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On the distance between some πps sampling designs
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0003-1524-0851
2007 (English)In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 97, no 1-3, 79-97 p.Article in journal (Refereed) Published
Abstract [en]

Asymptotic distances between probability distributions appearing in πps sampling theory are studied. The distributions are Poisson, Conditional Poisson (CP), Sampford, Pareto, Adjusted CP and Adjusted Pareto sampling. We start with the Kullback-Leibler divergence and the Hellinger distance and derive a simpler distance measure using a Taylor expansion of order two. This measure is evaluated first theoretically and then numerically, using small populations. The numerical examples are also illustrated using a multidimensional scaling technique called principal coordinate analysis (PCO). It turns out that Adjusted CP, Sampford, and adjusted Pareto are quite close to each other. Pareto is a bit further away from these, then comes CP and finally Poisson which is rather far from all the others.

Place, publisher, year, edition, pages
Dordrecht: Reidel , 2007. Vol. 97, no 1-3, 79-97 p.
Keyword [en]
Asymptotic distance, conditional poisson sampling, hellinger distance, inclusion probabilities, kullback-leibler divergence, pareto sampling, principal coordinate analysis, sampford sampling, target probabilities
URN: urn:nbn:se:umu:diva-19696DOI: 10.1007/s10440-007-9134-xOAI: diva2:202369
Available from: 2009-03-10 Created: 2009-03-10 Last updated: 2016-03-07Bibliographically approved
In thesis
1. Contributions to the theory of unequal probability sampling
Open this publication in new window or tab >>Contributions to the theory of unequal probability sampling
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five papers related to the theory of unequal probability sampling from a finite population. Generally, it is assumed that we wish to make modelassisted inference, i.e. the inclusion probability for each unit in the population is prescribed before the sample is selected. The sample is then selected using some random mechanism, the sampling design. Mostly, the thesis is focused on three particular unequal probability sampling designs, the conditional Poisson (CP-) design, the Sampford design, and the Pareto design. They have different advantages and drawbacks: The CP design is a maximum entropy design but it is difficult to determine sampling parameters which yield prescribed inclusion probabilities, the Sampford design yields prescribed inclusion probabilities but may be hard to sample from, and the Pareto design makes sample selection very easy but it is very difficult to determine sampling parameters which yield prescribed inclusion probabilities. These three designs are compared probabilistically, and found to be close to each other under certain conditions. In particular the Sampford and Pareto designs are probabilistically close to each other. Some effort is devoted to analytically adjusting the CP and Pareto designs so that they yield inclusion probabilities close to the prescribed ones. The result of the adjustments are in general very good. Some iterative procedures are suggested to improve the results even further. Further, balanced unequal probability sampling is considered. In this kind of sampling, samples are given a positive probability of selection only if they satisfy some balancing conditions. The balancing conditions are given by information from auxiliary variables. Most of the attention is devoted to a slightly less general but practically important case. Also in this case the inclusion probabilities are prescribed in advance, making the choice of sampling parameters important. A complication which arises in the context of choosing sampling parameters is that certain probability distributions need to be calculated, and exact calculation turns out to be practically impossible, except for very small cases. It is proposed that Markov Chain Monte Carlo (MCMC) methods are used for obtaining approximations to the relevant probability distributions, and also for sample selection. In general, MCMC methods for sample selection does not occur very frequently in the sampling literature today, making it a fairly novel idea.

Place, publisher, year, edition, pages
Umeå: Institutionen för Matematik och Matematisk Statistik, Umeå universitet, 2009. 26 p.
balanced sampling, conditional Poisson sampling, inclusion probabilities, maximum entropy, Markov chain Monte Carlo, Pareto sampling, Sampford sampling, unequal probability sampling.
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
urn:nbn:se:umu:diva-22459 (URN)978-91-7264-760-2 (ISBN)
Public defence
2009-06-04, MA121, MIT-huset, Umeå Universitet, 90187 Umeå, Umeå, 13:15 (English)
Available from: 2009-05-13 Created: 2009-05-11 Last updated: 2016-03-07Bibliographically approved

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Lundquist, Anders
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