Nonparametric Functional Central Limit Theorem for Time Series Regression with Application to Self-normalized...

Area of Science:

• Statistics
• Time Series Analysis
• Nonparametric Statistics

Background:

• Traditional kernel smoothing for nonparametric mean function inference relies on normal approximation.
• This method requires estimating asymptotic variance, involving extra nonparametric smoothing and bandwidth selection.
• Bandwidth sensitivity and poor small-sample performance limit the reliability of existing methods.

Purpose of the Study:

• To develop a robust, bandwidth-free inference procedure for nonparametric mean functions in time series.
• To construct accurate point-wise confidence intervals that overcome limitations of traditional methods.
• To establish the theoretical validity and practical superiority of the proposed approach.

Main Methods:

• Extending the self-normalized approach, originally for parametric inference, to the nonparametric setting.
• Establishing a functional central limit theorem for recursive nonparametric mean regression estimates.
• Developing a bandwidth-free method for constructing confidence intervals.

Main Results:

• The proposed self-normalized approach provides a bandwidth-free method for inference.
• A functional central limit theorem is established, showing the limiting process is a Gaussian process with non-stationary, dependent increments.
• Simulation studies demonstrate superior finite sample performance compared to traditional methods.

Conclusions:

• The self-normalized approach offers a more reliable and accurate method for nonparametric mean function inference in time series.
• This method alleviates issues related to bandwidth selection and small-sample approximations.
• The findings suggest a significant advancement in time series statistical inference.