Kolmogorov Entropy for Convergence Rate in Incomplete Functional Time Series: Application to Percentile and...

Area of Science:

• Statistics
• Functional Data Analysis
• Time Series Analysis

Background:

• Estimating convergence rates in functional data analysis (FDA) is complex, often requiring advanced mathematical tools.
• Existing methods for functional time series analysis face challenges with incomplete data and correlation structures.

Purpose of the Study:

• To establish Kolmogorov entropy as a key tool for characterizing convergence rates in FDA.
• To derive uniform convergence rates for local linear estimation of conditional distribution and quantile functions.
• To develop a central limit theorem for these estimators under general conditions.

Main Methods:

• Utilizing Kolmogorov entropy to analyze the convergence rates of local linear estimators.
• Developing theoretical proofs for uniform convergence and establishing a central limit theorem.
• Modeling time series correlation with the ergodic assumption and handling missing data using a missing at random mechanism.

Main Results:

• Kolmogorov entropy is shown to be fundamental for characterizing local linear estimation convergence rates.
• Uniform convergence rates are derived for conditional cumulative distribution and quantile functions.
• A central limit theorem is established for the proposed estimators, applicable to incomplete functional time series.

Conclusions:

• The proposed methods provide a robust framework for functional data analysis, particularly for incomplete time series.
• The use of Kolmogorov entropy offers a novel approach to understanding estimation convergence in FDA.
• Monte Carlo simulations confirm the practical utility and finite sample performance of the developed estimators.