Functional delta residuals and applications to simultaneous confidence bands of moment based statistics

TL;DR

This paper introduces functional delta residuals that mimic the covariance structure of limit processes, enabling the construction of valid simultaneous confidence bands for transformed functional parameters using bootstrap methods.

Contribution

It provides an explicit construction of functional delta residuals and a bootstrap approach for accurate confidence bands in moment-based functional statistics.

Findings

Bootstrap method accurately estimates quantiles of the limit process.

Constructed confidence bands achieve correct coverage rates in simulations.

Application to functional Cohen's d, skewness, and kurtosis demonstrates practical utility.

Abstract

Given a functional central limit (fCLT) for an estimator and a parameter transformation, we construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the limit process of the functional delta method. An explicit construction of these residuals for transformations of moment-based estimators and a multiplier bootstrap fCLT for the resulting functional delta residuals are proven. The latter is used to consistently estimate the quantiles of the maximum of the limit process of the functional delta method in order to construct asymptotically valid simultaneous confidence bands for the transformed functional parameters. Performance of the coverage rate of the developed construction, applied to functional versions of Cohen's d, skewness and kurtosis, is illustrated in simulations and their application to test Gaussianity is discussed.