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Lardin Pauline - One of the best experts on this subject based on the ideXlab platform.

  • Variance estimation and asymptotic confidence bands for the mean Estimator of sampled functional data with high entropy unequal probability sampling designs
    2013
    Co-Authors: Cardot Hervé, Goga Camelia, Lardin Pauline
    Abstract:

    For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson Estimator can be approximated by the H\'ajek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent Estimator of the variance function of the Horvitz-Thompson Estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the H\'ajek formula. Finally, the accuracy of the proposed variance Estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week.Comment: Revised for the Scandinavian Journal of Statistic

  • Variance estimation and asymptotic confidence bands for the mean Estimator of sampled functional data with high entropy unequal probability sampling designs
    HAL CCSD, 2013
    Co-Authors: Cardot Hervé, Goga Camelia, Lardin Pauline
    Abstract:

    En révision pour Scandinavian J. of StatisticsFor fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson Estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent Estimator of the variance function of the Horvitz-Thompson Estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the Hájek formula. Finally, the accuracy of the proposed variance Estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week

  • Estimation de synchrones de consommation électrique par sondage et prise en compte d'information auxiliaire
    HAL CCSD, 2012
    Co-Authors: Lardin Pauline
    Abstract:

    In this thesis, we are interested in estimating the mean electricity consumption curve. Since the study variable is functional and storage capacities are limited or transmission cost are high survey sampling techniques are interesting alternatives to signal compression techniques. We extend, in this functional framework, estimation methods that take into account available auxiliary information and that can improve the accuracy of the Horvitz-Thompson Estimator of the mean trajectory. The first approach uses the auxiliary information at the estimation stage, the mean curve is estimated using model-assisted Estimators with functional linear regression models. The second method involves the auxiliary information at the sampling stage, considering πps (unequal probability) sampling designs and the functional Horvitz-Thompson Estimator. Under conditions on the entropy of the sampling design the covariance function of the Horvitz-Thompson Estimator can be estimated with the Hájek approximation extended to the functional framework. For each method, we show, under weak hypotheses on the sampling design and the regularity of the trajectories, some asymptotic properties of the Estimator of the mean curve and of its covariance function. We also establish a functional central limit theorem.Next, we compare two methods that can be used to build confidence bands. The first one is based on simulations of Gaussian processes and is assessed rigorously. The second one uses bootstrap techniques in a finite population framework which have been adapted to take into account the functional nature of the dataDans cette thèse, nous nous intéressons à l'estimation de la synchrone de consommation électrique (courbe moyenne). Etant donné que les variables étudiées sont fonctionnelles et que les capacités de stockage sont limitées et les coûts de transmission élevés, nous nous sommes intéressés à des méthodes d'estimation par sondage, alternatives intéressantes aux techniques de compression du signal. Nous étendons au cadre fonctionnel des méthodes d'estimation qui prennent en compte l'information auxiliaire disponible afin d'améliorer la précision de l'estimateur de Horvitz-Thompson de la courbe moyenne de consommation électrique. La première méthode fait intervenir l'information auxiliaire au niveau de l'estimation, la courbe moyenne est estimée à l'aide d'un estimateur basé sur un modèle de régression fonctionnelle. La deuxième l'utilise au niveau du plan de sondage, nous utilisons un plan à probabilités inégales à forte entropie puis l'estimateur de Horvitz-Thompson fonctionnel. Une estimation de la fonction de covariance est donnée par l'extension au cadre fonctionnel de l'approximation de la covariance donnée par Hájek. Nous justifions de manière rigoureuse leur utilisation par une étude asymptotique. Pour chacune de ces méthodes, nous donnons, sous de faibles hypothèses sur les probabilités d'inclusion et sur la régularité des trajectoires, les propriétés de convergence de l'estimateur de la courbe moyenne ainsi que de sa fonction de covariance. Nous établissons également un théorème central limite fonctionnel. Afin de contrôler la qualité de nos estimateurs, nous comparons deux méthodes de construction de bande de confiance sur un jeu de données de courbes de charge réelles. La première repose sur la simulation de processus gaussiens. Une justification asymptotique de cette méthode sera donnée pour chacun des estimateurs proposés. La deuxième utilise des techniques de bootstrap qui ont été adaptées afin de tenir compte du caractère fonctionnel des donnée

  • Estimation de synchrones de consommation électrique par sondage et prise en compte d'information auxiliaire
    2012
    Co-Authors: Lardin Pauline, Cardot Hervé, Goga Camelia
    Abstract:

    Dans cette thèse, nous nous intéressons à l'estimation de la synchrone de consommation électrique (courbe moyenne). Etant donné que les variables étudiées sont fonctionnelles et que les capacités de stockage sont limitées et les coûts de transmission élevés, nous nous sommes intéressés à des méthodes d'estimation par sondage, alternatives intéressantes aux techniques de compression du signal. Nous étendons au cadre fonctionnel des méthodes d'estimation qui prennent en compte l'information auxiliaire disponible afin d'améliorer la précision de l'estimateur de Horvitz-Thompson de la courbe moyenne de consommation électrique. La première méthode fait intervenir l'information auxiliaire au niveau de l'estimation, la courbe moyenne est estimée à l'aide d'un estimateur basé sur un modèle de régression fonctionnelle. La deuxième l'utilise au niveau du plan de sondage, nous utilisons un plan à probabilités inégales à forte entropie puis l'estimateur de Horvitz-Thompson fonctionnel. Une estimation de la fonction de covariance est donnée par l'extension au cadre fonctionnel de l'approximation de la covariance donnée par Hájek. Nous justifions de manière rigoureuse leur utilisation par une étude asymptotique. Pour chacune de ces méthodes, nous donnons, sous de faibles hypothèses sur les probabilités d'inclusion et sur la régularité des trajectoires, les propriétés de convergence de l'estimateur de la courbe moyenne ainsi que de sa fonction de covariance. Nous établissons également un théorème central limite fonctionnel. Afin de contrôler la qualité de nos estimateurs, nous comparons deux méthodes de construction de bande de confiance sur un jeu de données de courbes de charge réelles. La première repose sur la simulation de processus gaussiens. Une justification asymptotique de cette méthode sera donnée pour chacun des estimateurs proposés. La deuxième utilise des techniques de bootstrap qui ont été adaptées afin de tenir compte du caractère fonctionnel des donnéesIn this thesis, we are interested in estimating the mean electricity consumption curve. Since the study variable is functional and storage capacities are limited or transmission cost are high survey sampling techniques are interesting alternatives to signal compression techniques. We extend, in this functional framework, estimation methods that take into account available auxiliary information and that can improve the accuracy of the Horvitz-Thompson Estimator of the mean trajectory. The first approach uses the auxiliary information at the estimation stage, the mean curve is estimated using model-assisted Estimators with functional linear regression models. The second method involves the auxiliary information at the sampling stage, considering pps (unequal probability) sampling designs and the functional Horvitz-Thompson Estimator. Under conditions on the entropy of the sampling design the covariance function of the Horvitz-Thompson Estimator can be estimated with the Hájek approximation extended to the functional framework. For each method, we show, under weak hypotheses on the sampling design and the regularity of the trajectories, some asymptotic properties of the Estimator of the mean curve and of its covariance function. We also establish a functional central limit theorem.Next, we compare two methods that can be used to build confidence bands. The first one is based on simulations of Gaussian processes and is assessed rigorously. The second one uses bootstrap techniques in a finite population framework which have been adapted to take into account the functional nature of the dataDIJON-BU Doc.électronique (212319901) / SudocSudocFranceF

Pauline Lardin - One of the best experts on this subject based on the ideXlab platform.

  • variance estimation and asymptotic confidence bands for the mean Estimator of sampled functional data with high entropy unequal probability sampling designs
    Scandinavian Journal of Statistics, 2014
    Co-Authors: Hervé Cardot, Camelia Goga, Pauline Lardin
    Abstract:

    For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson Estimator can be approximated by the Hajek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that it asymptotically provides a uniformly consistent Estimator of the variance function of the Horvitz-Thompson Estimator of the mean function. Rates of convergence to the true variance function are given for rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes whose variance function is given by the Hajek formula. Finally, the accuracy of the proposed variance Estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week.

  • Variance estimation and asymptotic confidence bands for the mean Estimator of sampled functional data with high entropy unequal probability sampling designs
    2013
    Co-Authors: Hervé Cardot, Camelia Goga, Pauline Lardin
    Abstract:

    For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson Estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent Estimator of the variance function of the Horvitz-Thompson Estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the Hájek formula. Finally, the accuracy of the proposed variance Estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week.

Hervé Cardot - One of the best experts on this subject based on the ideXlab platform.

  • variance estimation and asymptotic confidence bands for the mean Estimator of sampled functional data with high entropy unequal probability sampling designs
    Scandinavian Journal of Statistics, 2014
    Co-Authors: Hervé Cardot, Camelia Goga, Pauline Lardin
    Abstract:

    For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson Estimator can be approximated by the Hajek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that it asymptotically provides a uniformly consistent Estimator of the variance function of the Horvitz-Thompson Estimator of the mean function. Rates of convergence to the true variance function are given for rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes whose variance function is given by the Hajek formula. Finally, the accuracy of the proposed variance Estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week.

  • Variance estimation and asymptotic confidence bands for the mean Estimator of sampled functional data with high entropy unequal probability sampling designs
    2013
    Co-Authors: Hervé Cardot, Camelia Goga, Pauline Lardin
    Abstract:

    For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson Estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent Estimator of the variance function of the Horvitz-Thompson Estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the Hájek formula. Finally, the accuracy of the proposed variance Estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week.

  • Horvitz Thompson Estimators for functional data asymptotic confidence bands and optimal allocation for stratified sampling
    Biometrika, 2011
    Co-Authors: Hervé Cardot, Etienne Josserand
    Abstract:

    When dealing with very large datasets of functional data, survey sampling approaches are useful in order to obtain Estimators of simple functional quantities, without being obliged to store all the data. We propose a Horvitz--Thompson Estimator of the mean trajectory. In the context of a superpopulation framework, we prove, under mild regularity conditions, that we obtain uniformly consistent Estimators of the mean function and of its variance function. With additional assumptions on the sampling design we state a functional central limit theorem and obtain asymptotic confidence bands. Stratified sampling is studied in detail, and we also obtain a functional version of the usual optimal allocation rule, considering a mean variance criterion. These techniques are illustrated by a test population of Ne18 902 electricity meters for which we have individual electricity consumption measures every 30 minutes over one week. We show that stratification can substantially improve both the accuracy of the Estimators and reduce the width of the global confidence bands compared with simple random sampling without replacement. Copyright 2011, Oxford University Press.

Goga Camelia - One of the best experts on this subject based on the ideXlab platform.

  • Variance estimation and asymptotic confidence bands for the mean Estimator of sampled functional data with high entropy unequal probability sampling designs
    2013
    Co-Authors: Cardot Hervé, Goga Camelia, Lardin Pauline
    Abstract:

    For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson Estimator can be approximated by the H\'ajek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent Estimator of the variance function of the Horvitz-Thompson Estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the H\'ajek formula. Finally, the accuracy of the proposed variance Estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week.Comment: Revised for the Scandinavian Journal of Statistic

  • Variance estimation and asymptotic confidence bands for the mean Estimator of sampled functional data with high entropy unequal probability sampling designs
    HAL CCSD, 2013
    Co-Authors: Cardot Hervé, Goga Camelia, Lardin Pauline
    Abstract:

    En révision pour Scandinavian J. of StatisticsFor fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson Estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent Estimator of the variance function of the Horvitz-Thompson Estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the Hájek formula. Finally, the accuracy of the proposed variance Estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week

  • Estimation de synchrones de consommation électrique par sondage et prise en compte d'information auxiliaire
    2012
    Co-Authors: Lardin Pauline, Cardot Hervé, Goga Camelia
    Abstract:

    Dans cette thèse, nous nous intéressons à l'estimation de la synchrone de consommation électrique (courbe moyenne). Etant donné que les variables étudiées sont fonctionnelles et que les capacités de stockage sont limitées et les coûts de transmission élevés, nous nous sommes intéressés à des méthodes d'estimation par sondage, alternatives intéressantes aux techniques de compression du signal. Nous étendons au cadre fonctionnel des méthodes d'estimation qui prennent en compte l'information auxiliaire disponible afin d'améliorer la précision de l'estimateur de Horvitz-Thompson de la courbe moyenne de consommation électrique. La première méthode fait intervenir l'information auxiliaire au niveau de l'estimation, la courbe moyenne est estimée à l'aide d'un estimateur basé sur un modèle de régression fonctionnelle. La deuxième l'utilise au niveau du plan de sondage, nous utilisons un plan à probabilités inégales à forte entropie puis l'estimateur de Horvitz-Thompson fonctionnel. Une estimation de la fonction de covariance est donnée par l'extension au cadre fonctionnel de l'approximation de la covariance donnée par Hájek. Nous justifions de manière rigoureuse leur utilisation par une étude asymptotique. Pour chacune de ces méthodes, nous donnons, sous de faibles hypothèses sur les probabilités d'inclusion et sur la régularité des trajectoires, les propriétés de convergence de l'estimateur de la courbe moyenne ainsi que de sa fonction de covariance. Nous établissons également un théorème central limite fonctionnel. Afin de contrôler la qualité de nos estimateurs, nous comparons deux méthodes de construction de bande de confiance sur un jeu de données de courbes de charge réelles. La première repose sur la simulation de processus gaussiens. Une justification asymptotique de cette méthode sera donnée pour chacun des estimateurs proposés. La deuxième utilise des techniques de bootstrap qui ont été adaptées afin de tenir compte du caractère fonctionnel des donnéesIn this thesis, we are interested in estimating the mean electricity consumption curve. Since the study variable is functional and storage capacities are limited or transmission cost are high survey sampling techniques are interesting alternatives to signal compression techniques. We extend, in this functional framework, estimation methods that take into account available auxiliary information and that can improve the accuracy of the Horvitz-Thompson Estimator of the mean trajectory. The first approach uses the auxiliary information at the estimation stage, the mean curve is estimated using model-assisted Estimators with functional linear regression models. The second method involves the auxiliary information at the sampling stage, considering pps (unequal probability) sampling designs and the functional Horvitz-Thompson Estimator. Under conditions on the entropy of the sampling design the covariance function of the Horvitz-Thompson Estimator can be estimated with the Hájek approximation extended to the functional framework. For each method, we show, under weak hypotheses on the sampling design and the regularity of the trajectories, some asymptotic properties of the Estimator of the mean curve and of its covariance function. We also establish a functional central limit theorem.Next, we compare two methods that can be used to build confidence bands. The first one is based on simulations of Gaussian processes and is assessed rigorously. The second one uses bootstrap techniques in a finite population framework which have been adapted to take into account the functional nature of the dataDIJON-BU Doc.électronique (212319901) / SudocSudocFranceF

Chauvet Guillaume - One of the best experts on this subject based on the ideXlab platform.

  • A cautionary note on the Hanurav-Vijayan sampling algorithm
    2021
    Co-Authors: Chauvet Guillaume
    Abstract:

    We consider the Hanurav-Vijayan sampling design, which is the default method programmed in the SURVEYSELECT procedure of the SAS software. We prove that it is equivalent to the Sunter procedure, but is capable of handling any set of inclusion probabilities. We prove that the Horvitz-Thompson Estimator is not generally consistent under this sampling design. We propose a conditional Horvitz-Thompson Estimator, and prove its consistency under a non-standard assumption on the first-order inclusion probabilities. Since this assumption seems difficult to control in practice, we recommend not to use the Hanurav-Vijayan sampling design

  • Exponential inequalities for sampling designs
    2020
    Co-Authors: Chauvet Guillaume, Gerber Mathieu
    Abstract:

    In this work we introduce a general approach, based on the mar-tingale representation of a sampling design and Azuma-Hoeffding's inequality , to derive exponential inequalities for the difference between a Horvitz-Thompson Estimator and its expectation. Applying this idea, we establish such inequalities for Chao's procedure, Till{\'e}'s elimination procedure, the generalized Midzuno method as well as for Brewer's method. As a by-product, we prove that the first three sampling designs are (conditionally) negatively associated. For such sampling designs, we show that that the inequality we obtain is usually sharper than the one obtained by applying known results for negatively associated random variables

  • Exponential inequalities for sampling designs
    HAL CCSD, 2020
    Co-Authors: Chauvet Guillaume, Gerber Mathieu
    Abstract:

    In this work we introduce a general approach, based on the mar-tingale representation of a sampling design and Azuma-Hoeffding's inequality , to derive exponential inequalities for the difference between a Horvitz-Thompson Estimator and its expectation. Applying this idea, we establish such inequalities for Chao's procedure, Tillé's elimination procedure, the generalized Midzuno method as well as for Brewer's method. As a by-product, we prove that the first three sampling designs are (conditionally) negatively associated. For such sampling designs, we show that that the inequality we obtain is usually sharper than the one obtained by applying known results for negatively associated random variables

  • A comparison of pivotal sampling and unequal probability sampling with replacement
    'Elsevier BV', 2017
    Co-Authors: Chauvet Guillaume, Ruiz-gazen Anne
    Abstract:

    International audienceWe prove that any implementation of pivotal sampling is more efficient than multinomial sampling. This yields the weak consistency of the Horvitz Thompson Estimator and the existence of a conservative variance Estimator. A small simulation study supports our findings

  • Estimation Under Cross-Classified Sampling With Application to a Childhood Survey
    'Informa UK Limited', 2017
    Co-Authors: Juillard Hélène, Chauvet Guillaume, Ruiz-gazen Anne
    Abstract:

    International audienceThe cross-classified sampling design consists in drawing samples from a two-dimensional population, independently in each dimension. Such design is commonly used in consumer price index surveys and has been recently applied to draw a sample of babies in the French Longitudinal Survey on Childhood, by crossing a sample of maternity units and a sample of days. We propose to derive a general theory of estimation for this sampling design. We consider the Horvitz–Thompson Estimator for a total, and show that the cross-classified design will usually result in a loss of efficiency as compared to the widespread two-stage design. We obtain the asymptotic distribution of the Horvitz–Thompson Estimator and several unbiased variance Estimators. Facing the problem of possibly negative values, we propose simplified nonnegative variance Estimators and study their bias under a super-population model. The proposed Estimators are compared for totals and ratios on simulated data. An application on real data from the French Longitudinal Survey on Childhood is also presented, and we make some recommendations. Supplementary materials for this article are available online