Prediction intervals and bands with improved coverage for functional data under noisy discrete observation.

Area of Science:

• Statistics
• Functional Data Analysis

Background:

• Functional data analysis (FDA) involves analyzing curves observed at discrete, irregular points with noise.
• Reconstructing individual curves using prediction intervals and bands is a key FDA task.
• Standard FDA methods estimate curve properties and use Gaussian assumptions for prediction sets, but often fail to achieve nominal coverage.

Purpose of the Study:

• To investigate the cause of coverage failure in standard FDA prediction sets.
• To propose a computationally feasible method to improve prediction region coverage.
• To extend the method to covariate-adjusted functional models.

Main Methods:

• Estimating mean and covariance functions using penalized splines.
• Deriving conditional distributions under Gaussian assumptions.
• Developing a novel sandwich estimator for spline estimator covariance.
• Sampling from the approximate distribution of spline estimators to account for model uncertainty.

Main Results:

• Identified coverage failure in standard FDA prediction sets as a key issue.
• Proposed a new method that significantly improves prediction region coverage.
• Demonstrated the method's applicability to covariate-adjusted models.

Conclusions:

• The proposed method offers a reliable approach for constructing functional data prediction regions.
• Accounting for spline estimator uncertainty is crucial for accurate prediction intervals.
• This work advances practical applications of functional data analysis.