s00362-024-01656-9_about Vectors in the Physics.pptx

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)
2
+ ε with scalar response Y and covariates X from a separable Hilbert space,
e.g. L ([0, 1]). Structural changes in the distribution can appear, even when the parameters α and β do 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 Lnt ] and the last n
— Lnt ] error terms from a sample of n observations, 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 εˆ = Y — αˆ
— (X, βˆ). Similar tests have been considered by Bai (1994) in
1 Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany
1 3

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, βˆ — β) appear 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. 2 we define the test statistic and present
model assumptions to obtain the asymptotically distribution-free hypothesis test. In Sect. 3 we discuss the assumptions on the
parameter estimators and some examples. In Sect. 4 consistency 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. Section 6 concludes 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 H be 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
1 3

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 Yi and H-valued covariate Xi , and with parameters α e R, βin H . The covariates X1 ,..., Xn are assumed to
be iid with E Xi < ∞, and the errors ε1 ,..., εn are 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 αˆ and βˆ denote estimators for the parameters α e R and β e H. We build residuals εˆi = Yi — αˆ — (Xi , βˆ), i =
1 . . . , n. The test statistic
Tn = sup sup |Gˆ n (t, z)|
t e[0,1] zeR
based on the process
ˆ n
G (t, z) = Lnt] (n — Lnt])
n3/2
ˆ
F ˜
Lnt ] Lnt ]
(z) — F (z) ,
compares for each k = 1,..., n — 1 the empirical distribution functions
ˆk
F (z) = 1 Σ
k
i =1
˜k
I {εˆi ≤ z}, F (z) =
1
k n Σ
n — k i =k+1
i
I {εˆ ≤ z}
of the first k and last n — k residuals, respectively. Note that one can write
ˆ n
G (t, z) = Lnt]
n1/2
F
ˆ ˆ
Lnt ] n
(z) — F (z) .
For the asymptotic distribution of the test statistic under the null hypothesis we assume the following conditions. Let P
denote the distribution of (X1 , ε1).
(a.1) |αˆ — α|= oP(1), βˆ — β = oP(1)
(a.2) Let ε1 ,..., εn be independent and identically distributed with cdf F that is
Hölder-continuous of order γ e (0, 1] with Hölder-constant c.
(a.3) P βˆ — β e B → 1 as n → ∞ for a class B ⊂ H such that the function class
F = {(x, e) '→ I {e ≤ v + (x, b)} | v e R, b e 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 1 3

, L. Selk
(a.2) is fulfilled for γ = 1, but (a.2) is less restrictive in the cases γ e (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 β which is a typical assumption. If
1
2
γ e ( , 1 ] in assumption (a.2), and β is in a Sobolev-space with third derivatives,
(a.2) holds for the estimator βˆ from Yuan and Cai (2010). This estimator can also be
applied for smaller γ in (a.2) if higher smoothness of β is assumed.
Define the process Gn as Gˆ n , but based on the true errors instead of residuals, i.e.
n
G (t, z) = L nt]
n 1/2 L ]
F nt (z) — Fn(z)
with
Lnt ]
F (z) = 1
Lnt ]
Lnt ]
Σ
i =1
i
I {ε ≤ z}. (2.1)
Further let G be a completely tucked Brownian sheet, i.e. a centered Gaussian process on [0, 1]2 with 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 sup |Gˆ n (t, z) — Gn (t, z)|= oP(1), (2.2)
t e[0,1] zeR
and thus the process (Gˆ n (t, z))t e[0,1],zeR converges weakly to (G(t, F(z)))t e[0,1],zeR.
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 Tn under the null hypothesis of no change-point, which is the distribution of T = supt,ue[0,1] |G(t, u)| because F is
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¯ e (0, 1) and q be the (1 — α¯ )- quantile of T . Then the test that rejects the
null hypothesis if Tn > q has asymptotic level α¯ . Consistency is considered in Sect. 4.
Remark 2.3 The choice of Tn as a Kolmogorov–Smirnov type test statistic is not mandatory. In principle, any continuous
functional of the process Gˆ n can be con-
sidered. The most common ones, besides Tn , are of Cramér-von–Mises type, e. g.
Tn,2
= supt
. ˆ 2
|G (t, z)| dF 1
. .
e[0,1] R 0 R
ˆ
n n,3 n 2
(z) or T = |G (t, z)| dF(z)dt . The
asymptotic distribution of these test statistics under the null hypothesis also follows from Theorem 2.2 and with
ˆ n
|G (t, z)| dF (z) →
R R
2 2
∫ ∫ ∫ 1
0
2
|G(t, F(z))| dF(z) = |G(t, x)| dx,
1 3

and thus these test statistics are asymptotically distribution-free as well. However,
Tn,2 and Tn,3 contain the unknown quantity F and must therefore be modified in
order to be applied. This can be done by replacing the integral with the sample mean:
T˜n,2 = supt e[ , ] n
1 n
0 1 i =1
ˆ
|G n i 2 ˜
(t, εˆ )| and T n,3 = 1
0 n
Σ . Σ 1 n i =1 ˆ
|G n i 2
(t, εˆ )| dt .
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 β
that fulfill the conditions.
1. VC-class condition
Assumption (a.3) can be derived from a VC-function class condition formulated as
ˆ
follows. Assume that P β — β e B → 1 as n → ∞ for a class B ⊂ H such that the
class of maps
{H → R, x '→ (x, b)+ v | b e B,v e R}
is a VC-subgraph class. By definition then
(3.1)
{{(x, e) e H × R | e ≤ (x, b)+ v}| b e B,v e R}
is a VC-class of sets. The class F from (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
. .
1 1
0 0
2 1/2
(g, h)= g(t)h(t) dt and norm g = ( g (t) dt) . For the parameter function
β we assume sparsity as in Lee and Park (2012). Let (φ j ) j eN be a basis of H and
assume β = Σ
j eJ j j
β φ for some finite, but unknown index set J . Lee and Park
ˆ
(2012) consider the estimator β = Σ k
j =1 βˆj φ j with
(βˆ1,..., βˆk ) = arg
b
min
,
, 1
1,...,bk eR n
n
Σ
i =1
k
Σ
j =1
2
k
Σ
Yi — Yn — b j (Xi — Xn ,φj ) + wˆ j |b j | , , j =1
, 2
where k is a chosen dimension-cut-off, wˆ j are suitable weights based on initial esti-
n
mators, and Y =
n i =1
i n
Y , X = 1
n
Σ Σ
1 n n
Xi . Further, αˆ = Yn — (βˆ, Xn ). Under
i =1
2
suitable assumptions, in particular E X < ∞, and k is larger than the largest index
in J , Lee and Park (2012) show in their Theorem 2 that P(βˆj = 0 for j e/ J) → 1 for
1 3

, L. Selk
n → ∞. Thus we can set
B =
⎧
⎨ Σ
⎩
j eJ
j j j
b φ b e R ∀ j e J
⎫
⎬
⎭
⎨
⎩
Σ
j eJ
j j
H → R, x '→ b (x,φ )+ j
v b e R ∀ j e J,v e R
and obtain P(βˆ — β e B) → 1 for n → ∞. Further, the class of maps in (3.1), i.e.
⎧ ⎫
⎬
⎭
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. Fur- thermore, 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 H is 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, x ) for all x e H with x ≥ 0. Then assumption (a.3) can be replaced by the condition in the next lemma.
ˆ
Lemma 3.2 Assume (a.1), (a.2) and P β — β e B → 1 as n → ∞ for a function
class B ⊂ H such that the bracketing number fulfills log N[ ] (B,‹, ·
 ) ≤ K /‹1/k for some k > 1/γ . Here γ is 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
. .
1 1
0 0
2 1/2
(g, h) = g(t)h(t) dt and norm g = ( g (t) dt) . We assume β e
m
2
W ([0, 1 ]) for some m > 2 and the Sobolev-space
m
2
W ([0, 1 ]) = b : [ ( j)
0, 1]→ R | b is absolutely continuous for j = 0,..., m — 1,
b(m)
and < ∞
}
,
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.
ˆ
αˆ , β = arg min m
1
aeR,beW2 ([0,1]) n
n
Σ
i =1
n
Yi — a + (Xi , b) + λ b (m)
2 ¨ ¨ ¨ ¨ 2
1 3

for a suitable positive sequence λn converging to zero. Convergence rates of βˆ and its derivatives can be found in Corollaries 10
and 11 in Yuan and Cai (2010). Under
( j) ( j)
¨ ˆ ¨ P
suitable assumptions one obtains β — β = o (1 ) for j = 0, 1, 2, and thus
P(βˆ — β e B) → 1 for the function class
2
2
B = b e W [0,
b(2)
1] :b +
 ≤ 1
}
.
By Corollary 4.3.38 in Giné and Nickl (2021) and Lemma 9.21 in Kosorok (2008) the class B fulfills the bracketing number
condition in Lemma 3.2 for k = 2. Thus the 1
2
assumptions (a.1)–(a.3) are fulfilled if F is Hölder-continuous of order γ e ( , 1 ]. Less
1
2
restrictive assumptions on F , i.e. γ ≤ , 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 kn
∗ = Lnϑ∗] with ϑ∗ e (0, 1). We write the
functional linear model as in Sect. 2 under the following assumption.
(a.2)’ Assume ε1 ,..., εkn
∗ are iid with cdf F1, and εkn
∗+1,..., εn are iid with cdf
F2 /= F1. Let F1 and F2 be Hölder-continuous of order γ1, γ2 e (0, 1] with
Hölder-constant c1, c2, respectively.
Let further P1 denote the distribution of (X1 , ε1) (before the change) and P2 denote the distribution of (Xn , εn ) (after the
change). For the empirical distribution functions Fˆk and F˜k as in Sect. 2 we obtain the following asymptotic result.
Lemma 4.1 Under assumptions (a.1) and (a.2)’ and if (a.3) is valid for P = P1 and P = P2, it holds that
sup |Fˆkn
∗ (z) — F1(z)|= oP(1) and sup |F˜kn
∗ (z) — F2(z)|= oP(1).
zeR zeR
The proof is given in the appendix. Now note that
≥ n
∗ ∗
n
Tn k (n — k )
n1/2 n2 zeR
sup F
k ∗
n
ˆ ˜ k ∗
n
(z) — F (z) ,
and by Lemma 4.1 the right hand side converges in probability to the positive constant
ϑ∗(1 — ϑ∗) sup |F1(z) — F2(z)| .
zeR
From this it follows that tests that reject the null hypothesis of no change-point if
Tn > q for some q > 0 (see Sect. 2) are consistent. 1 3

n
ˆ ˆ
ϑ = min t : sup |G ˆ
(t, z)|= sup sup |G
n n r
, L. Selk
The estimator for the change point ϑ∗ is based on the process Gˆ n and is defined as
}
(t , z)| .
zeR t re[0,1] zeR
Lemma 4.2 Under assumptions (a.1), (a.2)’ and if (a.3) holds for P = P1 and P = P2, the change point estimator is consistent, i. e.
|ϑˆn — ϑ∗
|= oP(1).
The proof is given in the appendix.
5 Finite sample properties
We consider the Hilbert space H = L2([0, 1]). For i = 1 ,..., n the functional
i
X (t) = 1
2
observations Xi (t), t e [0, 1], are generated according to
5
Σ
l=1
i ,l i ,l i ,l i ,l i ,l i ,l
B sin t(5 — B )2π — M — E [B sin (5 — B )2π — M ] ,
where Bi,l ∼ U [0, 5] and Mi,l ∼ U [0, 2π ] for l = 1,..., 5, i = 1 ,..., n. U stands for the (continuous) uniform distribution.
The functional linear model is built as
∫
i i 3, 3 i
Y = X (t)γ 1 (t)dt + ε ,
where the coefficient function γa,b(t) = ba/ Г(a)ta—1e—bt I {t > 0} is the density of the Gamma distribution. Furthermore, we
assume that each Xi is 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 λn chosen by generalized cross- validation as described in Yuan and Cai (2010).
We model three similar types of change points, such that
1 L
n
2
ε , . . . , ε ∼ N(0, 1), ε n
2
] L ]+ ˜ ˜ ˜
1 n 1,δ 2,δ 3,δ
, . . . , ε ∼ F (respectively F , F ),
where F˜1,
δ , F˜2,
δ , F˜3,
δ have in common that the mean remains zero and the variance remains one. In particular
• F˜1,
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.
• F˜2,
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.
1 3

0
2
0
4
0
6
0
8
0
1
0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
delta
Rejection
in
percent
n=200
n=100
0
2
0
4
0
6
0
8
0
1
0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
delta
Rejection
in
percent
n=200
n=100
0
2
0
4
0
6
0
8
0
1
0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
delta
Rejection
in
percent
n=200
n=100
Fig. 1 Rejection probabilities with a change in the error distribution from N(0, 1) to F˜1,
δ (left), to F˜2,δ (middle) and to F˜3,
δ (right). The dotted line marks
the 5% level
• F˜3,
δ is the “skew-normal”-distribution
SN —
,
,
, ,
2π (10δ)2 + (10δ)4 π 1 + (10δ)
2
, ,
10δ .
π 2 + 2π 2 — 2π · (10δ)2 + π 2 — 2π · (10δ)4 π + (π — 2) · (10δ)2
,
,
A random variable Z is distributed SN(λ1 , λ2 , λ3) if Z = λ1 + λ2 · Z0 and Z0
has the density 2φ(x)Ф(λ3 x), where φ is the density and Ф is the distribution
function of the standard normal distribution [see Azzalini and Capitanio (1999)].
1 2
,
π
2 λ
3
The expected value of such a random variable Z is calculated as λ +λ ·, 1+λ3
2
2 π
and the variance as λ 1 — · 2 2 λ3
2
1+λ3
2 . This results in the parameters for the
distribution after the change point, such that the the expected value of the errors remains 0 and the variance remains 1.
So δ = 0 represents the null hypothesis of no change point, and the difference between the distribution before and after the change
point grows with δ in all three cases.
In Fig. 1 the rejection probabilities for 500 repetitions, level 5% [critical value tabled in Picard (1985)] and sample sizes n e
{100, 200} are shown. In all three cases it can be seen that the level is approximated well and the power increases for increasing
parameter δ as well as for increasing sample size n. In the case of a change in skewness, the increase with δ is not as pronounced
as in the other two cases. This is because the distributions for different values of δ become more similar as δ increases. The same
types of changes (from N(0, 1) to F˜1,
δ and to F˜2,
δ ) were also simulated in Selk and Neumeyer (2013) for a real-valued
nonparametric autoregression model with lag 1. The results are comparable with an even higher power in the paper at hand.
In addition, we model a more distinct change, that is
ε , . . . , ε n
2
∼ N(0, 0. 2
1 L ] L ]+1 n
2 2
5 ), ε n , . . . , ε ∼ N(0,(0.5 + δ) ).
As expected for a change in the variance the power grows faster with increasing δ than in the other three cases, especially for
small δ. The results are shown in Fig. 2. This kind of change point was also simulated in Neumeyer and Van Keilegom (2009)
1 3

, L. Selk
Fig. 2 Rejection probabilities with a change in
the error distribution from N(0, 0.52) to
N(0, (0.5 + δ)2). The dotted line marks the
5% level
0
2
0
4
0
6
0
8
0
1
0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
delta
Rejection
in
percent
n=200
n=100
for a nonparametric regression model with one-dimensional regressor. The results are similar.
Next to the Kolmogorov–Smirnov type test with test statistic Tn , we also applied Cramér-von-Mises type tests with test
statistics Tn,2 and Tn,3. The results are very similar and are not presented here for the sake of brevity.
6 Concluding remarks
To detect structural changes in functional linear models, we considered the classical test by Bickel and Wichura (1971) for a
change in the distribution, but based on estimated errors. We gave simple assumptions under which the asymptotic distribution of
the test statistic under the null is the same as for iid data. The test as well as the corresponding change point estimators are
consistent in one-change point models. The same test can be considered in more complex regression models with functional
covariates, e.g. a quadratic model as in Boente and Parada (2023) or nonparametric models, see Ferraty and Vieu (2006). We only
consider independent data, but testing for change points in the innovation distribution in times series models that include functional
covariates is a very interesting topic. However, the proofs for asymptotic distributions will be more complicated. In future work
we are planning to consider a time series model Yt = m(Xt ) + εt , where Yt and εt are real-valued and Xt contains a functional
part, but can also contain past values Yt —1 , . .. , Yt — p . We presume that the proofs as in Selk and Neumeyer (2013) (for
nonparametric autoregression time series with independent errors) and of Sect. 4.2 in Neumeyer and Omelka (2025) (for linear
models with finite- dimensional covariates and beta-mixing errors) can be combined with the proofs in the paper at hand to
consider change-point tests for the innovation distribution under the assumption that (Xt , εt ), t e Z, is a strictly stationary beta-
mixing time series.

In the proof of Theorem 2.2 we derive an expansion for the sequential residual-based empirical distribution function,
FˆLn
t ] (z) = FLnt ] (z) + Rn(z) + oP(n—1/2
)
uniformly in t e [0, 1], z e R, where FLnt ] is defined in (2.1), and the term
Rn(z) = EX [F(z + αˆ — α + (X, βˆ — β))]— F(z)
appears from estimating the parameters [see (A.1) in the appendix]. Here EX denotes the expectation with respect to X , which
has the same distribution as Xi , but is independent of αˆ , βˆ. For change-point testing the remainder term Rn cancels when
considering the test statistic Tn . For other testing procedures, e. g. goodness-of-fit tests for the error distribution, this typically
nonnegligible term is of relevance, see Koul (2002) and Neumeyer et al. (2006) for linear models and Akritas and Van Keilegom (
2001) for nonparametric regression. Under more restrictive assumptions one can further expand the remainder term as follows.
Assume that F is twice differentiable with density F r = f and bounded f r and further |αˆ — α|+ βˆ — β = oP(n—1/4), and
E X 2 < ∞. Then by Taylor’s expansion one obtains
FˆLn
t ] (z) = FLnt ] (z) + f (z) αˆ — α + (E [X ], βˆ — β) + oP(n—1/2
).
In models with intercept α, where the estimator for α is chosen as αˆ = Yn — (Xn , βˆ)
the remainder term is
Rn(z) = f (z) εn + (Xn — E [X ], βˆ — β) + oP(n—1/2
).
n
(Here, X = n Σ
—1 n i =1 Xi and analogous for Y n and εn .) By Cauchy–Schwarz-
1/2
inequality and the central limit theorem for n (Xn — E [X ]) one obtains that the
dominating part of the remainder term is f (z)εn . This is the same as in homoscedastic finite-dimensional linear models with intercept
and nonparametric regression models. Note that α and β are identifiable if the kernel of the covariance operator of the covariate
X is {0}. But often functional linear models without intercept are considered in the literature. So in our model assume α = αˆ = 0.
Then the remainder term is
Rn(z) = f (z)(E [X ], βˆ — β)+ oP(n—1/2
),
and (x, βˆ — β) for fixed x e H typically has a slower rate than n—1/2, see Cardot et al. (2007), Shang and Cheng (2015), Yeon et al.
(2023). If one assumes E [X ]= 0, then this problematic term cancels [similar as e.g. for centered ARMA-processes, see Bai (1994
)], but otherwise f (z)(E [X ], βˆ — β) will dominate the asymptotic distribution of the process (FˆLn
t ] (z) — F(z))t e[0,1],zeR. For
our change-point test this dominating term vanishes. The same holds when estimating the conditional copula of the response in
multidimensional functional linear models, given the covariate, see Theorem 5 in Neumeyer and Omelka (2025). But e.g. for
1 3

, L. Selk
would be of relevance. We consider goodness-of-fit tests for the error distribution in the different cases explained above in
future work. With the derived expansion for residual empirical distribution functions one can also develop other tests for the error
distribution as e.g. for symmetry, or equality of error distributions in different models, see e.g. Pierce and Kopecky (1979),
Neumeyer et al. (2005), Pardo Fernandez (2007), among many others, in the cases of regression models with finite-dimensional
covariates.
A Proofs
For ease of notation let (X, Y, ε) be some generic random variable with the same distribution as (X1 , Y1, ε1) under the null, but
independent from the sample (Xi , Yi ), i = 1,..., n. Let P denote the distribution of (X, ε). Further let EX denote the
expectation with respect to X , which in the context below is the conditional expectation given (Xi , Yi ), i = 1,..., n.
The proofs of Theorem 2.2 and Lemma 3.2 are similar as a part of the proof of Theorem 5 in Neumeyer and Omelka (2025),
but under less restrictive assumptions.
A.1 Proof of Theorem 2.2
From the Donsker-property in assumption (a.3) and Corollary 9.31 in Kosorok (2008) it follows that
{(x, e) '→ I {e ≤ z + a + (x, b)} — I {e ≤ z}| z e R, a e R, b e B}
is also P-Donsker. From Theorem 2.12.1 in van der Vaart and Wellner (1996) it follows that also the centered sequential process
1 Lnt ]
Σ
n i
H (t, z, a, b) = ,
n
I {ε ≤ z + a + ( i
X , b)}
i =1
—I {εi ≤ z}— E [F(z + a + (X, b)] + F(z) ,
indexed in t e [0, 1], z e R, a e R, b e B, converges weakly to a centered Gaussian process. Thus the process Hn is
asymptotically stochastic equicontinuous with respect to the semi-metric
ρ((t1, z1, a1, b1), (t2, z2, a2, b2)) = |t1 — t2|+ Var(I {ε ≤ z1 + a1 + (X, b1)}
—I {ε ≤ z2 + a2 + (X, b2)}),
see van der Vaart and Wellner (1996), problem 2, p. 93, and Sect 2.12. In particular we need

≤ |E [F(z + a + (X, b)) — F(z)]|
≤ cE
!
|a + (X, b)|γ
"
≤ c |a|+ b E X γ
by Cauchy–Schwarz and Jensen’s inequality. Now setting a = αˆ —α and b = βˆ—β we
obtain convergence to zero in probability by assumption (a.1). Thus from asymptotic stochastic equicontinuity of the process Hn ,
and Hn(t, z, 0, 0) = 0 we obtain that
sup
|Hn(t, z, αˆ — α, βˆ — β)|= oP(1),
t e[0,1],zeR
which means that
Lnt] ˆ
,
n
F Lnt ]
Lnt]
(z) = ,
n
F Lnt ]
Lnt] X
(z) + ,
n
E [F(z + αˆ — α + ( ˆ
X, β — β) )]— F(z)
+oP(1) (A.1)
uniformly in t e [0, 1], z e R, where FLnt ] was defined in (2.1) and is based on the true errors. In particular for t = 1 we have
Fˆn (z) = Fn(z) + EX [F(z + αˆ — α + (X, βˆ — β))]— F(z) + oP(n—1/2)
uniformly in z e R. From those expansions we obtain
ˆ n
Lnt] ˆ
G (t, z) = ,
n
F Lnt ]
Lnt] ˆn
(z) — ,
n
F (z)
Lnt]
= ,
n
F Lnt ]
Lnt] n P
(z) — ,
n
F (z) + o (1 )
= Gn (t, z) + oP(1)
uniformly in t e [0, 1], z e R.
A.2 Proof of Lemma 3.2
(A.2)
п
Let ‹ > 0 and let
L U
i i
! "
b , b , i = 1,..., N(‹) = O exp(‹—2/(kγ))
2/γ
be brackets for B of ·-length ‹ [see assumption (a.3)]. Now for b e [ L U
i i
b , b ]
the indicator function I {e ≤ v + (x, b)} is contained in the bracket
!
i
I {e ≤ v + (x I {x ≥ 0}, b )+ ( L U
i
x I {x < 0}, b )}, I {e ≤ v + ( U
i
x I {x ≥ 0}, b )
L
i
+(xI {x < 0}, b )}
"
1 3

, L. Selk
for each v e R. Further the above bracket has L2 (P)-length
!
E I {ε ≤ v + (X I U
i
{X ≥ 0}, b )+ ( L
i
X I {X < 0}, b )}
—I {ε ≤ v + (X I L
i
{X ≥ 0}, b )+ ( U
i
X I {X < 0}, b )}
" 2 1/2
!
≤ E F v + ( U
i
X, b ) — F v + ( L
i
X, b )
" 1/2
!
i i
≤ E c|(X, b — b )|
U L γ " 1/2
1/2
bi — b i
γ /2 U L γ /2
≤ c (E X )
= O(‹),
by assumption (a.2), Cauchy–Schwarz and Jensen’s inequality. Similar to the proof of Lemma 1 in Akritas and Van Keilegom (
2001) one obtains an upper bound O(‹—2 exp(‹—2/k )) for the L2(P)-bracketing number of the class F. Thus F is P- Donsker
by the bracketing integral condition in Theorem 19.5 of van der Vaart (1998).
п
To show the assertion for Fˆkn
∗ we use the arguments as in the proof of Theorem 2.2 for the process Hn , but based on the iid
sample (X1 , Y1), . . . , (Xkn
∗ , Ykn
∗ ) before the change. Then as in the proof of Theorem 2.2 asymptotic stochastic equicontinuity of
the process Hkn
∗ holds and thus
sup |Hkn
∗ (t, z, αˆ — α, βˆ — β)|= oP(1).
t e[0,1],zeR
Here, αˆ and βˆ depend on the whole sample and assumption (a.1) is used. Thus as in Eq. (A.1) we obtain
ˆ
F k ∗
n k ∗
n 1
X 1
(z) = F (z) + E [F (z + αˆ — α + ( 1 ˆ
X , β — β) 1 P
1
)]— F (z) + o , n
uniformly in z e R. By the classical Glivenko–Cantelli result Fkn
∗ converges uniformly almost surely to F1. By assumptions (a.1),
(a.2)’ and E X1 < ∞ the remainder term is oP(1) and the assertion for Fˆkn
∗ follows. The assertion for F˜kn
∗ can be shown
analogously. п
1 3

First note that
zeR
ˆ ˆ
n n
t e[0,1] t e[0,1]
ϑ e arg max sup |G (t, z)| = arg max sup
#
zeR
ˆ
G (t, z)
n1/2
n
$
.
Further it holds
Gˆ n (t, z)
n1/2
=
n2
Lnt] (n — Lnt]) 1
Lnt ]
Lnt ]
Σ
i =1
1
I {εî ≤ z}—
n — Lnt ]
n
Σ
i =Lnt ]+1
I {εî ≤ z}
= Lnt](n
n2
#
— Lnt]) 1
Lnt ]
∗
Lnt ] L
∧ nϑ ]
Σ
i =1
1
I {εî ≤ z}+ I {t > ϑ∗}
Lnt ]
Lnt ]
Σ
i =Lnϑ∗]+1
I {εî ≤ z}
1
—
n — Lnt ]
n
Σ
i =Lnt ] L
∨ nϑ∗]+1
1
I {εî ≤ z}— I {t < ϑ∗
}
n — Lnt ]
∗
Lnϑ ]
Σ
i =Lnt ]+1
I {εî ≤ z}
$
= Lnt] (n — L
n2
#
nt]) L ∗
nt ]∧ Lnϑ ]
Lnt ] 1 ∗
F (z) + I {t > ϑ }
∗
Lnt ]— Lnϑ ]
Lnt ] 2
F (z)
—
∗
n — Lnt ]∨ Lnϑ ]
n — Lnt ] 2 ∗
F (z) — I {t < ϑ }
∗
Lnϑ ]— Lnt ]
n — Lnt ]
$
F1(z) + oP(1),
since we have
sup sup
t e[0,ϑ∗] zeR
Lnt] 1 Lnt ]
Σ
n Lnt ] i =1
i 1
I {εˆ ≤ z}— F (z)
≤ sup sup
t e[0,ϑ ] zeR ∗
Lnt] 1
∗ Lnϑ ] Lnt ]
Lnt ]
Σ
i =1
I {εî ≤ z}—
1
∗
Lnϑ ]
∗
Lnϑ ]
Σ
i =1
i
I {εˆ ≤ z}
c ˛' I
= 1
Lnϑ∗]
˜
1/2 Lnϑ ]
G ∗ (t,z)
(A.3)
+ Lnt] 1
∗ ∗
Lnϑ ] Lnϑ ]
∗
Lnϑ ]
Σ
i =1
i 1
I {εˆ ≤ z}— F (z) (A.4)
= oP(1).
Here we have used Lemma 4.1 for the term (A.4). Further G˜ Lnϑ∗] is defined as Gˆ n
based on the iid-sample (X1 , Y1), . . . , (Xkn
∗ , Ykn
∗ ), but where the residuals are built
with αˆ , βˆ based on the whole sample. With the same argument as in the proof of Theorem 2.2 it holds that
G˜ Lnϑ∗](t, z) = GLnϑ∗](t, z) + oP(1) 1 3

, L. Selk
Analogously one can show that sup
∗ sup
t e[ϑ ,1] zeR
n—Lnt] 1
n n—Lnt ]
Σ n
i =Lnt ]+ 1 I {εˆi ≤
2 P
z}— F (z) = o (1 ).
Thus, it holds uniformly in t e [0, 1]
ˆ n
G (t , z)
n1/2
∗
= I {t > ϑ } ∗
Lnϑ ](n — Lnt ])
n2
1 2
(F (z) — F (z))
∗
+ I {t ≤ ϑ }
∗
Lnt ](n — Lnϑ ] )
n2
1 2 P
(F (z) — F (z)) + o (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) as ϑ∗ is well-separated maximum of t '→ I {t > ϑ∗}ϑ∗(1 — t) + I {t
≤ ϑ∗}t(1 — ϑ∗). п
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/.
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