Tez Türü: Doktora
Tezin Yürütüldüğü Kurum: İngiltere
Tez Danışmanı: Nikolai Leonenko
Tezin Onay Tarihi: 2022
Tezin Dili: İngilizce
Desteklendiği Program: Diğer
Özet:
Entropy is one of the most basic and significant descriptors of a probability distribution.
It is still a commonly used measure of uncertainty and randomness in information
theory and mathematical statistics. We study statistical inference for Shannon
and Rényi’s entropy-related functionals of multivariate Gaussian and Student-t distributions.
This thesis investigates three themes. First, we provide a non-parametric
test of goodness-of-fit for a class of multivariate generalized Gaussian distributions
based on maximum entropy principle via using the k-th nearest neighbour (NN) distance
estimator of the Shannon entropy. Its asymptotic unbiasedness and consistency
are demonstrated. Second, we show a class of estimators of the Rényi entropy based
on an independent identical distribution sample drawn from an unknown distribution
f on Rm. We focus on the maximum Rényi entropy principle for multivariate
Student-t and Pearson type II distributions. We also consider the entropy-based test
for multivariate Student-t distribution using the k-th NN distances estimator of entropy
and employ the properties of entropy estimators derived from NN distances.
Third, we introduce a new classes of unimodal rotational invariant directional distributions,
which generalize von Mises-Fisher distribution. We propose three types of
distributions in which one of them represents the axial data. We provide all of the
formula together with a short computational study of parameter estimators for each
new type via the method of moments and method of maximum likelihood. We also
offer the goodness-of-fit test to detect that the sample entries follow one of the introduced
generalized von Mises-Fisher distribution based on the maximum entropy
principle using the k-th NN distances estimator of Shannon entropy and to prove its
L2-consistence.