems and examine their properties. Finally, we treat the parameter estimation problem in the waiting time distributions with a numerical example.     

Ranked set sampling,coherent rankings and size-biased permutations

The paper examines the effect of the set size upon the performance (reciprocal variance) of balanced ranked set sampling for estimation of a population mean. Performance is shown to be monotone increasing with the set size for the wide class of ranking models that satisfy a property called coherence. This class includes perfect ranking as well as ranking by concomitant variable. Stochastic ranking models based upon size-biased permutations are also shown to satisfy coherence and, consequently, monotonicity.     

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.     

Horvitz-Thompson estimator and generalized πPS designs

Generalized πPS designs were defined by T.J. Rao (1972). Working with a general super-population model θ(g), the strategy consisting of GπPS design together with the associated Horvitz-Thompson estimator of the population total was shown to be better than two other well known strategies in T.J. Rao (1971,1972). In this note we prove the θ(g)-optimality of the strategy consisting of GπPS design together with the associated Horvitz-Thompson estimator in the entire class of p-unbiased strategies of the population total with expected sample size fixed. In view of our theorem the results of T.J. Rao follow as special cases.     

Inference in the Pareto distribution based on progressive Type II censoring with random removals

This study considers the estimation problem for the Pareto distribution based on progressive Type II censoring with random removals. The number of units removed at each failure time has a discrete uniform distribution. We are going to use the maximum likelihood method to obtain the estimator of parameter. The expectation and variance of the maximum likelihood estimator will be derived. The expected time required to complete such an experiment will be computed. Some numerical results of expected test times are carried out for this type of progressive censoring and other sampling schemes.     

On triple sampling scremes for estimating from binomial data with misclassification errors ∗

Previous work has been carried out on the use of double sampling schemes for inference from binomial data which are subject to misclassification. The double sampling scheme utilizes a sample of n units which are classified by both a fallible and a true device and another sample of n₂ units which are classified only by a fallible device. A triple sampljng scheme incorporates an additional sample of n_l units which are classified only by the true device. In this paper we apply this triple sampling to estimation from binomialdata. First estimation of a binomial proportion is discussed under different misclassification structures. Then, the problem of optimal allocation of sample sizes is discussed.     

A two stage procedure for selecting δ∗-optimal means in the normal model

Suppose π₁,…,π_k are k normal populations with π_i having unknown mean μ_i and unknown variance σ². The population π_i will be called δ^?-optimal (or good) if μ_i is within a specified amountδ? of the largest mean. A two stage procedure is proposed which selects a subset of the k populations and guarantees with probability at least P^? that the selected subset contains only δ?-optimal π_i ’s. In addition to screening out non-good populations the rule guarantees a high proportion of sufficiently good π_i’S will be selected.     

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.     

HORVITZ-THOMPSON AND DES RAJ ESTIMATORS REVISITED

In this paper it is shown that the generalized πPS sampling strategy consisting of the design with π_i, the probability of inclusion of the ith unit in the sample, proportional to the modified size together with the corresponding Horvitz-Thompson estimator (Rao, 1971), is superior to the symmetrized Des Raj strategy under a general super-population set-up for all values of the super-population parameter g, when the samples are of size two.     

Designing of a variables two-plan system by minimizing the average sample number

This article proposes a variables two-plan sampling system called tightened-normal-tightened (TNT) sampling inspection scheme where the quality characteristic follows a normal distribution or a lognormal distribution and has an upper or a lower specification limit. The TNT variables sampling inspection scheme will be useful when testing is costly and destructive. The advantages of the variables TNT scheme over variables single and double sampling plans and attributes TNT scheme are discussed. Tables are also constructed for the selection of parameters of known and unknown standard deviation variables TNT schemes for a given acceptable quality level (AQL) and limiting quality level (LQL). The problem is formulated as a nonlinear programming where the objective function to be minimized is the average sample number and the constraints are related to lot acceptance probabilities at AQL and LQL under the operating characteristic curve.     

Parameter estimation in nonlinear regression models 1

Fisher information contained in record values, inter-record times and their concomitants from a sample of fixed size is derived in general and explicit expressions are deduced for some specific known bivariate classes of distributions. A comparison between fixed sampling and inverse sampling schemes with equal number of records and concomitants is also carried out. We also consider parameter estimation based on bivariate records and a small simulation study is done.     

TWO-STAGE SAMPLING OVER TWO OCCASIONS

We propose a general sampling strategy for estimating a finite population total for the current occasion based on a two-stage sampling scheme adopted on the current and a previous occasion. This is in generalization of sampling schemes considered earlier by, among others, Singh (1968), Abraham, Khosla & Kathuria (1969), Singh & Kathuria (1969) and Kathuria (1975). Applications of the general strategy in specific situations are illustrated.     

On convergence of properly weighted samples to the target distribution

We consider importance sampling as well as other properly weighted samples with respect to a target distribution π$\pi$ from a different point of view. By considering the associated weights as sojourn times until the next jump, we define appropriate jump processes. When the original sample sequence forms an ergodic Markov chain, the associated jump process is an ergodic semi-Markov process with stationary distribution π$\pi$. In this respect, properly weighted samples behave very similarly to standard Markov chain Monte Carlo (MCMC) schemes in that they exhibit convergence to the target distribution as well. Indeed, some standard MCMC procedures like the Metropolis–Hastings algorithm are included in this context. Moreover, when the samples are independent and the mean weight is bounded above, we describe a slight modification in order to achieve exact (weighted) samples from the target distribution.     

Procedures and tables for the selection of tightened-normal-tightened (TNT-( n ; c 1 , c 2 )) sampling schemes

Procedures for the selection of tightened-normal-tightened (TNT) sampling schemes of type TNT-( n ; c 1 , c 2 ) are presented. It is shown that the TNT scheme provides a smaller sample size than the matched single-sampling plan.