Detecting relevant differences in the covariance operators of functional time series: a sup-norm approach

Abstract

In this paper we propose statistical inference tools for the covariance operators of functional time series in the two sample and change point problem. In contrast to most of the literature, the focus of our approach is not testing the null hypothesis of exact equality of the covariance operators. Instead, we propose to formulate the null hypotheses in the form that “the distance between the operators is small”, where we measure deviations by the sup-norm. We provide powerful bootstrap tests for these type of hypotheses, investigate their asymptotic properties and study their finite sample properties by means of a simulation study.

Appendix: Proofs of main results

1.1 A.1: Proof of Theorem 1

We apply the central limit theorem as formulated in Theorem 2.1 in Dette et al. (2020a) to the sequence of $C(T^2)$-valued random variables $((Z_{j} - \mu )^{\check{\otimes }2})_{j\in {\mathbb {N}}} = (\eta _{j}^{\check{\otimes }2})_{j\in {\mathbb {N}}}$.

It can be easily seen that conditions (A1), (A2) and (A4) in this reference are satisfied. In order to see that the remaining condition (A3) also holds, we use the triangle inequality and Assumption 1 of the present work to obtain, for any $j\in {\mathbb {N}}$ and $s,t,s^\prime ,t^\prime \in T$,

$$\begin{aligned} |\eta _{j}(s)\eta _{j}(t) - \eta _{j}(s^\prime )\eta _{j}(t^\prime )|&\le |\eta _{j}(s)(\eta _{j}(t) - \eta _{j}(t^\prime ))| + |\eta _{j}(t^\prime )(\eta _{j}(s) - \eta _{j}(s^\prime ))| \\&\le \Vert \eta _{j}\Vert _\infty \, \big (|\eta _{j}(t) - \eta _{j}(t^\prime )| + |\eta _{j}(s) - \eta _{j}(s^\prime )| \big ) \\&\le \Vert \eta _{j}\Vert _\infty \, M \, \big ( \rho (t, t^\prime ) + \rho (s, s^\prime ) \big ) \\&\lesssim \Vert \eta _{j}\Vert _\infty \, M \, \rho _{\max } \big ((t,s), (t^\prime , s^\prime ) \big ) \end{aligned}$$

where $\mathbb {E}\big [ (\Vert \eta _{j}\Vert _\infty \, M )^J \big ] \le \tilde{K} < \infty$ by (A3). Now observe that

$$\begin{aligned} \frac{1}{\sqrt{n}} \sum _{j=1}^n (Z_{j} - \bar{Z}_{n})^{\check{\otimes }2} = \frac{1}{\sqrt{n}} \sum _{j=1}^n \eta _{j}^{\check{\otimes }2} -\frac{1}{\sqrt{n}} \bigg (\frac{1}{\sqrt{n}} \sum _{j=1}^n \eta _{j} \bigg )^{\check{\otimes }2} = \frac{1}{\sqrt{n}} \sum _{j=1}^n \eta _{j}^{\check{\otimes }2} + o_{{\mathbb {P}}}(1). \end{aligned}$$

Here the error $o_{\mathbb {P}}(1)$ refers to the supremum norm, because by Theorem 2.1 in Dette et al. (2020a) the sequence $\big (\frac{1}{\sqrt{n}} \sum _{j=1}^n \eta _{j} \big )_{n \in {\mathbb {N}}}$ converges weakly in C([0, 1]) and by continuous mapping $\Vert \frac{1}{\sqrt{n}} \sum _{j=1}^n \eta _{j}^{\check{\otimes }2} \Vert _\infty$ is of order $O_{\mathbb {P}}(1)$, which yields $\frac{1}{\sqrt{n}}\Vert (\frac{1}{\sqrt{n}} \sum _{j=1}^n \eta _{j})^{\check{\otimes }2} \Vert _\infty = o_{{\mathbb {P}}} (1)$. Moreover, as shown above, Theorem 2.1 in Dette et al. (2020a) can also be applied to the sequence $( \eta _{j}^{\check{\otimes }2} )_{j\in {\mathbb {N}}}$, which yields the claim of Theorem 1. $\square$

1.2 A.2: Proof of Proposition 1

As the samples are independent, it directly follows from Theorem 1 that

$$\begin{aligned} \sqrt{m+n}&\bigg ( \frac{1}{m} \sum _{j=1}^m ({\tilde{X}}_{m,j}^{\check{\otimes }2} - C_1 ), \, \frac{1}{n} \sum _{j=1}^n (\tilde{Y}_{n,j}^{\check{\otimes }2} - C_2 ) \bigg ) \\&= \sqrt{m+n} \bigg ( \frac{1}{m} \sum _{j=1}^m (\eta _{1,j}^{\check{\otimes }2} - C_1 ), \, \frac{1}{n} \sum _{j=1}^n (\eta _{2,j}^{\check{\otimes }2} - C_2 ) \bigg ) \\&\quad + o_{\mathbb {P}}(1) \rightsquigarrow \bigg ( \frac{1}{\sqrt{\lambda }} ~ Z_1, \frac{1}{\sqrt{1-\lambda }} ~ Z_2 \bigg ) \end{aligned}$$

in $C([0,1]^2)^2$ as $m,n\rightarrow \infty$, where $Z_1$ and $Z_2$ are independent, centred Gaussian processes defined by their long-run covariance operators (14) and (15). By the continuous mapping theorem it follows that

$$\begin{aligned} Z_{m,n} = \sqrt{m+n} \, \bigg ( \frac{1}{m} \sum _{j=1}^m {\tilde{X}}_{m,j}^{\check{\otimes }2} - \, \frac{1}{n} \sum _{j=1}^n \tilde{Y}_{n,j}^{\check{\otimes }2} - (C_1 - C_2) \bigg ) \rightsquigarrow Z \end{aligned}$$

(50)

in $C([0,1]^2)$ as $m,n\rightarrow \infty$ (the error $o_{\mathbb {P}}(1)$ refers again to the supremum norm), where Z is again a centred Gaussian process with covariance operator (13).

If $d_\infty = 0$, the convergence in (50) together with the continuous mapping yield (12). If $d_\infty > 0$, the asymptotic distribution of ${\hat{d}}_\infty$ can be deduced from Theorem B.1 in the online supplement of Dette et al. (2020a) or alternatively from the results in Cárcamo et al. (2020). $\square$

1.3 A.3: Proof of Theorem 2 and 3

Proof of Theorem 2. Using similar arguments as in the proof of Theorem 1, it follows that the process ${\hat{B}}^{(r)}_{m,n}$ in (18) admits the stochastic expansion

$$\begin{aligned} {\hat{B}}^{(r)}_{m,n}&= \sqrt{n+m} \bigg \{ \frac{1}{m} \sum _{k=1}^{m-l_1+1} \frac{1}{\sqrt{l_1}}\bigg ( \sum _{j=k}^{k+l_1-1} \eta _{1,j}^{\check{\otimes }2} -\frac{l_1}{m}\sum _{i=1}^m \eta _{1,j}^{\check{\otimes }2} \bigg ) \xi _k^{(r)} \\&\quad - \frac{1}{n} \sum _{k=1}^{n-l_2+1} \frac{1}{\sqrt{l_2}}\bigg ( \sum _{j=k}^{k+l_2-1} \eta _{2,j}^{\check{\otimes }2} -\frac{l_2}{n}\sum _{i=1}^n \eta _{2,j}^{\check{\otimes }2} \bigg ) \zeta _k^{(r)} \bigg \} { ~+ ~O \big ( R_{m,n} \big ) }, \end{aligned}$$

where the remainder is defined by $R_{m,n}=R^{(1)}_m - R^{(2)}_n$ with

$$\begin{aligned} R^{(1)}_m= & {} \frac{1}{\sqrt{m}} \sum _{k=1}^{m-l_1+1} \frac{1}{\sqrt{l_1}} \left( - \sum _{j=k}^{k+l_1-1} \eta _{1,j} \check{\otimes }\bar{\eta }_1 - \bar{\eta }_1 \check{\otimes }\sum _{j=k}^{k+l_1-1} \eta _{1,j} + 2l_1 \bar{\eta }_1^{\check{\otimes }2} \right) \xi ^{(r)}_k , \end{aligned}$$

(51)

$$\begin{aligned} R^{(2)}_n= & {} \frac{1}{\sqrt{n}} \sum _{k=1}^{n-l_2+1} \frac{1}{\sqrt{l_2}} \left( - \sum _{j=k}^{k+l_2-1} \eta _{2,j} \check{\otimes }\bar{\eta }_2 - \bar{\eta }_2 \check{\otimes }\sum _{j=k}^{k+l_2-1} \eta _{2,j} + 2l_2 \bar{\eta }_2^{\check{\otimes }2} \right) \xi ^{(r)}_k. \end{aligned}$$

(52)

Because both terms have a similar structure, we consider only the first one. Note that it is easy to see that $\Vert \bar{\eta }_1^{\check{\otimes }2}\Vert = O_{\mathbb {P}}(\frac{1}{m})$ and therefore the third term in (51) is of order $O_{\mathbb {P}} \big (\sqrt{\frac{l_1}{m}}\big ) = o_{\mathbb {P}}(1)$. The first and second term can be treated in the same way and we only consider the first one. It follows from the proof of Theorem 4.3 in Dette et al. (2020a) that the term

$$\begin{aligned} \left\| \frac{1}{\sqrt{m}} \sum _{k=1}^{m-l_1+1} \frac{1}{\sqrt{l_1}}\left( \sum _{j=k}^{k+l_1-1} \eta _{1,j} \right) \xi ^{(r)}_k \right\| _\infty \end{aligned}$$

is of order $O_{\mathbb {P}}(1)$. By Theorem 2.1 in the same reference the second factor of the tensor satisfies $\Vert \bar{\eta }_1\Vert _\infty = O_\mathbb {P}(\frac{1}{\sqrt{m}})$, and therefore the first term in (51) is of order $o_\mathbb {P}(1)$. Using similar arguments for the second term in (51) and the summand $R^{(2)}_n$ yields

$$\begin{aligned} R_{m,n}= o_\mathbb {P}(1). \end{aligned}$$

Next, note that the sequences $(\eta _{1,j}^{\check{\otimes }2})_{j\in {\mathbb {N}}}$ and $(\eta _{2,j}^{\check{\otimes }2})_{j\in {\mathbb {N}}}$ satisfy Assumption 2.1 in Dette et al. (2020a).

Thus, similar arguments as in the proof of Theorem 3.3 and 4.3 in the same reference yield

$$\begin{aligned} \big ( Z_{m,n} , {\hat{B}}_{m,n}^{(1)},\dots ,{\hat{B}}_{m,n}^{(R)}\big ) \rightsquigarrow (Z, Z^{(1)},\ldots ,Z^{(R)}) \end{aligned}$$

(53)

in $C([0,1]^2)^{R+1}$ as $m,n \rightarrow \infty$ where the process $Z_{m,n}$ is defined in (50) and the random functions $Z^{(1)},\ldots ,Z^{(R)}$ are independent copies of Z which is also defined in (50). Note that in this paper the authors prove weak convergence of a vector in $C([0,1])^{R+1}$. The proof of weak convergence of the finite dimensional distributions can be directly transferred to vectors in $C([0,1]^2)^{R+1}$ , while the proof of equicontinuity requires condition (A1) in Assumption 1, which reduces for the space $C([0,1]^2)$ to (6). If $d_\infty = 0$, the continuous mapping theorem implies

$$\begin{aligned} \big ( \sqrt{m+n} \, \hat{d}_{\infty } ,~ T_{m,n}^{(1)},\ldots ,T_{m,n}^{(R)}\big ) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} (T,~ T^{(1)},\ldots ,T^{(R)}) \end{aligned}$$

(54)

in ${\mathbb {R}}^{R+1}$ as $m,n \rightarrow \infty$ where the statistic $\hat{d}_\infty$ is defined by (11), the bootstrap statistics $T_{m,n}^{(1)},\ldots ,T_{m,n}^{(R)}$ are defined by (19) and the random variables $T^{(1)},\ldots ,T^{(R)}$ are independent copies of T which is defined by (12). Now, Lemma 4.2 in Bücher and Kojadinovic (2019) directly implies (21), that is,

$$\begin{aligned} \lim _{m,n,R\rightarrow \infty } \mathbb {P}\bigg ( \hat{d}_{\infty } > \frac{T_{m,n}^{\{\lfloor R(1-\alpha )\rfloor \}}}{\sqrt{m+n}} \bigg ) = \alpha . \end{aligned}$$

For the application of this result, it is required that the distribution of the random variable T has a continuous distribution function, which follows from Gaenssler et al. (2007). In order to show the consistency of test (20) in the case $d_\infty >0$, write

$$\begin{aligned} \mathbb {P}\bigg ( \hat{d}_{\infty }> \frac{T_{m,n}^{\{\lfloor R(1-\alpha )\rfloor \}}}{\sqrt{m+n}} \bigg )&= \mathbb {P}\big ( \sqrt{m+n} \, (\hat{d}_{\infty } - d_{\infty }) + \sqrt{m+n} \, d_{\infty } > T_{m,n}^{\{\lfloor R(1-\alpha )\rfloor \}} \big ) \end{aligned}$$

and note that, given (54) and (16), the assertion in (22) follows by simple arguments. $\square$

Proof of Theorem 3. First note that the same arguments as in the proof of Theorem 3.6 in Dette et al. (2020a) show that the estimators of the extremal sets defined by (23) are consistent that is

$$\begin{aligned} d_H( \hat{\mathcal {E}}_{m,n}^\pm , \mathcal {E}^\pm ) \xrightarrow [m,n\rightarrow \infty ]{{\mathbb {P}}} 0, \end{aligned}$$

where $d_H$ denotes the Hausdorff distance. Thus, given the convergence in (53), the arguments in the proof of Theorem 3.7 in the same reference yield

$$\begin{aligned} \big ( \sqrt{n+m} ~ (\hat{d}_\infty - d_\infty ) ,~ K_{m,n}^{(1)},\ldots ,K_{m,n}^{(R)}\big ) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} (T(\mathcal {E}),~ T^{(1)}(\mathcal {E}),\ldots ,T^{(R)}(\mathcal {E})) \end{aligned}$$

(55)

in ${\mathbb {R}}^{R+1}$ as $m,n \rightarrow \infty$ where the statistic $\hat{d}_\infty$ is defined by (11), the bootstrap statistics $K_{m,n}^{(1)},\ldots ,K_{m,n}^{(R)}$ are defined by (24) and the random variables $T^{(1)}({\mathcal {E}}),\ldots ,T^{(R)}({\mathcal {E}})$ are independent copies of $T(\mathcal {E})$ which is defined by (16). Note that this convergence holds true under the null and the alternative hypothesis.

If $\varDelta = d_\infty$, Lemma 4.2 in Bücher and Kojadinovic (2019) directly implies (26) and again the results in Gaenssler et al. (2007) ensure that the limit $T(\mathcal {E})$ has a continuous distribution function.

If $\varDelta \ne d_\infty$, write

$$\begin{aligned} \mathbb {P}\bigg ( \hat{d}_{\infty }> \varDelta + \frac{K_{m,n}^{\{\lfloor R(1-\alpha )\rfloor \}}}{\sqrt{n+m}} \bigg ) = \mathbb {P}\big ( \sqrt{m+n} \, (\hat{d}_{\infty } - d_{\infty }) + \sqrt{m+n} \, (d_{\infty } - \varDelta ) > K_{m,n}^{\{\lfloor R(1-\alpha )\rfloor \}} \big ). \end{aligned}$$

Then, it follows from (55) and simple arguments that, for any $R\in {\mathbb {N}}$,

$$\begin{aligned} \lim _{m,n\rightarrow \infty } \mathbb {P}\bigg ( \hat{d}_{\infty }> \varDelta + \frac{K_{m,n}^{\{\lfloor R(1-\alpha )\rfloor \}}}{\sqrt{n+m}} \bigg ) = 0 \quad \text {and} \quad \liminf _{m,n\rightarrow \infty } \mathbb {P}\bigg ( \hat{d}_{\infty } > \varDelta + \frac{K_{m,n}^{\{\lfloor R(1-\alpha )\rfloor \}}}{\sqrt{n+m}} \bigg ) = 1 \end{aligned}$$

if $\varDelta > d_\infty$ and $\varDelta < d_\infty$, respectively. This proves the remaining assertions of Theorem 3. $\square$

1.4 A.4: Proof of Proposition 2

Let $C_{n,j}$ denote the covariance operator of $X_{n,j}$ defined by $C_{n,j}(s,t) = \text {Cov}(X_{n,j}(s),X_{n,j}(t))$ and consider the sequential process

$$\begin{aligned} \hat{\mathbb {V}}_n (s)&= \frac{1}{\sqrt{n}} \sum ^{\lfloor s n \rfloor }_{j=1} ({\tilde{X}}_{n,j}^{\check{\otimes }2} - C_{n,j}) + \sqrt{n} \left( s \, - \frac{\lfloor s n \rfloor }{n} \right) \left( {\tilde{X}}_{n, \lfloor s n \rfloor +1}^{\check{\otimes }2} - C_{n,\lfloor s n \rfloor +1} \right) \, \\&= \frac{1}{\sqrt{n}} \sum ^{\lfloor s n \rfloor }_{j=1} ({\tilde{\eta }}_{n,j}^{\check{\otimes }2} - C_{n,j}) + \sqrt{n} \left( s \, - \frac{\lfloor s n \rfloor }{n} \right) \left( {\tilde{\eta }}_{n, \lfloor s n \rfloor +1}^{\check{\otimes }2} - C_{n,\lfloor s n \rfloor +1} \right) + o_{\mathbb {P}}(1) \end{aligned}$$

which is an element of $C([0,1], C([0,1]^2))$. Here the order $o_{\mathbb {P}}(1)$ for the remainder is obtained by similar arguments as given at the beginning of the proof of Theorem 2 and the details are omitted for the sake of brevity. Note that $\{\hat{\mathbb {V}}_n(s)\}_{s\in [0,1]}$ can equivalently be regarded as an element of $C([0,1]^3)$ and we have the representation

$$\begin{aligned} \hat{\mathbb {V}}_n = \tilde{\mathbb {V}}_{1,n} + \tilde{\mathbb {V}}_{2,n} , \end{aligned}$$

(56)

where the processes $\tilde{\mathbb {V}}_{1,n}, \tilde{\mathbb {V}}_{2,n} \in C([0,1]^3)$ are defined by

$$\begin{aligned} \tilde{\mathbb {V}}_{1,n}(s,t,u)&= \, \hat{\mathbb {V}}_{1,n}(s,t,u) \mathbb {1}\{\lfloor sn \rfloor < \lfloor s^* n \rfloor \} + \hat{\mathbb {V}}_{1,n}(\lfloor s^* n \rfloor /n,t,u) \mathbb {1}\{\lfloor sn \rfloor \ge \lfloor s^* n \rfloor \} \\ \tilde{\mathbb {V}}_{2,n}(s,t,u)&= \, (\hat{\mathbb {V}}_{2,n}(s,t,u) - \hat{\mathbb {V}}_{2,n}(\lfloor s^* n \rfloor /n,t,u)) \mathbb {1}\{\lfloor sn \rfloor \ge \lfloor s^* n \rfloor \} \end{aligned}$$

($s,t,u\in [0,1]$) and

$$\begin{aligned} \hat{\mathbb {V}}_{l,n}(s) = \frac{1}{\sqrt{n}} \sum ^{\lfloor s n \rfloor }_{j=1} (\eta _{n,j}^{\check{\otimes }2} - C_{l}) + \sqrt{n} \Big (s \, - \frac{\lfloor s n \rfloor }{n} \Big ) \big (\eta _{n, \lfloor s n \rfloor +1}^{\check{\otimes }2} - C_{l} \big ) \quad (l = 1,2). \end{aligned}$$

Recall the definition of the array ($\tilde{\eta }_{n,j} :n\in {\mathbb {N}}, j = 1,\ldots , n$) in (27). By Theorem 2.2 in Dette et al. (2020a) it follows that

$$\begin{aligned} \hat{\mathbb {V}}_{l,n} \rightsquigarrow \mathbb {V}_l \quad (l = 1,2) \end{aligned}$$

in $C([0,1]^3)$, where $\mathbb {V}_l$ is a centred Gaussian measure on $C([0,1]^3)$ characterized by the covariance operator

$$\begin{aligned} \text {Cov}\big (\mathbb {V}_l(s,t,u), \mathbb {V}_l(s^\prime ,t^\prime ,u^\prime ) \big )&= (s \wedge s^\prime ) \, \mathbb {C}_l((t,u),(t^\prime , u^\prime )), \quad l = 1,2 \end{aligned}$$

and the long-run covariance operator $\mathbb C_l$ is defined in (32). From the continuous mapping theorem, we obtain

$$\begin{aligned} \tilde{\mathbb {V}}_{l,n} \rightsquigarrow \tilde{\mathbb {V}}_l \quad \quad (l = 1,2) \end{aligned}$$

(57)

in $C([0,1]^3)$, where $\tilde{\mathbb {V}}_1, \tilde{\mathbb {V}}_2$ are centred Gaussian measures on $C([0,1]^3)$ characterized by

$$\begin{aligned} \tilde{\mathbb {V}}_1 (s,t,u) = \mathbb {V}_{1}(s \wedge s^*,t,u) \, , \quad \tilde{\mathbb {V}}_2 (s,t,u) = (\mathbb {V}_{2}(s,t,u) - \mathbb {V}_{2}(s^*,t,u)) \mathbb {1}\{ s \ge s^* \} \end{aligned}$$

with covariance operators

$$\begin{aligned} \text {Cov}\big (\tilde{\mathbb {V}}_1(s,t,u), \tilde{\mathbb {V}}_1(s^\prime ,t^\prime ,u^\prime ) \big )&= (s \wedge s^\prime \wedge s^*) \, \mathbb {C}_1((t,u),(t^\prime , u^\prime )) \\ \text {Cov}\big (\tilde{\mathbb {V}}_2(s,t,u), \tilde{\mathbb {V}}_2(s^\prime ,t^\prime ,u^\prime ) \big )&= (s\wedge s^\prime - s^*)_+ \, \mathbb {C}_2((t,u),(t^\prime , u^\prime )) \, . \end{aligned}$$

In the following we will show the weak convergence

$$\begin{aligned} \hat{\mathbb {V}}_n \rightsquigarrow \mathbb {V} \end{aligned}$$

(58)

in $C([0,1]^3)$ as $n\rightarrow \infty$, where $\mathbb {V}\in C([0,1]^3)$ is a centred Gaussian random variable characterized by its covariance operator

$$\begin{aligned} \text {Cov}(\mathbb {V}(s,t,u), \mathbb {V}(s^\prime ,t^\prime ,u^\prime )) = (s\wedge s^\prime \wedge s^*) \, \mathbb {C}_1((t,u), (t^\prime , u^\prime )) + (s\wedge s^\prime - s^*)_+ \, \mathbb {C}_2((t,u), (t^\prime , u^\prime )) \end{aligned}$$

and the long-run covariance operators $\mathbb {C}_1, \mathbb {C}_2$ are defined by (32). The convergence in (57) implies that the processes $\tilde{\mathbb {V}}_{1,n}, \tilde{\mathbb {V}}_{2,n}$ are asymptotically tight and the representation in (56) yields that $\hat{\mathbb {V}}_{n}$ is asymptotically tight as well (see Section 1.5 in Van der Vaart and Wellner 1996). In order to prove the convergence in (58), it consequently remains to show the convergence of the finite-dimensional distributions. For this, we utilize the Crámer–Wold device and show that

$$\begin{aligned} \tilde{Z}_n&=\sum _{j=1}^q c_j \hat{\mathbb {V}}_n (s_j,t_j,u_j)&= \sum _{j=1}^q c_j \big \{ \tilde{\mathbb {V}}_{1,n}(s_j,t_j,u_j) + \tilde{\mathbb {V}}_{2,n}(s_j,t_j,u_j) \big \} \\&{\mathop {\longrightarrow }\limits ^{\mathcal {D}}} \tilde{Z} = \sum _{j=1}^q c_j \mathbb {V}(s_j ,t_j,u_j) \end{aligned}$$

for any $(s_1,t_1,u_1),\dots ,(s_q,t_q,u_q) \in [0,1]^3$, $c_1,\ldots ,c_q \in {\mathbb {R}}$ and $q\in {\mathbb {N}}$. Asymptotic normality of $\tilde{Z}_n$ can be proved by the same arguments as in the proof of Theorem 2.1 in Dette et al. (2020a), and it remains to show that the variance of the random variable $\tilde{Z}_n$ converges to the variance of $\tilde{Z}$. Using (3.17) in Dehling and Philipp (2002) and assumptions (A2) and (A4) we obtain for any $(s,t,u),(s^\prime , t^\prime , u^\prime ) \in [0,1]^3$

$$\begin{aligned} \begin{aligned}&\text {Cov}(\tilde{\mathbb {V}}_{1,n} (s,t,u), \tilde{\mathbb {V}}_{2,n} (s^\prime ,t^\prime ,u^\prime ) ) \\&\quad = \frac{1}{n} \sum ^{\lfloor (s \wedge s^*) n \rfloor }_{j=1} \sum ^{\lfloor s^\prime n \rfloor }_{i=\lfloor s^* n \rfloor + 1} \text {Cov}(\tilde{\eta }_{n,j}^{\check{\otimes }2}(t,u), \tilde{\eta }_{n,i}^{\check{\otimes }2}(t^\prime ,u^\prime )) + o(1) \\&\quad \lesssim \frac{1}{n} \sum ^{\lfloor (s \wedge s^*) n \rfloor }_{j=1} \sum ^{\lfloor s^\prime n \rfloor }_{i=\lfloor s^* n \rfloor + 1} \Vert \tilde{\eta }_{n,j}^{\check{\otimes }2}(t,u)\Vert _2 \, \Vert \tilde{\eta }_{n,i}^{\check{\otimes }2}(t^\prime ,u^\prime )\Vert _2 \, \varphi (i-j)^{1/2} + o(1) \\&\quad \lesssim \frac{1}{n} \sum ^{\lfloor (s \wedge s^*) n \rfloor }_{j=1} \sum ^{\lfloor s^\prime n \rfloor }_{i=\lfloor s^* n \rfloor + 1} \varphi (i-j)^{1/2} + o(1) \lesssim \frac{1}{n} \sum ^{\lfloor s^\prime n \rfloor - 1}_{i = 1} i \varphi (i)^{1/2} + o(1){\longrightarrow _{n\rightarrow \infty }} 0, \end{aligned} \end{aligned}$$

(59)

where the symbol “$\lesssim$” means less or equal up to a constant independent of n, and $\Vert X\Vert _2 = \mathbb {E}[X^2]^{1/2}$ denotes the $L^2$-norm of a real-valued random variable X (also note that we implicitly assume $\sum _{i=j}^k a_i = 0$ if $k<j$). Furthermore, assuming without loss of generality that $s \le s^\prime$, we have

$$\begin{aligned}&\text {Cov}(\tilde{\mathbb {V}}_{1,n} (s,t,u), \tilde{\mathbb {V}}_{1,n} (s^\prime ,t^\prime ,u^\prime ) ) = \frac{1}{n} \sum ^{\lfloor (s\wedge s^*) n \rfloor }_{j=1} \sum ^{\lfloor (s^\prime \wedge s^*) n \rfloor }_{i= 1} \text {Cov}(\eta _{1,j}^{\check{\otimes }2}(t,u), \eta _{1,i}^{\check{\otimes }2}(t^\prime ,u^\prime )) + o(1) \\&\quad = \frac{1}{n} \sum ^{\lfloor (s\wedge s^*) n \rfloor }_{j=1} \left( \sum ^{\lfloor (s \wedge s^*) n \rfloor }_{i= 1} + \sum ^{\lfloor (s^\prime \wedge s^*) n \rfloor }_{i=\lfloor (s \wedge s^*) n \rfloor + 1} \right) \text {Cov}(\eta _{1,j}^{\check{\otimes }2}(t,u), \eta _{1,i}^{\check{\otimes }2}(t^\prime ,u^\prime )) + o(1) \\&\quad = \frac{1}{n} \sum ^{\lfloor (s\wedge s^*) n \rfloor }_{j=1} \sum ^{\lfloor (s \wedge s^*) n \rfloor }_{i= 1} \text {Cov}(\eta _{1,j}^{\check{\otimes }2}(t,u), \eta _{1,i}^{\check{\otimes }2}(t^\prime ,u^\prime )) + o(1), \end{aligned}$$

where the last equality follows by the same arguments as used in (59). For the remaining expression we use the dominated convergence theorem to obtain

$$\begin{aligned}&\frac{1}{n} \sum ^{\lfloor (s\wedge s^*) n \rfloor }_{j=1} \sum ^{\lfloor (s \wedge s^*) n \rfloor }_{i= 1} \text {Cov}(\eta _{1,j}^{\check{\otimes }2}(t,u), \eta _{1,i}^{\check{\otimes }2}(t^\prime ,u^\prime )) \\&= \sum ^{\lfloor (s \wedge s^*) n \rfloor -1}_{i= -(\lfloor (s \wedge s^*) n \rfloor -1)} \frac{\lfloor (s \wedge s^*) n \rfloor - |i|}{n} \, \text {Cov}(\eta _{1,0}^{\check{\otimes }2}(t,u), \eta _{1,i}^{\check{\otimes }2}(t^\prime ,u^\prime )) {\longrightarrow _{{n \rightarrow \infty }}} (s \wedge s^*) \, \mathbb {C}_1((t,u), (t^\prime , u^\prime )) \end{aligned}$$

which means that for any $(s,t,u),(s^\prime , t^\prime , u^\prime ) \in [0,1]^3$

$$\begin{aligned} \text {Cov}(\tilde{\mathbb {V}}_{1,n} (s,t,u), \tilde{\mathbb {V}}_{1,n} (s^\prime ,t^\prime ,u^\prime ) ) {\longrightarrow }{}{n\rightarrow \infty } (s \wedge s^\prime \wedge s^*) \, \mathbb {C}_1((t,u), (t^\prime , u^\prime )) . \end{aligned}$$

By similar arguments we obtain

$$\begin{aligned} \text {Cov}(\tilde{\mathbb {V}}_{2,n} (s,t,u), \tilde{\mathbb {V}}_{2,n} (s^\prime ,t^\prime ,u^\prime ) ) {\longrightarrow }{}{n\rightarrow \infty } (s \wedge s^\prime - s^*)_+ \, \mathbb {C}_2((t,u), (t^\prime , u^\prime )) \end{aligned}$$

and therefore we have

$$\begin{aligned} \text {Var}(\tilde{Z}_n )&= \sum _{j=1}^q \sum _{j^\prime =1}^q c_j c_{j^\prime } \text {Cov}(\hat{\mathbb {V}}_n (s_j,t_j,u_j), \hat{\mathbb {V}}_n (s_{j^\prime },t_{j^\prime },u_{j^\prime }) ) \\&= \sum _{j=1}^q \sum _{j^\prime =1}^q c_j c_{j^\prime } \big \{ \text {Cov}(\tilde{\mathbb {V}}_{1,n} (s_j,t_j,u_j), \tilde{\mathbb {V}}_{1,n} (s_{j^\prime },t_{j^\prime },u_{j^\prime }) ) \\&\quad + \text {Cov}(\tilde{\mathbb {V}}_{2,n} (s_j,t_j,u_j), \tilde{\mathbb {V}}_{2,n} (s_{j^\prime },t_{j^\prime },u_{j^\prime }) ) \big \} + o(1) \\&{\longrightarrow }_{n\rightarrow \infty } \sum _{j=1}^q \sum _{j^\prime =1}^q c_j c_{j^\prime } \text {Cov}(\mathbb {V} (s_j,t_j,u_j), \mathbb {V} (s_{j^\prime },t_{j^\prime },u_{j^\prime }) ) = \text {Var}(\tilde{Z}) \end{aligned}$$

which finally proves (58).

Next we define the $C([0,1]^3)$-valued process

$$\begin{aligned} \hat{\mathbb {W}}_n(s,t,u) = \hat{\mathbb {V}}_n(s,t,u) - s \hat{\mathbb {V}}_n(1,t,u) \, , \qquad s,t,u \in [0,1] \, , \end{aligned}$$

(60)

then the convergence in (58) and the continuous mapping theorem yield

$$\begin{aligned} \hat{\mathbb {W}}_n \rightsquigarrow \mathbb {W} \end{aligned}$$

(61)

in $C([0,1]^3)$, where $\mathbb {W}$ is centred Gaussian defined by $\mathbb {W}(s,t,u)= \mathbb {V}(s,t,u) - s \mathbb {V}(1,t,u)$ with covariance operator given by (31). Finally, recall the definition of the process $(\hat{\mathbb {U}}_{n}:n\in \mathbb {N})$ in (28) and note that, in contrast to $\hat{\mathbb {W}}_n$, this process is not centred. Consequently, if $d_\infty = 0$, we have $\sqrt{n}\, \mathbb {U}_n = \hat{\mathbb {W}}_n$ and the convergence in (61) and the continuous mapping theorem directly yield (30).

If $d_\infty > 0$, assertion (33) is a consequence of the weak convergence in (61) and Theorem B.1 in the online supplement of Dette et al. (2020a) and also of the results in Cárcamo et al. (2020). $\square$

1.5 A.5: Proof of Theorem 4 and 5

Proof of Theorem 4. Recalling the definition of the bootstrap processes in (34) it can be shown by similar arguments as given at beginning of the proof of Theorem 2 that

$$\begin{aligned} \sup _{s,t,u \in [0,1[^{3}} | \hat{B}_n^{(r)}(s,t,u) - \hat{C}_n^{(r)}(s,t,u) | = o_{\mathbb {P}}(1) \end{aligned}$$

(62)

(for $r = 1,\dots , R$), where

$$\begin{aligned} \begin{aligned} \hat{C}_n^{(r)}(s,t,u)&= \frac{1}{\sqrt{n}} \sum _{k=1}^{\lfloor sn \rfloor } \frac{1}{\sqrt{l}} \left( \sum _{j=k}^{k+l-1} \tilde{Y}_{n,j}(t,u) - \frac{l}{n} \sum _{j=1}^n \tilde{Y}_{n,j}(t,u) \right) \xi _k^{(r)} \\&+ \sqrt{n}\left( s - \frac{\lfloor sn \rfloor }{n} \right) \frac{1}{\sqrt{l}} \left( \sum _{j=\lfloor sn \rfloor +1}^{\lfloor sn \rfloor +l} \tilde{Y}_{n,j}(t,u) - \frac{l}{n} \sum _{j=1}^n \tilde{Y}_{n,j}(t,u) \right) \xi _{\lfloor sn \rfloor +1}^{(r)} \end{aligned}, \end{aligned}$$

$\tilde{Y}_{n,j} = \tilde{\eta }_{n,j}^{\check{\otimes }2}(t,u) - (\hat{C}_2 - \hat{C}_1) \mathbb {1}\{j > \lfloor \hat{s}n \rfloor \}$ ($j=1,\dots ,n$). The array $(\tilde{\eta }_{n,j}^{\check{\otimes }2} \, :n\in {\mathbb {N}}, ~ j = 1,\ldots , n)$ satisfies (A1), (A3) and (A4) of Assumption 2.1 in Dette et al. (2020a). The convergence in (61) and similar arguments as in the proof of Theorem 4.3 in the same reference show

$$\begin{aligned} (\hat{\mathbb {V}}_n, \hat{B}_n^{(1)},\ldots , \hat{B}_n^{(R)}) \rightsquigarrow (\mathbb {V}, \mathbb {V}^{(1)},\ldots ,\mathbb {V}^{(R)}) \, \end{aligned}$$

(63)

in $C([0,1]^3)^{R+1}$ as $n \rightarrow \infty$, where the process $\mathbb {V}$ is defined in (57) and $\mathbb {V}^{(1)},\dots ,\mathbb {V}^{(R)}$ are independent copies of $\mathbb {V}$. For the sake of completeness we repeat the necessary main steps here, which are proved using analogous arguments as given in Dette et al. (2020a). First we define ${Y}_{n,j}(t,u) = \tilde{\eta }_{n,j}^{\check{\otimes }2}(t,u) - ({C}_2 - {C}_1) \mathbb {1}\{j > \lfloor {s}^{*} n \rfloor \}$ and show the approximation

$$\begin{aligned} \sup _{s,t,u \in [0,1[^{3}} \big | \hat{C}_n^{(r)}(s,t,u) - \bar{C}_n^{(r)}(s,t,u) \big | = o_{\mathbb {P}}(1) , \end{aligned}$$

(64)

where the process $\bar{C}_n^{(r)}$ is defined by

$$\begin{aligned} \begin{aligned} {\bar{C}}_n^{(r)}(s,t,u)&= \frac{1}{\sqrt{n}} \sum _{k=1}^{\lfloor sn \rfloor } \frac{1}{\sqrt{l}} \left( \sum _{j=k}^{k+l-1} {Y}_{n,j}(t,u) - \frac{l}{n} \sum _{j=1}^n {Y}_{n,j}(t,u) \right) \xi _k^{(r)} \\&+ \sqrt{n}\left( s - \frac{\lfloor sn \rfloor }{n} \right) \frac{1}{\sqrt{l}} \left( \sum _{j=\lfloor sn \rfloor +1}^{\lfloor sn \rfloor +l} {Y}_{n,j}(t,u) - \frac{l}{n} \sum _{j=1}^n {Y}_{n,j}(t,u) \right) \xi _{\lfloor sn \rfloor +1}^{(r)} \end{aligned}. \end{aligned}$$

In a second step we show

$$\begin{aligned} \sup _{s,t,u \in [0,1[^{3}} \big | \bar{C}_n^{(r)}(s,t,u) - \tilde{C}_n^{(r)}(s,t,u) \big | = o_{\mathbb {P}}(1) , \end{aligned}$$

(65)

where the process $\tilde{C}_n^{(r)}$ is defined by

$$\begin{aligned} \begin{aligned} \tilde{C}_n^{(r)}(s,t,u)&= \frac{1}{\sqrt{n}} \sum _{k=1}^{\lfloor sn \rfloor } \frac{1}{\sqrt{l}} \left( \sum _{j=k}^{k+l-1} \left( {Y}_{n,j}(t,u) - C_{1} (t,u) \right) \right) \xi _k^{(r)} \\&+ \sqrt{n}\left( s - \frac{\lfloor sn \rfloor }{n} \right) \frac{1}{\sqrt{l}} \left( \sum _{j=\lfloor sn \rfloor +1}^{\lfloor sn \rfloor +l} \left( {Y}_{n,j}(t,u) - C_{1} (t,u) \right) \right) \xi _{\lfloor sn \rfloor +1}^{(r)} \end{aligned}. \end{aligned}$$

In a third step one notes that

$$\begin{aligned} {Y}_{n,j}(t,u) = {\left\{ \begin{array}{ll} \tilde{\eta }_{n,j}^{\check{\otimes }2}(t,u) - {C}_1 &{} \text { if } j \le \lfloor {s}^{*} n \rfloor \ \\ \tilde{\eta }_{n,j}^{\check{\otimes }2}(t,u) - {C}_2 &{} \text { if } j > \lfloor {s}^{*} n \rfloor \end{array}\right. } \end{aligned}$$

and shows the weak convergence

$$\begin{aligned} (\hat{\mathbb {V}}_n,\tilde{C}_n^{(1)},\ldots ,\tilde{C}_n^{(R)}) \rightsquigarrow (\mathbb {V}, \mathbb {V}^{(1)},\ldots ,\mathbb {V}^{(R)}) \, \end{aligned}$$

in $C([0,1]^3)^{R+1}$ as $n \rightarrow \infty$, where the process $\mathbb {V}$ is defined in (57) and $\mathbb {V}^{(1)},\ldots ,\mathbb {V}^{(R)}$ are independent copies of $\mathbb {V}$. Observing (62), (64) and (65) then proves the weak convergence in (63) Finally, this result and the continuous mapping theorem yield

$$\begin{aligned} \big ( \hat{\mathbb {W}}_n , \hat{\mathbb {W}}_n^{(1)},\ldots ,\hat{\mathbb {W}}_n^{(R)}\big ) \rightsquigarrow (\mathbb {W}, \mathbb {W}^{(1)},\ldots , \mathbb {W}^{(R)}) \end{aligned}$$

(66)

in $C([0,1]^3)^{R+1}$ as $n \rightarrow \infty$ where the process $\hat{\mathbb {W}}_n$ is defined by (60), the bootstrap counterparts $\hat{\mathbb {W}}_{n}^{(1)},\ldots ,\hat{\mathbb {W}}_{n}^{(R)}$ are defined by (36) and the random variables $\mathbb {W}^{(1)},\ldots ,\mathbb {W}^{(R)}$ are independent copies of $\mathbb {W}$ which is defined by its covariance operator (31).

If $d_\infty = 0$, the continuous mapping theorem directly implies

$$\begin{aligned} \big ( \hat{\mathbb {M}}_n , \check{T}_{n}^{(1)},\ldots ,\check{T}_{n}^{(R)}\big ) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} (\check{T}, \check{T}^{(1)},\ldots , \check{T}^{(R)}) \end{aligned}$$

in ${\mathbb {R}}^{R+1}$ as $n \rightarrow \infty$ where the statistic $\hat{\mathbb {M}}_n$ is defined by (29), the bootstrap statistics $\check{T}_{n}^{(1)},\ldots ,\check{T}_{n}^{(R)}$ are defined by (37) and the random variables $\check{T}^{(1)},\ldots ,\check{T}^{(R)}$ are independent copies of the random variable $\check{T}$ defined by (30). Now the same arguments as in the discussion starting from Eq. (54) imply the assertions of Theorem 4. $\square$

Proof of Theorem 5. We first mention that it follows by similar arguments as given in the proof of Theorem 4.2 in Dette et al. (2020a) that the estimator of the unknown change location defined by (35) satisfies

$$\begin{aligned} |\hat{s}-s^*| = O_{\mathbb {P}}(n^{-1}) \end{aligned}$$

whenever $d_\infty >0$. Whenever $d_\infty =0$, suppose that the estimate $\hat{s}$ converges weakly to a $[\vartheta ,1-\vartheta ]$-valued random variable which is denoted by $s_{\max }$. Then, if $d_\infty > 0$, the convergence in (33) and Slutsky’s theorem yield

$$\begin{aligned} \sqrt{n}\big ( \hat{d}_\infty - d_\infty \big ) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} D (\mathcal {E})= {\tilde{D} (\mathcal {E}) }/[{s^*(1-s^*)}] \, , \end{aligned}$$

(67)

where $\tilde{D} (\mathcal {E})$ is the same as in (33) and the statistic $\hat{d}_\infty$ is defined by (39).

The same arguments as in the proof of Theorem 3.6 in Dette et al. (2020a) again yield that the estimators of the extremal sets defined by (40) are consistent. The convergence in (66) and similar arguments as in the proof of Theorem 4.4 in the same reference then yield

$$\begin{aligned} \big ( \sqrt{n} ~ (\hat{d}_\infty - d_\infty ) ,~ \check{K}_{n}^{(1)},\ldots ,\check{K}_{n}^{(R)}\big ) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} (D(\mathcal {E}),~ D^{(1)}(\mathcal {E}),\ldots , D^{(R)}(\mathcal {E})) \end{aligned}$$

(68)

in ${\mathbb {R}}^{R+1}$ as $n \rightarrow \infty$ where the bootstrap statistics $\check{K}_{n}^{(1)},\ldots ,\check{K}_{n}^{(R)}$ are defined by (41) and the random variables $D^{(1)}(\mathcal {E}),\ldots ,D^{(R)}(\mathcal {E})$ are independent copies of $D(\mathcal {E})$ which is defined by (67). The convergence in the preceding equation holds true under the null and the alternative hypothesis and now the same arguments as in the discussion starting from Eq. (55) imply the assertions made in Theorem 5. $\square$