Testing for changes in the error distribution in functional linear models
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Neumeyer, Natalie; Selk, Leonie Article — Published Version Testing for changes in the error distribution in functional linear models Statistical Papers Provided in Cooperation with: Springer Nature Suggested Citation: Neumeyer, Natalie; Selk, Leonie (2025) : Testing for changes in the error distribution in functional linear models, Statistical Papers, ISSN 1613-9798, Springer, Berlin, Heidelberg, Vol. 66, Iss. 2, https://doi.org/10.1007/s00362-024-01656-9 This Version is available at: https://hdl.handle.net/10419/318563 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
Statistical Papers (2025) 66:33 https://doi.org/10.1007/s00362-024-01656-9 REGULAR ARTICLE Testing for changes in the error distribution in functional linear models Natalie Neumeyer1·Leonie Selk1 Received: 3 April 2024 / Revised: 6 November 2024 © The Author(s) 2025 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. Keywords Change-points ·Functional data analysis ·Regularized function estimators ·Regression ·Residual processes Mathematics Subject Classification Primary 62R10; Secondary 62G10 ·62G30 1 Introduction We consider a functional linear model Y=α+X,β+εwith scalar response Y and covariates Xfrom a separable Hilbert space, e.g. L2([0,1]). Structural changes in the distribution can appear, even when the parameters αand βdo not change. For this reason we focus on detecting changes in the error distribution. If the errors were observable one could use the classical test (and change point estimators) based on the difference of the sequential empirical distribution functions of the first nt and the last n−nterror terms from a sample of nobservations, see Csörgö et al. (1997), Picard (1985), Carlstein (1988), Dümbgen (1991), Hariz et al. (2005) and Hariz et al. (2007). In a regression model those tests have to be based on estimated residuals ˆε=Y−ˆα−X,ˆ β. Similar tests have been considered by Bai (1994)in BNatalie Neumeyer [email protected] Leonie Selk [email protected] 1Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany 0123456789().: V,-vol 123
33 Page 2 of 17 N. Neumeyer, L. Selk the context of ARMA-models, by Koul (1996) in the context of nonlinear time series, by Ling (1998) for nonstationary autoregressive models, and by Neumeyer and Van Keilegom (2009) and Selk and Neumeyer (2013) for nonparametric independent and time series regression models. Typically the asymptotic distribution is derived using asymptotic expansions of residual-based empirical distribution functions. For models with functional covariates those expansions can be problematic because inner products X,ˆ β−βappear and those can have a slow rate of convergence [see Cardot et al. (2007), Shang and Cheng (2015), Yeon et al. (2023)]. However, we show that under very simple non-restrictive assumptions those terms cancel for the suggested change point test statistic and thus the asymptotic distribution is the same as based on true (unobserved) errors. Change point testing and estimation for functional data, and for the parameter in functional linear models have been considered in the literature, but not for the error distribution. Tests for changes in the functional mean and in the parameter function of autoregressive models are considered in chapters 6 and 14 in Horváth and Kokoszka (2012). Berkes et al. (2009) propose a CUSUM testing procedure to detect a change in the mean of functional observations. They apply projections on principal components of the data to estimate the mean. Aue et al. (2009) extend this result and introduce an estimator for the change point in this model and derive its limit distribution. Aston and Kirch (2012) consider the same type of model with epidemic changes and dependent data. Aue et al. (2018) consider how to detect and date structural breaks in the mean of functional observations without the application of dimension reduction techniques (as functional principal component analysis). Aue et al. (2014) propose a monitoring procedure to detect structural changes in functional linear models with functional response, allowing for dependence in the data, including functional autoregressive processes. They test for a change in the regression operator, which is the analogue to our β, based on functional principal component analysis. A linear regression model with scalar response is considered in Horváth et al. (2024) who propose a tests for the detection of multiple change points in the regression parameter. The regressors in their model can be functional and can include lagged values of the response. The paper is organized as follows. In Sect. 2we define the test statistic and present model assumptions to obtain the asymptotically distribution-free hypothesis test. In Sect. 3we discuss the assumptions on the parameter estimators and some examples. In Sect. 4consistency of the test as well as of a change point estimator is considered in the context of one change point. Finite sample properties are shown in Sect. 5. Section6concludes the paper, in particular with an outlook on goodness-of-fit testing. The proofs are given in the appendix. 2 Model, test statistic and main result under the null Let Hbe a separable Hilbert space with inner product ·,·, corresponding norm · and Borel-sigma field. Let (Xi,Yi),i=1,...,n, be an independent sample of (H× R)-valued random variables defined on the same probability space with probability 123
Testing for changes in the error distribution… Page 3 of 17 33 measure P. The data are modeled as functional linear model Yi=α+Xi,β+εi,i=1,...,n, with scalar response Yiand H-valued covariate Xi, and with parameters α∈R,βinH. The covariates X1,...,Xnare assumed to be iid with EXi<∞, and the errors ε1,...,ε nare independent, centered, and independent of the covariates. Our aim is to test for change-points in the error distribution. In this section we consider the test statistic under the null hypothesis, where the errors are identically distributed. Let ˆαand ˆ βdenote estimators for the parameters α∈Rand β∈H. We build residuals ˆεi=Yi−ˆα−Xi,ˆ β,i=1...,n. The test statistic Tn=sup t∈[0,1] sup z∈R|ˆ Gn(t,z)| based on the process ˆ Gn(t,z)=nt(n−nt) n3/2ˆ Fnt(z)−˜ Fnt(z), compares for each k=1,...,n−1 the empirical distribution functions ˆ Fk(z)=1 k k i=1 I{ˆεi≤z},˜ Fk(z)=1 n−k n i=k+1 I{ˆεi≤z} of the first kand last n−kresiduals, respectively. Note that one can write ˆ Gn(t,z)=nt n1/2ˆ Fnt(z)−ˆ Fn(z). For the asymptotic distribution of the test statistic under the null hypothesis we assume the following conditions. Let Pdenote the distribution of (X1,ε 1). (a.1) |ˆα−α|=oP(1),ˆ β−β=oP(1) (a.2) Let ε1,...,ε nbe independent and identically distributed with cdf Fthat is Hölder-continuous of order γ∈(0,1]with Hölder-constant c. (a.3) Pˆ β−β∈B→1asn→∞for a class B⊂Hsuch that the function class F={(x,e)→ I{e≤v+x,b} | v∈R,b∈B} is P-Donsker. Remark 2.1 The assumptions are very mild and in particular less restrictive than typical assumptions for asymptotic distribution of residual-based empirical processes, even for finite-dimensional covariates. In assumption (a.1) only consistency is needed, no rates of convergence. Typically in the literature about residual-based procedures a bounded error density is assumed, see e.g. Akritas and Van Keilegom (2001). Then 123
33 Page 4 of 17 N. Neumeyer, L. Selk (a.2) is fulfilled for γ=1, but (a.2) is less restrictive in the cases γ∈(0,1). Suitable conditions for the general assumption (a.3) are discussed in Sect.3. One possibility for H=L2([0,1])is to assume smoothness of βwhich is a typical assumption. If γ∈(1 2,1]in assumption (a.2), and βis in a Sobolev-space with third derivatives, (a.2) holds for the estimator ˆ βfrom Yuan and Cai (2010). This estimator can also be applied for smaller γin (a.2) if higher smoothness of βis assumed. Define the process Gnas ˆ Gn, but based on the true errors instead of residuals, i.e. Gn(t,z)=nt n1/2Fnt(z)−Fn(z) with Fnt(z)=1 nt nt i=1 I{εi≤z}.(2.1) Further let Gbe a completely tucked Brownian sheet, i.e. a centered Gaussian process on [0,1]2with covariance structure Cov(G(s,u), G(t,v))=(s∧t−st)(u∧v−uv). Theorem 2.2 Under the assumptions (a.1)–(a.3), sup t∈[0,1] sup z∈R|ˆ Gn(t,z)−Gn(t,z)|=oP(1), (2.2) and thus the process (ˆ Gn(t,z))t∈[0,1],z∈Rconverges weakly to (G(t,F(z)))t∈[0,1],z∈R. The proof of (2.2) in the theorem is given in the appendix. The weak convergence of Gn is a classical result, see Bickel and Wichura (1971), Shorack and Wellner (1986). With the continuous mapping theorem one obtains the asymptotic distribution of the test statistic Tnunder the null hypothesis of no change-point, which is the distribution of T=supt,u∈[0,1]|G(t,u)|because Fis continuous. The test statistic is asymptotically distribution-free with the same limit distribution as for corresponding changepoint tests based on iid observations (not residuals). Let ¯α∈(0,1)and qbe the (1−¯α)- quantile of T. Then the test that rejects the null hypothesis if Tn>qhas asymptotic level ¯α. Consistency is considered in Sect.4. Remark 2.3 The choice of Tnas a Kolmogorov–Smirnov type test statistic is not mandatory. In principle, any continuous functional of the process ˆ Gncan be considered. The most common ones, besides Tn, are of Cramér-von–Mises type, e. g. Tn,2=supt∈[0,1]R|ˆ Gn(t,z)|2dF(z)or Tn,3=1 0R|ˆ Gn(t,z)|2dF(z)dt.The asymptotic distribution of these test statistics under the null hypothesis also follows from Theorem 2.2 and with R|ˆ Gn(t,z)|2dF(z)→R|G(t,F(z))|2dF(z)=1 0|G(t,x)|2dx, 123
Testing for changes in the error distribution… Page 5 of 17 33 and thus these test statistics are asymptotically distribution-free as well. However, Tn,2and Tn,3contain the unknown quantity Fand must therefore be modified in order to be applied. This can be done by replacing the integral with the sample mean: ˜ Tn,2=supt∈[0,1]1 nn i=1|ˆ Gn(t,ˆεi)|2and ˜ Tn,3=1 0 1 nn i=1|ˆ Gn(t,ˆεi)|2dt. 3 Discussion of assumptions and examples To show validity of the Donsker-class assumption (a.3) there are sufficient conditions on covering numbers or bracketing numbers. We discuss some specific conditions on the class B, examples for Hilbert spaces H, and estimators for the parameter function βthat fulfill the conditions. 3.1 VC-class condition Assumption (a.3) can be derived from a VC-function class condition formulated as follows. Assume that Pˆ β−β∈B→1asn→∞for a class B⊂Hsuch that the class of maps {H→R,x→x,b+v|b∈B,v ∈R}(3.1) is a VC-subgraph class. By definition then {{(x,e)∈H×R|e≤x,b+v}|b∈B,v ∈R} is a VC-class of sets. The class Ffrom (a.3) is the class of the corresponding indicator functions and (a.3) is fulfilled by Theorems 8.19 and 9.2 in Kosorok (2008). Example 3.1 We consider the Hilbert space H=L2([0,1])with inner product g,h=1 0g(t)h(t)dt and norm g=(1 0g2(t)dt)1/2. For the parameter function βwe assume sparsity as in Lee and Park (2012). Let (φj)j∈Nbe a basis of Hand assume β=j∈Jβjφjfor some finite, but unknown index set J. Lee and Park (2012) consider the estimator ˆ β=k j=1ˆ βjφjwith (ˆ β1,..., ˆ βk)=arg min b1,...,bk∈R⎛ ⎝1 n n i=1Yi−Yn− k j=1 bjXi−Xn,φj2+ k j=1ˆwj|bj|⎞ ⎠ 2 , where kis a chosen dimension-cut-off, ˆwjare suitable weights based on initial estimators, and Yn=1 nn i=1Yi,Xn=1 nn i=1Xi. Further, ˆα=Yn−ˆ β, Xn. Under suitable assumptions, in particular EX2<∞, and kis larger than the largest index in J, Lee and Park (2012) show in their Theorem 2 that P(ˆ βj=0for j/∈J)→1for 123
33 Page 6 of 17 N. Neumeyer, L. Selk n→∞. Thus we can set B=⎧ ⎨ ⎩ j∈J bjφjbj∈R∀j∈J⎫ ⎬ ⎭ and obtain P(ˆ β−β∈B)→1forn→∞. Further, the class of maps in (3.1), i.e. ⎧ ⎨ ⎩ H→R,x→ j∈J bjx,φj+vbj∈R∀j∈J,v ∈R⎫ ⎬ ⎭ is a finite dimensional vector space and thus a VC-class, see Lemma 2.6.15 in van der Vaart and Wellner (1996). Then as discussed above validity of (a.3) follows. Furthermore, from Theorem 2 in Lee and Park (2012) it also follows that our assumption (a.1) is fulfilled, and thus under assumption (a.2) the assertion of Theorem 2.2 holds. 3.2 Bracketing number condition In this subsection we assume that His a separable Hilbert space of real-valued functions (or vectors with real components) and the inner product is increasing in the sense that from h≤g(pointwise for functions; componentwise for vectors) it follows that h,x≤g,xfor all x∈Hwith x≥0. Then assumption (a.3) can be replaced by the condition in the next lemma. Lemma 3.2 Assume (a.1),(a.2) and Pˆ β−β∈B→1as n →∞for a function class B⊂Hsuch that the bracketing number fulfills log N[](B,,·)≤K/1/kfor some k >1/γ .Hereγis the Hölder-order from assumption (a.2). Then assumption (a.3) holds. The proof is given in the appendix. Example 3.3 We consider the Hilbert space H=L2([0,1])with inner product g,h=1 0g(t)h(t)dt and norm g=(1 0g2(t)dt)1/2. We assume β∈ Wm 2([0,1])for some m>2 and the Sobolev-space Wm 2([0,1])=b:[0,1]→R|b(j)is absolutely continuous for j=0,...,m−1, and b(m)<∞, where b(0)=b, and b(j)denotes the j-th derivative of b,j≥1. We consider the regularized estimators in Yuan and Cai (2010), i.e. ˆα, ˆ β=arg min a∈R,b∈Wm 2([0,1])1 n n i=1Yi−a+Xi,b2+λn b(m) 2 123
Testing for changes in the error distribution… Page 7 of 17 33 for a suitable positive sequence λnconverging to zero. Convergence rates of ˆ βand its derivatives can be found in Corollaries 10 and 11 in Yuan and Cai (2010). Under suitable assumptions one obtains ˆ β(j)−β(j) =oP(1)for j=0,1,2, and thus P(ˆ β−β∈B)→1 for the function class B=b∈W2 2[0,1]:b+b(2)≤1. By Corollary 4.3.38 in Giné and Nickl (2021) and Lemma 9.21 in Kosorok (2008)the class Bfulfills the bracketing number condition in Lemma 3.2 for k=2. Thus the assumptions (a.1)–(a.3) are fulfilled if Fis Hölder-continuous of order γ∈(1 2,1].Less restrictive assumptions on F, i.e. γ≤1 2, require for this concept higher smoothness of β. 4 Fixed one-change point alternative: consistency of the test and change point estimator In this section we consider fixed alternatives with one change point at index k∗ n= nϑ∗with ϑ∗∈(0,1). We write the functional linear model as in Sect.2under the following assumption. (a.2)’ Assume ε1,...,ε k∗ nare iid with cdf F1, and εk∗ n+1,...,ε nare iid with cdf F2= F1.LetF1and F2be Hölder-continuous of order γ1,γ 2∈(0,1]with Hölder-constant c1,c2, respectively. Let further P1denote the distribution of (X1,ε 1)(before the change) and P2denote the distribution of (Xn,ε n)(after the change). For the empirical distribution functions ˆ Fkand ˜ Fkas in Sect. 2we obtain the following asymptotic result. Lemma 4.1 Under assumptions (a.1) and (a.2)’ and if (a.3) is valid for P =P1and P=P2, it holds that sup z∈R|ˆ Fk∗ n(z)−F1(z)|=oP(1)and sup z∈R|˜ Fk∗ n(z)−F2(z)|=oP(1). The proof is given in the appendix. Now note that Tn n1/2≥k∗ n(n−k∗ n) n2sup z∈Rˆ Fk∗ n(z)−˜ Fk∗ n(z), and by Lemma 4.1 the right hand side converges in probability to the positive constant ϑ∗(1−ϑ∗)sup z∈R|F1(z)−F2(z)|. From this it follows that tests that reject the null hypothesis of no change-point if Tn>qfor some q>0 (see Sect. 2) are consistent. 123
33 Page 8 of 17 N. Neumeyer, L. Selk The estimator for the change point ϑ∗is based on the process ˆ Gnand is defined as ˆ ϑn=min t:sup z∈R|ˆ Gn(t,z)|= sup t∈[0,1] sup z∈R|ˆ Gn(t,z)|. Lemma 4.2 Under assumptions (a.1),(a.2)’ and if (a.3) holds for P =P1and P =P2, the change point estimator is consistent, i.e. |ˆ ϑn−ϑ∗|=oP(1). The proof is given in the appendix. 5 Finite sample properties We consider the Hilbert space H=L2([0,1]).Fori=1,...,nthe functional observations Xi(t),t∈[0,1], are generated according to Xi(t)=1 2 5 l=1Bi,lsin t(5−Bi,l)2π−Mi,l−E[Bi,lsin (5−Bi,l)2π−Mi,l], where Bi,l∼U[0,5]and Mi,l∼U[0,2π]for l=1,...,5, i=1,...,n.Ustands for the (continuous) uniform distribution. The functional linear model is built as Yi=Xi(t)γ3,1 3(t)dt +εi, where the coefficient function γa,b(t)=ba/(a)ta−1e−bt I{t>0}is the density of the Gamma distribution. Furthermore, we assume that each Xiis observed on a dense, equidistant grid of 300 evaluation points. The parameter estimators are the regularized estimators described in Example 3.3 with m=3 and a data-driven tuning parameter λnchosen by generalized crossvalidation as described in Yuan and Cai (2010). We model three similar types of change points, such that ε1,...,ε n 2∼N(0,1), εn 2+1,...,ε n∼˜ F1,δ (respectively ˜ F2,δ,˜ F3,δ), where ˜ F1,δ,˜ F2,δ,˜ F3,δ have in common that the mean remains zero and the variance remains one. In particular •˜ F1,δ is the distribution function of a random variable that is N(−2δ, 1)distributed with probability 0.5 and N(2δ,1)distributed with probability 0.5. •˜ F2,δ is the distribution function of a random variable that is N(0,(1−δ)2)distributed with probability 0.5 and N(0,2−(1−δ)2)distributed with probability 0.5. 123
Testing for changes in the error distribution… Page 15 of 17 33 A.4 Proof of Lemma 4.2 First note that ˆ ϑn∈arg max t∈[0,1]sup z∈R|ˆ Gn(t,z)|=arg max t∈[0,1]#sup z∈R ˆ Gn(t,z) n1/2$. Further it holds ˆ Gn(t,z) n1/2=nt(n−nt) n21 nt nt i=1 I{ˆεi≤z}− 1 n−nt n i=nt+1 I{ˆεi≤z} =nt(n−nt) n2#1 nt nt∧nϑ∗ i=1 I{ˆεi≤z}+I{t>ϑ ∗}1 nt nt i=nϑ∗+1 I{ˆεi≤z} −1 n−nt n i=nt∨nϑ∗+1 I{ˆεi≤z}−I{t<ϑ ∗}1 n−nt nϑ∗ i=nt+1 I{ˆεi≤z}$ =nt(n−nt) n2#nt∧nϑ∗ ntF1(z)+I{t>ϑ ∗}nt−nϑ∗ ntF2(z) −n−nt∨nϑ∗ n−ntF2(z)−I{t<ϑ ∗}nϑ∗−nt n−ntF1(z)$+oP(1), since we have sup t∈[0,ϑ∗] sup z∈R nt n 1 nt nt i=1 I{ˆεi≤z}−F1(z) ≤sup t∈[0,ϑ∗] sup z∈R nt nϑ∗ 1 nt nt i=1 I{ˆεi≤z}− 1 nϑ∗ nϑ∗ i=1 I{ˆεi≤z} %&' ( =1 nϑ∗1/2˜ Gnϑ∗(t,z) (A.3) +nt nϑ∗ 1 nϑ∗ nϑ∗ i=1 I{ˆεi≤z}−F1(z)(A.4) =oP(1). Here we have used Lemma 4.1 for the term (A.4). Further ˜ Gnϑ∗is defined as ˆ Gn based on the iid-sample (X1,Y1), . . . , (Xk∗ n,Yk∗ n), but where the residuals are built with ˆα,ˆ βbased on the whole sample. With the same argument as in the proof of Theorem 2.2 it holds that ˜ Gnϑ∗(t,z)=Gnϑ∗(t,z)+oP(1) uniformly in t∈[0,1],z∈R,see(A.2), and thus the term (A.3)isoP(1). 123
33 Page 16 of 17 N. Neumeyer, L. Selk Analogously one can show that supt∈[ϑ∗,1]supz∈R n−nt n1 n−ntn i=nt+1I{ˆεi≤ z}−F2(z)=oP(1). Thus, it holds uniformly in t∈[0,1] ˆ Gn(t,z) n1/2=I{t>ϑ ∗}nϑ∗(n−nt) n2(F1(z)−F2(z)) +I{t≤ϑ∗}nt(n−nϑ∗) n2(F1(z)−F2(z))+oP(1) =I{t>ϑ ∗}ϑ∗(1−t)+I{t≤ϑ∗}t(1−ϑ∗)(F1(z)−F2(z)) +oP(1). The assertion then follows by Theorem 2.12 in Kosorok (2008)asϑ∗is well-separated maximum of t→ I{t>ϑ ∗}ϑ∗(1−t)+I{t≤ϑ∗}t(1−ϑ∗). Acknowledgements The authors are grateful to the Editors and Guest Editors for the organization of the Special Issue “Goodness-of-Fit, Change-Point, and Related Problems”, and to the referees, the Associate Editor and the Guest Editor Simos Meintanis for their constructive comments and interesting ideas to expand the topic. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References Akritas MG, Van Keilegom I (2001) Non-parametric estimation of the residual distribution. Scand J Stat 28(3):549–567 Aston JAD, Kirch C (2012) Detecting and estimating changes in dependent functional data. J Multivar Anal 109:204–220 Aue A, Gabrys R, Horváth L, Kokoszka P (2009) Estimation of a change-point in the mean function of functional data. J Multivar Anal 100(10):2254–2269 Aue A, Hörmann S, Horváth L, Hušková M (2014) Dependent functional linear models with applications to monitoring structural change. Stat Sin 24:1043–1073 Aue A, Rice G, Sönmez O (2018) Detecting and dating structural breaks in functional data without dimension reduction. J R Stat Soc Ser B Stat Methodol 80(3):509–529 Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew-normal distribution. J R Stat Soc Ser B Stat Methodol 61:579–602 Bai J (1994) Weak convergence of the sequential empirical processes of residuals in ARMA models. Ann Stat 22:2051–2061 Berkes I, Gabrys R, Horváth L, Kokoszka P (2009) Detecting changes in the mean of functional observations. J R Stat Soc Ser B Stat Methodol 71(5):927–946 Bickel PJ, Wichura MJ (1971) Convergence criteria for multiparameter stochastic processes and some applications. Ann Math Stat 42:1656–1670 Boente G, Parada D (2023) Robust estimation for functional quadratic regression models. Comput Stat Data Anal 187:107798 123
Testing for changes in the error distribution… Page 17 of 17 33 Cardot H, Mas A, Sarda P (2007) Clt in functional linear regression models. Probab Theory Relat Fields 138:325–361 Carlstein E (1988) Nonparametric change-point estimation. Ann Stat 16(1):188–197 Csörgö M, Horváth L, Szyszkowicz B (1997) Integral tests for suprema of kiefer processes with application. Stat Decis 15:365–377 Dümbgen L (1991) The asymptotic behavior of some nonparametric change-point estimators. Ann Stat 19(3):1471–1495 Ferraty F, Vieu P (2006) Nonparametric functional data analysis, Springer series in statistics. Springer, New York Giné E, Nickl R (2021) Mathematical foundations of infinite-dimensional statistical models. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press Hariz SB, Wylie JJ, Zhang Q (2005) Nonparametric change-point estimation for dependent sequences. CR Math 341(10):627–630 Hariz SB, Wylie JJ, Zhang Q (2007) Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences. Ann Stat 35(4):1802–1826 Horváth L, Kokoszka P (2012) Inference for functional data with applications, Springer series in statistics. Springer, New York Horvath et al (2024) Variable selection based testing for parameter changes in regression with autoregressive dependence. J Bus Econ Stat 42(4):1331–1343 Kosorok MR (2008) Introduction to empirical processes and semiparametric inference. Springer, New York Koul HL (1996) Asymptotics of some estimators and sequential residual empiricals in nonlinear time series. Ann Stat 24:380–404 Koul HL (2002) Weighted empirical processes in dynamic nonlinear models, Lecture notes in statistics. Springer, New York Lee ER, Park BU (2012) Sparse estimation in functional linear regression. J Multivar Anal 105(1):1–17 Ling S (1998) Weak convergence of the sequential empirical processes of residuals in nonstationary autoregressive models. Ann Stat 26:741–754 Neumeyer N, Van Keilegom I (2009) Change-point tests for the error distribution in non-parametric regression. Scand J Stat 36(3):518–541 Neumeyer N, Dette H, Nagel E-R (2005) A note on testing symmetry of the error distribution in linear regression models. J Nonparametr Stat 17(6):697–715 Neumeyer N, Dette H, Nagel E-R (2006) Bootstrap tests for the error distribution in linear and nonparametric regression models. Austr New Zealand J Stat 48(2):129–156 Neumeyer N, Omelka M (2025) Generalized Hadamard differentiability of the copula mapping and its applications. Bernoulli (to appear) Pardo Fernandez JC (2007) Comparison of error distributions in nonparametric regression. Stat Probab Lett 77:350–356 Picard D (1985) Testing and estimating change-points in time series. Adv Appl Probab 17:841–867 Pierce DA, Kopecky KJ (1979) Testing goodness of fit for the distribution of errors in regression models. Biometrika 66(1):1–5 Selk L, Neumeyer N (2013) Testing for a change of the innovation distribution in nonparametric autoregression: the sequential empirical process approach. Scand J Stat 40(4):770–788 Shang Z, Cheng G (2015) Nonparametric inference in generalized functional linear models. Ann Stat 43(4):1742–1773 Shorack GR, Wellner JA (1986) Empirical processes with applications to statistics. Wiley van der Vaart A, Wellner JA (1996) Weak convergence and empirical processes. Springer, New York van der Vaart A (1998) Asymptotic statistic. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press Yeon H, Dai X, Nordman DJ (2023) Bootstrap inference in functional linear regression models with scalar response. Bernoulli 29(4):2599–2626 Yuan M, Cai T (2010) A reproducing kernel Hilbert space approach to functional linear regression. Ann Stat 38(6):3412–3444 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123