Topics in multifractal measures, nonparametrics and biostatistics
Attila Frigyesi
Centre for Mathematical Sciences
Mathematical Statistics
Lund University,
2004
ISBN 9162859935
LUNFMS10142004

Abstract:


This thesis consists of four papers. The first two papers, which comprise
the main part of the thesis, deal with an unexpected connection between kernel
density estimators and dimension spectra for multifractal measures. The third
paper presents a fully automated expert system for the diagnosis of pulmonary
embolism from ventilation/perfusion scintigraphy. The final paper concerns
statistical properties of the parameters of the operational model of
pharmacological agonism, a widely applied model for doseresponse curves
in pharmacology.


In the first paper kernel density estimators for singular distributions are
studied. The density estimator f is a function of the sample size and the
bandwidth. It was found that the integral of H(f), where H is a suitable
"magnifying" functional, diverges as the sample increases to infinity and
the bandwidth goes to 0.


In the second paper it is shown that, for a particular choice of H, the velocity
with which the integral of H(f) diverges depends on the q:th generalized
HentschelProcaccia dimension of the measure from which the sample is drawn.
This gives a new way to estimate dimension spectra for multifractal measures.
An alternative kernelbased method that gives the correlation integral as
a special case is also studied, which enables the estimation of the correlation
dimension.


The classic way of estimating generalized fractal dimensions with the aid
of grids gives the generalized Rényi dimension. For q>1 this is
proved to be equivalent to the generalized HentschelProcaccia dimension.
For q<1 the Rényi dimension may depend on the choice of grid and
thus be different from the uniquely defined HentschelProcaccia dimension.
Examples of such measures are given.



Key words:

singular distribution functions, absolute continuity, kernel density estimates,
multifractal measures, dimension spectrum, fractal dimension estimation,
generalized dimensions, Rényi dimension, HentschelProcaccia dimension,
correlation dimension, box counting, automated method, pulmonary embolism,
V/Qscan, ventilation/perfusion scintigraphy, doseresponse curves,
concentrationresponse curves, operational model of pharmacological agonism,
confidence interval, mixed effects