The concept of energy in nonparametric statistics: Goodness-of-Fit problems and deconvolution

In this thesis the concept of energy is introduced from physics into statistics.
The energy of samples, which are drawn from statistical distributions, is defined in
a similar way as for discrete charge density distributions in electrostatics.
A system of two sets of point charges with opposite sign is in a state of minimum
energy if they are equally distributed. This property is used to construct new
nonparametric, multivariate Goodness-of-Fit tests, to check whether two samples
belong to the same parent distribution and to deconvolute distributions distorted
by measurement.
The statistical minimum energy configuration does not depend on the application
of the one-over-distance power law of the electrostatic potential. To increase the
power of the new approach other monotonic decreasing distance functions may be
chosen. We prove that the new energy technique is applicable to all distance functions
which have positive Fourier transforms. The proposed approach is binning-free.
It is especially powerfull in multidimensional applications and superior to most of
the common statistical methods in many concrete situations.