Testing for the Markov property in time series via deep conditional generative learning.

Area of Science:

• Statistics
• Machine Learning
• Time Series Analysis

Background:

• The Markov property is fundamental in time series analysis, crucial for modeling sequential data.
• Testing this property and determining the order of Markov models are essential tasks in statistical inference.

Purpose of the Study:

• To propose a novel nonparametric test for the Markov property in high-dimensional time series.
• To extend this test for sequential determination of Markov model order.
• To establish theoretical guarantees for the test's performance.

Main Methods:

• Utilizing deep conditional generative learning to estimate conditional density functions.
• Developing a doubly robust test statistic with nonparametric estimation and parametric convergence rate.
• Employing sample splitting and cross-fitting for enhanced test consistency.

Main Results:

• The proposed test asymptotically controls the type-I error rate.
• The test demonstrates power approaching one, indicating high detection capability.
• Theoretical analysis provides sharp upper bounds on estimation errors.

Conclusions:

• The deep learning-based nonparametric test is effective for high-dimensional time series.
• The method offers a robust approach for Markov property testing and model order selection.
• Demonstrated efficacy through simulations and real-world data applications.