Scalar-on-Function Mode Estimation Using Entropy and Ergodic Properties of Functional Time Series Data

Area of Science:

• Statistics
• Time Series Analysis
• Functional Data Analysis

Background:

• Standard estimators for conditional mode often rely on mixing processes, which can be mathematically complex.
• Ergodicity offers a more tractable assumption, characterized by Kolmogorov-Sinai entropy, reflecting process dynamics and fluctuations.
• Functional time series (fts) present unique challenges due to their complex mathematical properties.

Purpose of the Study:

• To develop and analyze a novel recursive L1 estimator for the conditional mode.
• To investigate the estimator's properties when the input variable is in a pseudo-metric space.
• To provide a robust alternative to existing methods using an ergodicity assumption.

Main Methods:

• Construction of a recursive L1 estimator under an ergodicity assumption for functional time series.
• Derivation of asymptotic properties, including convergence rate and Borel-Cantelli (BC) consistency.
• Specialization of convergence rates to independent cases, kernel methods, and vector-valued scenarios.

Main Results:

• The proposed recursive L1 estimator is shown to be asymptotically consistent (BC consistent).
• Specific convergence rates are derived for various functional time series settings.
• Numerical experiments confirm the estimator's superiority over existing methods.

Conclusions:

• The novel recursive L1 estimator provides a robust and effective approach for conditional mode estimation in pseudo-metric spaces.
• Ergodicity offers a practical and mathematically sound alternative to mixing assumptions in functional time series analysis.
• The estimator's strong performance on simulated and real data validates its utility.