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Random sampling with repulsion
Cette thèse vise à explorer le concept de répulsion et à l’exploiter afin de développer de nouveaux plans de sondages.
Le premier chapitre est une revue de la littérature qui porte sur l’échantillonnage en population finie. Il est constitué de deux parties distinctes. Dans la première, trois principes visant à guider les praticiens dans leur conception d’un plan de sondage sont préconisés : la maximisation de l’entropie, la surreprésentation et la restriction. Dans la seconde partie, on fait une revue de la littérature concernant l’échantillonnage équilibré ainsi que de ses différentes variantes spatiales.
Le deuxième chapitre de cette thèse porte sur l’échantillonnage en population finie. En cherchant à modéliser l’écart entre deux unités sélectionnées, on obtient une famille de plans de sondage dont on peut ajuster la répulsion. Les cas de tailles fixes et aléatoires sont traités séparément et dans les deux cas, on propose des plans dont les probabilités d’inclusion d’ordre un sont constantes et celles d’ordre deux sont connues sous forme close.
Le troisième chapitre traite un problème similaire mais dans le cas d’un intervalle de la droite réelle. Le plan d’échantillonnage est en fait un processus ponctuel que l’on désire stationnaire et on modélise les distances entre deux occurrences du processus. Lorsque les distances entre deux unités successives sont indépendantes, le processus est de renouvellement et la taille de l’échantillon est en général aléatoire. On peut imposer la taille fixe et, dans ce cas, les intervalles entre deux unités ne sont plus indépendants mais seulement échangeables. Dans les deux cas, de taille fixe et de taille aléatoire, la famille de processus dévelopée englobe les processus les plus basiques, respectivement de Poisson et binomial ainsi que le tirage systématique comme cas limite. Ainsi, on dispose d’une famille de processus dépendant d’un paramètre qui permet de faire varier continument la répulsion.
Finalement, on traite du problème de l’échantillonnage spatial à intensité variable. En particulier, on développe un processus ponctuel répulsif basé sur les processus déterminantaux, que l’on peut échantillonner sur un domaine géométriquement complexe et dont on connaît analytiquement les intensités jointes d’ordre un et deux. L’estimateur d’Horvitz-Thompson est sans biais et on peut estimer sans biais sa variance., In this thesis, we explore the concept of repulsion and exploit it to develop new sampling designs.
The first chapter is a review of the literature about sampling in finite population and consists of two distinct parts. In the first one, we suggest three principles as guidelines for designing a survey: randomization, overrepresentation and restriction. In the second part, we review the literature about balanced sampling and its spatial versions. We intend to give the reader insights about some models and the corresponding optimal sampling design.
The second chapter of this thesis relates to sampling in finite population. By modelling the distribution of the number of units between two selected units, we obtain a family of sampling designs with a tunable repulsion. The cases of fixed and random sample size are considered separately. In both cases, the first order inclusion probabilities are equal and the second order inclusion probabilities are known under closed-form.
The third chapter is about repulsive sampling in an interval of the real line. In continuous spaces, a sampling design is a point process. We aim at imposing the process to be stationary which is similar to imposing equal first order inclusion probabilities in finite population. We focus on the intervals between two successive occurrences of the process, called spacings. If spacings between units are independent, the point process is a renewal process and the sample size is random. We can impose a fixed size sample but the spacings are only exchangeable and not independent. In both cases, the proposed family of point processes encompasses the most basic processes, that is, binomial in the case of fixed sample size and Poisson in the case of binomial sample size; it also includes the systematic sampling as a limiting case. Thus we obtain a family of processes depending on a parameter that allows us to continuously tune the repulsion between selections.
Finally, we consider the problem of developing a repulsive spatial point process with varying prescribed intensity. In particular, we propose a repulsive point process based on Determinantal processes from which we can easily sample on a geometrically complex domain. The first and second order inclusion densities are known under closed-form and are strictly positive. The Horvitz-Thompson estimator is unbiased and we can unbiasedly estimate its variance.
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• Publication
Quasi-Systematic Sampling From a Continuous Population
(2017)
A specific family of point processes are introduced that allow to select samples for the purpose of estimating the mean or the integral of a function of a real variable. These processes, called quasi-systematic processes, depend on a tuning parameter $r>0$ that permits to control the likeliness of jointly selecting neighbor units in a same sample. When $r$ is large, units that are close tend to not be selected together and samples are well spread. When $r$ tends to infinity, the sampling design is close to systematic sampling. For all $r > 0$, the first and second-order unit inclusion densities are positive, allowing for unbiased estimators of variance. Algorithms to generate these sampling processes for any positive real value of $r$ are presented. When $r$ is large, the estimator of variance is unstable. It follows that $r$ must be chosen by the practitioner as a trade-off between an accurate estimation of the target parameter and an accurate estimation of the variance of the parameter estimator. The method's advantages are illustrated with a set of simulations.
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Sampling Designs From Finite Populations With Spreading Control Parameters
(2018-1-10)
We present a new family of sampling designs in finite population based on the use of chain processes and of multivariate discrete distributions. In Bernoulli sampling, the number of non-selected units between two selected units has a geometric distribution, while, in simple random sampling, it has a negative hypergeometric distribution. We propose to replace these distributions by more general ones, which enables us to include a tuning parameter for the joint inclusion probabilities that have a relatively simple form. An effect of repulsion or attraction can then be added in the selection of the units in such a way that a large set of new designs are defined that include Bernoulli sampling, simple random sampling and systematic sampling. A set of simulations show the interest of the method.
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Quasi systematic sampling
(2015-6-25)
We present a family of sampling designs depending on a integer parameter r. Then, simple random sampling is a particular case of this sampling design, namely when r = 1 and the systematic sampling design is the limiting case when r tends to the infinity. For every r > 0, this sampling design has the important property to have first and second order densities which are tractable and positive. Thus, the Horvitz-Thompson estimator is unbiased and the estimator of the variance of the Horvitz-Thompson estimator is also unbiased. This family of sampling design can be used in finite population or on any finite interval of R.
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