Thesis Defence: Adaptive Penalized Likelihood method for Markov Chain and its Oracle Properties

Yining Zhou, supervised by Dr. Xiaoping Shi, will defend their thesis titled “Adaptive Penalized Likelihood method for Markov Chain and its Oracle Properties” in partial fulfillment of the requirements for the degree of Master of Science in Mathematics.

An abstract for Yining Zhou’s thesis is included below.

Defences are open to all members of the campus community as well as the general public. Registration is not required for in person defences.

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ABSTRACT

The Maximum Likelihood Estimation (MLE) and Likelihood Ratio Test (LRT) are widely used methods for estimating the transition probability matrix in Markov chains and identifying significant relationships between transitions, such as equality. However, the estimated transition probability matrix derived from MLE lacks accuracy compared to the real one, and LRT is inefficient in high-dimensional Markov chains.

In this thesis, we extended the adaptive Lasso technique from linear models to Markov chains and proposed a novel model by applying penalized maximum likelihood estimation to optimize the estimation of the transition probability matrix. We name this novel methodology McALasso. Additionally, we demonstrated that McALasso enjoys oracle properties, meaning the estimated transition probability matrix performs as well as the true one when given. Simulations show that our new method performs very well overall in comparison with various competitors. Real data analysis further convinces us of the value of our proposed method.