Adaptive Penalized Likelihood method for Markov Chains

TL;DR

This paper introduces an adaptive penalized likelihood method for Markov chains that improves transition matrix estimation accuracy, especially in high-dimensional cases, by extending adaptive Lasso to Markov models.

Contribution

The paper develops a novel penalized maximum likelihood approach for Markov chains, demonstrating oracle properties and superior performance over existing methods.

Findings

The new method achieves high accuracy in estimating transition matrices.

Simulation results show better performance compared to competitors.

Real data analysis confirms practical effectiveness.

Abstract

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.