Supervisors info:
Τρέβεζας Σάμης, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Summary:
The present thesis aims at presenting the application of the Bootstrap method on dependent data. The Bootstrap method was introduced in the late ’70s by the eminent statistician Bradley Efron, bringing a revolution in Statistics and many other related fields. In its initial formulation, this method considered independent data. Here we are dealing with data that exhibit a particular type of dependence, that of a discrete-time Markov chain. In the first part of the thesis, we provide a theorem-proof type of presentation of the basic Markov chain theory, both with discrete and arbitrary state space. In the second part, we focus on the problem of estimating the transition matrix of a Markov chain based on an observed path of the chain. We first examine how we can use asymptotic methods to tackle this problem, presenting both the classical and the Bayesian framework. Then, we show how we can exploit the Bootstrap method to approach this problem. We delve into both the
frequentist and the Bayesian frameworks of tackling this problem, and we give detailed
proofs of the main asymptotic results that validate these procedures. Finally, we apply the
above theoretical methods in simulated and real data.
Keywords:
Discrete-Time Markov Chains, Estimation of Transition Probability Matrix, Bootstrap Method, Bayesian Bootstrap
