ContinuousTime Models in Kernel Smoothing
Martin Sköld
Centre for Mathematical Sciences
Mathematical Statistics
Lund University,
1999
ISBN 9162838121
LUNFMS10091999

Abstract:

This thesis consists of five papers (Papers AE) treating problems in
nonparametric statistics, especially methods of kernel smoothing applied
to density estimation for stochastic processes (Papers AD) and regression
analysis (Paper E). A recurrent theme is to, instead of treating highly
positively correlated data as "asymptotically independent", take advantage
of local dependence structures by using continuoustime models.


In Papers A and B we derive expressions for the asymptotic variance of the
kernel density estimator of continuoustime multivariate stationary process
and relate convergence rates to the local character of the sample paths.
This is in Paper B applied to automatic selection of smoothing parameter
of the estimators. In Paper C we study a continuoustime version of a
leastsquares crossvalidation approach to selecting smoothing parameter,
and the impact the dependence structure of data has on the algorithm. A
correction factor is introduced to improve the methods performance for dependent
data. Papers D and E treat two statistical inverse problems where the interesting
data are not directly observable. In Paper D we consider the problem of
estimating the density of a stochastic process from noisy observations. We
introduce a method of smoothing the errors and show that by a suitably chosen
sampling scheme the convergence rate of independent data methods can be improved
upon. Finally in Paper E we treat a problem of nonparametric regression
analysis when data is sampled with a sizebias. Our method covers a wider
range of practical situations than previously studied methods and by viewing
the problem as a locally weighted leastsquares regression problem, extensions
to higher order polynomial estimators are straightforward.

Key words:

Density estimation, kernel smoothing, asymptotic variance, bandwidth selection,
dependent data, continuous time, errorsinvariables, deconvolution, size
bias.