Adaptive Penalized Likelihood method for Markov Chains

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 study, 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. Meanwhile, we demonstrated that the new model enjoys oracle properties, which means the estimated transition probability matrix has the same performance as the real one when given. Simulations show that our new method behave very well overall in comparison with various competitors. Real data analysis further convince the value of our proposed method.