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Testing for changes in the error distribution in functional linear models

Natalie Neumeyer, Leonie Selk

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.

Testing for changes in the error distribution in functional linear models

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.

Paper Structure

This paper contains 13 sections, 4 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model, test statistic and main result under the null
  3. Discussion of assumptions and examples
  4. VC-class condition
  5. Bracketing number condition
  6. Fixed one-change point alternative: consistency of the test and change point estimator
  7. Finite sample properties
  8. Concluding remarks
  9. Proofs
  10. Proof of Theorem \ref{['theo1']}
  11. Proof of Lemma \ref{['lem1']}
  12. Proof of Lemma \ref{['lem-consistency']}
  13. Proof of Lemma \ref{['lem-est-consistency']}

Key Result

Theorem 2.2

Under the assumptions (a.1)--(a.3), and thus the process $(\hat{G}_n(t,z))_{t\in[0,1],z\in\mathbb{R}}$ converges weakly to $( G(t,F(z)))_{t\in[0,1],z\in\mathbb{R}}$.

Figures (2)

  • Figure 1: Rejection probabilities with a change in the error distribution from $\mathcal{N}(0,1)$ to $\tilde{F}_{1,\delta}$ (left), to $\tilde{F}_{2,\delta}$ (middle) and to $\tilde{F}_{3,\delta}$ (right). The dotted line marks the $5\%$ level.
  • Figure 2: Rejection probabilities with a change in the error distribution from $\mathcal{N}(0,0.5^2)$ to $\mathcal{N}(0,(0.5+\delta)^2)$. The dotted line marks the $5\%$ level.

Theorems & Definitions (8)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Example 3.1
  • Lemma 3.2
  • Example 3.3
  • Lemma 4.1
  • Lemma 4.2