29,419 research outputs found
Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling
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 here 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 deduce 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 means of a test population of
N=18902 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 to simple random sampling without
replacement.Comment: Accepted for publication in Biometrik
A simple variance estimator of change for rotating repeated surveys: an application to the EU-SILC household surveys
A common problem is to compare two cross-sectional estimates for the same study variable taken on two different waves or occasions, and to judge whether the change observed is statistically significant. This involves the estimation of the sampling variance of the estimator of change. The estimation of this variance would be relatively straightforward if cross-sectional estimates were based on the same sample. Unfortunately, samples are not completely overlapping, because of rotations used in repeated surveys. We propose a simple approach based on a multivariate (general) linear regression model. The variance estimator proposed is not a model-based estimator. We show that the estimator proposed is design consistent when the sampling fractions are negligible. It can accommodate stratified and two-stage sampling designs. The main advantage of the approach proposed is its simplicity and flexibility. It can be applied to a wide class of sampling designs and can be implemented with standard statistical regression techniques. Because of its flexibility, the approach proposed is well suited for the estimation of variance for the European Union Statistics on Income and Living Conditions surveys. It allows us to use a common approach for variance estimation for the different types of design. The approach proposed is a useful tool, because it involves only modelling skills and requires limited knowledge of survey sampling theory
On Improvement in Estimating Population Parameter(s) Using Auxiliary Information
The purpose of writing this book is to suggest some improved estimators using
auxiliary information in sampling schemes like simple random sampling and
systematic sampling.
This volume is a collection of five papers. The following problems have been
discussed in the book:
In chapter one an estimator in systematic sampling using auxiliary
information is studied in the presence of non-response. In second chapter some
improved estimators are suggested using auxiliary information. In third chapter
some improved ratio-type estimators are suggested and their properties are
studied under second order of approximation.
In chapter four and five some estimators are proposed for estimating unknown
population parameter(s) and their properties are studied.
This book will be helpful for the researchers and students who are working in
the field of finite population estimation.Comment: 63 pages, 8 tables. Educational Publishing & Journal of Matter
Regularity (Beijing
Properties of Design-Based Functional Principal Components Analysis
This work aims at performing Functional Principal Components Analysis (FPCA)
with Horvitz-Thompson estimators when the observations are curves collected
with survey sampling techniques. One important motivation for this study is
that FPCA is a dimension reduction tool which is the first step to develop
model assisted approaches that can take auxiliary information into account.
FPCA relies on the estimation of the eigenelements of the covariance operator
which can be seen as nonlinear functionals. Adapting to our functional context
the linearization technique based on the influence function developed by
Deville (1999), we prove that these estimators are asymptotically design
unbiased and consistent. Under mild assumptions, asymptotic variances are
derived for the FPCA' estimators and consistent estimators of them are
proposed. Our approach is illustrated with a simulation study and we check the
good properties of the proposed estimators of the eigenelements as well as
their variance estimators obtained with the linearization approach.Comment: Revised version for J. of Statistical Planning and Inference (January
2009
Confidence bands for Horvitz-Thompson estimators using sampled noisy functional data
When collections of functional data are too large to be exhaustively
observed, survey sampling techniques provide an effective way to estimate
global quantities such as the population mean function. Assuming functional
data are collected from a finite population according to a probabilistic
sampling scheme, with the measurements being discrete in time and noisy, we
propose to first smooth the sampled trajectories with local polynomials and
then estimate the mean function with a Horvitz-Thompson estimator. Under mild
conditions on the population size, observation times, regularity of the
trajectories, sampling scheme, and smoothing bandwidth, we prove a Central
Limit theorem in the space of continuous functions. We also establish the
uniform consistency of a covariance function estimator and apply the former
results to build confidence bands for the mean function. The bands attain
nominal coverage and are obtained through Gaussian process simulations
conditional on the estimated covariance function. To select the bandwidth, we
propose a cross-validation method that accounts for the sampling weights. A
simulation study assesses the performance of our approach and highlights the
influence of the sampling scheme and bandwidth choice.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ443 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Coupling methods for multistage sampling
Multistage sampling is commonly used for household surveys when there exists
no sampling frame, or when the population is scattered over a wide area.
Multistage sampling usually introduces a complex dependence in the selection of
the final units, which makes asymptotic results quite difficult to prove. In
this work, we consider multistage sampling with simple random without
replacement sampling at the first stage, and with an arbitrary sampling design
for further stages. We consider coupling methods to link this sampling design
to sampling designs where the primary sampling units are selected
independently. We first generalize a method introduced by [Magyar Tud. Akad.
Mat. Kutat\'{o} Int. K\"{o}zl. 5 (1960) 361-374] to get a coupling with
multistage sampling and Bernoulli sampling at the first stage, which leads to a
central limit theorem for the Horvitz--Thompson estimator. We then introduce a
new coupling method with multistage sampling and simple random with replacement
sampling at the first stage. When the first-stage sampling fraction tends to
zero, this method is used to prove consistency of a with-replacement bootstrap
for simple random without replacement sampling at the first stage, and
consistency of bootstrap variance estimators for smooth functions of totals.Comment: Published at http://dx.doi.org/10.1214/15-AOS1348 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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