Testing for changes in the error distribution in functional linear models

TL;DR

This paper develops a change point detection method for error distributions in functional linear models with infinite-dimensional covariates, ensuring asymptotic distribution-freeness and consistency.

Contribution

It introduces a novel change point test for error distributions in functional linear models, addressing challenges from infinite-dimensional covariates and proving its asymptotic properties.

Findings

Test is asymptotically distribution-free.

Test is consistent for one-change point alternatives.

Estimator of the change point is consistent.

Abstract

We consider linear models with scalar responses and covariates from a separable Hilbert space. The aim is to detect change points in the error distribution, based on sequential residual empirical distribution functions. Expansions for those estimated functions are more challenging in models with infinite-dimensional covariates than in regression models with scalar or vector-valued covariates due to a slower rate of convergence of the parameter estimators. Yet the suggested change point test is asymptotically distribution-free and consistent for one-change point alternatives. In the latter case we also show consistency of a change point estimator.