Deconvolution problems in density estimation [Elektronische Ressource] / vorgelegt von Christian Wagner

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• Mathematics

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• Publié par: universitat_ulm
• Publié le: 01 janvier 2009
• Nombre de lectures: 19
• Poids de l'ouvrage: 1 Mo

Extrait

Deconvolution problems
in density estimation
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakultät für Mathematik und Wirtschaftswissenschaften
der Universität Ulm
vorgelegt von
Christian Wagner
aus
Öhringen
2009Amtierender Dekan: Prof. Dr. Werner Kratz
Erstgutachter: Prof. Dr. Ulrich Stadtmüller
Zweitgutachter: Prof. Dr. Volker Schmidt
Tag der Promotion: 8. Juni 2009Contents
Introduction and Summary iii
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Concepts in density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Focus of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Deconvolution of densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Results about approximative deconvolution estimators . . . . . . . . . . . . . . . . . vii
Results about contaminated-data-only models . . . . . . . . . . . . . . . . . . . . . viii
Aggregated data models and corresponding results . . . . . . . . . . . . . . . . . . . . . . x
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Density Estimation Methods 1
1.1 Direct density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Quality measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Asymptotic properties of the direct density estimation . . . . . . . . . . . . . 4
1.2 Density deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Classical consistency results . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Supersmooth target densities . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Approximative deconvolution methods . . . . . . . . . . . . . . . . . . . . . 14
1.3 Aggregated data models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Unknown Error Density 19
2.1 TAYLEX and SIMEX estimators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Justification of the appearing bias reduction . . . . . . . . . . . . . . . . . . 19
2.1.2 Consistency results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Modified variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.1 Model and estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.2 Consistency results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.4 Proof of Theorem 2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3 Additional error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.1 Model and estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.2 Consistency results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.3.3 Proof of Theorem 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
iii Contents
3 Aggregated Data Models 103
3.1 Estimators and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2 Consistency and minimax rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.3 Properties of the unweighted estimator . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.4 Aggregated data models in density deconvolution . . . . . . . . . . . . . . . . . . . 112
3.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.5.1 Proofs of Lemmas 3.2.1 and 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . 113
3.5.2 Proofs of Theorems 3.2.1 and 3.2.2. . . . . . . . . . . . . . . . . . . . . . . 116
3.5.3 Proof of Theorem 3.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.5.4 Proof of Theorem 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.5.5 Proof of Theorem 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A Appendix 141
A.1 Spaces of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.2 Integration theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.3 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.4 Characteristic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.5 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.6 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
B Auxiliary Results 149
List of Figures 155
List of Symbols 157
Bibliography 161
German Summary 165Introduction and Summary
Motivation
In many circumstances repeated measurements of a quantity are observed and one would like to
gain as much information as possible from these observations. Examples are the log return per
day of a company share, or the policy holders’ lifespans for life insurances. Other quantities of
interest could be participants’ blood pressures in a clinical study or households’ consumptions
of electricity per year. There are a variety of other situations where repeated observations are
possibleforinstanceinbiology, geology, otherfieldsofscienceorsocialscience. Instatisticsthese
repeated measurements are often modeled as realisations of random objects, so-called random
variables. Under this assumption the first characteristics of the data one might consider are the
mean or the variance. Yet, both values contain only partial information about the distribution
of a random variable, whereas complete information is given by the cumulative distribution
function, with its empirical counterpart, the so-called empirical distribution function. However,
when plotting the empirical distribution function, one only receives a step function and it is
hard to obtain more detailed information from this graph. In case of an absolutely continuous
random variable, the density function of the quantity of interest can be estimated to circumvent
this downside. Using an appropriate estimate of the density function allows for information
about modes, symmetry, and frequent values of the random variable to be gathered. Even more
information about modes or the change in frequency of the random variable’s values should be
attainable through an estimate of the density’s derivatives. Both the estimation of a density and
its derivatives will be addressed in this work. However, since both problems lead to comparable
considerations, the focus is on density estimation in the introduction.
Concepts in density estimation
There are two basically different approaches for estimating a density. The first is the parametric
density estimation approach, where one assumes that the observations come from a parametric
family of densities that has to be specified in advance. Then, the task is to estimate the pa-
rameters that fit the data best. The second approach is the nonparametric density estimation,
where one does not impose a certain functional form on the density. Instead, one tries to find an
estimate using only minor assumptions such as some smoothness of the density. The parametric
estimation procedure has advantages but also large drawbacks. Foremost, it is difficult to specify
an appropriate parametric family since this information might often be not directly accessible
from the situation, where the data was observed. However, a misspecification of the family of
densities will lead to an estimate that does not capture important structures of the density; a
fact that contradicts the objective of obtaining as much information from the data as possible.
Additionally, even if one insists on using a parametric approach, a nonparametric estimate will
give a good starting point to find an appropriate model. For this reason, nonparametric density
estimation will be studied in more detail here.
iiiiv Introduction and Summary
There are various density estimation procedures in nonparametric density estimation like
orthogonal series methods and histograms with fixed or random partition, among others. His-
tograms with fixed partition were for example studied in Révész [1971], for an introduction to
the other mentioned methods see for instance Prakasa Rao [1983]. However, the estimators most
commonly used in nonparametric density estimation are the so-called kernel density estimators
introduced in Rosenblatt [1956] and studied in more detail in Parzen [1962]. An estimator of this
type has already been studied in a less general setting in Akaike [1954]. Due to its importance
the explicit formula will be given here. For n independent random variables X ,...,X that1 n
are identically distributed as X, the kernel density estimator of the density f at the point ξ isX
given by nX1 ξ−Xjˆf (ξ) = K ,X nh h
j=1
wherehisapositiverealnumber,theso-calledbandwidth,andK(y)isaso-calledkernelfunction.
For instance, the kernel could be a standard normal density.
The kernel density estimator is not very sensitive to the choice of the kernel, in contrast
to the choice of the band