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On unequal probability sampling designs
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The main objective in sampling is to select a sample from a population in order to estimate some unknown population parameter, usually a total or a mean of some interesting variable. When the units in the population do not have the same probability of being included in a sample, it is called unequal probability sampling. The inclusion probabilities are usually chosen to be proportional to some auxiliary variable that is known for all units in the population. When unequal probability sampling is applicable, it generally gives much better estimates than sampling with equal probabilities. This thesis consists of six papers that treat unequal probability sampling from a finite population of units.

A random sample is selected according to some specified random mechanism called the sampling design. For unequal probability sampling there exist many different sampling designs. The choice of sampling design is important since it determines the properties of the estimator that is used. The main focus of this thesis is on evaluating and comparing different designs. Often it is preferable to select samples of a fixed size and hence the focus is on such designs.

It is also important that a design has a simple and efficient implementation in order to be used in practice by statisticians. Some effort has been made to improve the implementation of some designs. In Paper II, two new implementations are presented for the Sampford design.

In general a sampling design should also have a high level of randomization. A measure of the level of randomization is entropy. In Paper IV, eight designs are compared with respect to their entropy. A design called adjusted conditional Poisson has maximum entropy, but it is shown that several other designs are very close in terms of entropy.

A specific situation called real time sampling is treated in Paper III, where a new design called correlated Poisson sampling is evaluated. In real time sampling the units pass the sampler one by one. Since each unit only passes once, the sampler must directly decide for each unit whether or not it should be sampled. The correlated Poisson design is shown to have much better properties than traditional methods such as Poisson sampling and systematic sampling.

Place, publisher, year, edition, pages
Umeå: Department of Mathematics and Mathematical Statistics, Umeå University , 2010. , 31 + 6 papers p.
Keyword [en]
conditional Poisson sampling, correlated Poisson sampling, entropy, extended Sampford sampling, Horvitz-Thompson estimator, inclusion probabilities, list-sequential sampling, non-rejective implementation, Pareto sampling, Poisson sampling, probability functions, ratio estimator, real-time sampling, repeated Poisson sampling, Sampford sampling, sampling designs, splitting method, unequal probability sampling
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:umu:diva-33701ISBN: 978-91-7264-999-6 (print)OAI: oai:DiVA.org:umu-33701DiVA: diva2:317506
Public defence
2010-05-28, MIT-huset, MA 121, Umeå universitet, Umeå, 13:15 (English)
Opponent
Supervisors
Available from: 2010-05-07 Created: 2010-05-03 Last updated: 2010-05-18Bibliographically approved
List of papers
1. Repeated poisson sampling
Open this publication in new window or tab >>Repeated poisson sampling
2009 (English)In: Statistics and Probability Letters, ISSN 0167-7152, Vol. 79, 760-764 p.Article in journal (Refereed) Published
Abstract [en]

The repeated Poisson sampling design is a new design for selecting a sample of fixed size with unequal inclusion probabilities. The design is very close to the conditional Poisson sampling design, but the implementation of the RP-design is much more efficient.

Place, publisher, year, edition, pages
Elsevier, 2009
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-19665 (URN)10.1016/j.spl.2008.10.027 (DOI)
Available from: 2009-03-09 Created: 2009-03-09 Last updated: 2010-05-07Bibliographically approved
2. Non-rejective implementations of the Sampford sampling design
Open this publication in new window or tab >>Non-rejective implementations of the Sampford sampling design
2009 (English)In: Journal of Statistical Planning and Inference, ISSN 0378-3758, Vol. 139, 2111-2114 p.Article in journal (Refereed) Published
Abstract [en]

Sampford sampling is a method for unequal probability sampling. There exist several implementations of the Sampford sampling design which all are rejective methods, i.e. 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 list-sequential method, which works in all situations, is presented. The method is a modification of a previously existing rejective list-sequential method. Another list-sequential implementation of Sampford sampling is also presented.

Place, publisher, year, edition, pages
Elsevier Bv., 2009
Keyword
Sampford sampling, Unequal probability sampling, List-sequential sampling, Non-rejective implementation
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-19667 (URN)10.1016/j.jspi.2008.09.015 (DOI)
Available from: 2009-03-09 Created: 2009-03-09 Last updated: 2010-05-18Bibliographically approved
3. On a generalization of Poisson sampling
Open this publication in new window or tab >>On a generalization of Poisson sampling
2010 (English)In: Journal of Statistical Planning and Inference, ISSN 0378-3758, Vol. 140, no 4, 982-991 p.Article in journal (Other academic) Published
Abstract [en]

In real-time sampling, the units of a population pass a sampler one by one. Alternatively the sampler may successively visit the units of the population. Each unit passes only once and at that time it is decided whether or not it should be included in the sample. The goal is to take a sample and efficiently estimate a population parameter. The list sequential sampling method presented here is called correlated Poisson sampling. The method is an alternative to Poisson sampling, where the units are sampled independently with given inclusion probabilities. Correlated Poisson sampling uses weights to create correlations between the inclusion indicators. In that way it is possible to reduce the variation of the sample size and to make the samples more evenly spread over the population. Simulation shows that correlated Poisson sampling improves the efficiency in many cases.

Keyword
Correlated Bernoulli sampling, Correlated Poisson sampling, Horvitz–Thompson ratio estimator, Inclusion probabilities, List sequential sampling, Real-time sampling, Simulation, Splitting method
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-8389 (URN)10.1016/j.jspi.2009.09.024 (DOI)000273659900011 ()
Note

Även utgiven som: Research Report in Mathematical Statistics, ISSN 1653-0829, 2007:2.

Available from: 2008-01-20 Created: 2008-01-20 Last updated: 2015-10-02Bibliographically approved
4. Entropy of unequal probability sampling designs
Open this publication in new window or tab >>Entropy of unequal probability sampling designs
2010 (English)In: Statistical Methodology, ISSN 1572-3127, E-ISSN 1878-0954, Vol. 7, no 2, 84-97 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Elsevier Bv., 2010
Keyword
Entropy, Probability functions, Sampling designs, Unequal probability sampling
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-31085 (URN)10.1016/j.stamet.2009.10.005 (DOI)
Available from: 2010-01-28 Created: 2010-01-28 Last updated: 2010-05-07Bibliographically approved
5. An extension of Sampford's method for unequal probability sampling
Open this publication in new window or tab >>An extension of Sampford's method for unequal probability sampling
2011 (English)In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 38, no 2, 377-392 p.Article in journal (Other academic) Published
Abstract [en]

Sampford's (1967) 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 (1997a,b), is also extended. The extended Pareto design is shown to be close to the extended Sampford design.

Place, publisher, year, edition, pages
John Wiley & Sons, 2011
Keyword
conditional Poisson sampling, Pareto sampling, pivotal method, Sampford sampling, unequal probability sampling
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-33700 (URN)10.1111/j.1467-9469.2010.00707.x (DOI)
Available from: 2010-05-03 Created: 2010-05-03 Last updated: 2014-03-19Bibliographically approved
6. Efficient sampling when the inclusion probabilities do not sum to an integer
Open this publication in new window or tab >>Efficient sampling when the inclusion probabilities do not sum to an integer
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Usually when sampling with unequal inclusion probabilities a fixed sample size is desirable. Then it is required that the sum of the inclusion probabilities should equal the sample size. Sometimes it is not suitable to re-scale the inclusion probabilities to sum to an integer. It is shown that every unequal probability sampling design for fixed sample size can be extended to this situation. The cost for not re-scaling the inclusion probabilities is that we have to accept a small variation in sample size. Different strategies for estimation under these circumstances are also given.

Keyword
conditional inclusion probabilities, extended Sampford sampling, Horvitz-Thompson estimator, ratio estimator, unequal probability sampling
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-33699 (URN)
Available from: 2010-05-03 Created: 2010-05-03 Last updated: 2010-05-07Bibliographically approved

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