Abrupt change in mean using block bootstrap and avoiding variance estimation

Abstract

We deal with sequences of weakly dependent observations that are naturally ordered in time. Their constant mean is possibly subject to change at most once at some unknown time point. The aim is to test whether such an unknown change has occurred or not. The change point methods presented here rely on ratio type test statistics based on maxima of the cumulative sums. These detection procedures for the abrupt change in mean are also robustified by considering a general score function. The main advantage of the proposed approach is that the variance of the observations neither has to be known nor estimated. The asymptotic distribution of the test statistic under the no change null hypothesis is derived. Moreover, we prove the consistency of the test under the alternatives. A block bootstrap method is developed in order to obtain better approximations for the test’s critical values. The validity of the bootstrap algorithm is shown. The results are illustrated through a simulation study, which demonstrates computational efficiency of the procedures. A practical application to real data is presented as well.

Proofs

Proof of Theorem 1

The proof is analogous in several steps with the proof of Theorem 1.1 in Horváth et al. (2008). Without loss on generality, we assume that $\mu =0$. Let

$$\begin{aligned} Z_n(t)=\frac{1}{\sqrt{n}}\sum _{1\le j \le nt}\psi (\varepsilon _j)\quad \text{ and }\quad \widetilde{Z}_n(t)=\frac{1}{\sqrt{n}}\sum _{nt< j \le n}\psi (\varepsilon _j). \end{aligned}$$

By applying Theorem 1 from Doukhan (1994, Section 1.5.1) with the consequent remark justified by Herrndorf (1983) and Bulinskii (1987, 1989), we get

$$\begin{aligned} \left( Z_n(t),\widetilde{Z}_n(t)\right) \xrightarrow [n\rightarrow \infty ]{\mathscr {D}^2[0,1]}\sigma (\psi )\,\left( \mathcal {W}(t),\widetilde{\mathcal {W}}(t)\right) , \end{aligned}$$

(10)

where $\{\mathcal {W}(t),\,0\le t\le 1\}$ is a standard Wiener process and $\widetilde{\mathcal {W}}(t)=\mathcal {W}(1)-\mathcal {W}(t)$. Consequently, Lemma 4.3 and Lemma 4.4 by Hušková and Marušiaková (2012) together with Assumptions A1–A5 lead to

$$\begin{aligned}&\sup _{1\le i \le nt} \left\{ n^{\kappa }\sqrt{\frac{[nt]}{i([nt]-i)}}\ \left| \sum _{1\le j \le i}\psi \left( Y_j-\widehat{\mu }_{1,[nt]}(\psi )\right) \right. \right. \nonumber \\&\quad \left. \left. -\left( \sum _{1\le j \le i}\psi (\varepsilon _j) -\frac{i}{[nt]}\sum _{1\le j \le nt}\psi (\varepsilon _j)\right) \right| \right\} \xrightarrow [n\rightarrow \infty ]{\mathsf {P}\,} 0, \end{aligned}$$

for some $\kappa >0$, where [a] denotes the integer part of $a\in \mathbb {R}$. Hence,

$$\begin{aligned}&\frac{1}{\sqrt{n}}\sup _{1<i\le nt}\left| \sum _{1\le j \le i}\psi \left( Y_j-\widehat{\mu }_{1,[nt]}(\psi )\right) \right| \\&\quad =\sup _{1\le i \le nt}\left| Z_n\left( \frac{i}{n}\right) -\frac{i}{[nt]}Z_n(t)\right| + o_{\mathsf {P}\,}(1),\quad n\rightarrow \infty . \end{aligned}$$

Similarly, we get

$$\begin{aligned}&\frac{1}{\sqrt{n}}\sup _{nt<i\le n}\left| \sum _{i\le j \le n}\psi \left( Y_j-\widehat{\mu }_{2,[nt]}(\psi )\right) \right| \\&\quad =\sup _{nt< i \le n}\left| \widetilde{Z}_n\left( \frac{i}{n}\right) -\frac{n-i}{n-[nt]}\widetilde{Z}_n(t)\right| + o_{\mathsf {P}\,}(1),\quad n\rightarrow \infty . \end{aligned}$$

With respect to (10), we get for all $0<\gamma <1/2$

$$\begin{aligned}&\Bigg (\frac{1}{\sqrt{n}}\sup _{1<i\le nt}\left| \sum _{1\le j \le i}\psi \left( Y_j-\widehat{\mu }_{1,[nt]}(\psi )\right) \right| ,\\&\quad \frac{1}{\sqrt{n}}\sup _{nt-1<i\le n-1}\left| \sum _{i+1\le j \le n}\psi \left( Y_j-\widehat{\mu }_{2,[nt]}(\psi )\right) \right| \Bigg )\\&\xrightarrow [n\rightarrow \infty ]{\mathscr {D}^2[\gamma ,1-\gamma ]}\sigma (\psi ) \left( \displaystyle \underset{0\le u\le t}{\sup }\left| \mathcal {W}(u)-\frac{u}{t}\mathcal {W}(t)\right| ,\right. \\&\left. \quad \underset{t\le u\le 1}{\sup }\left| \widetilde{\mathcal {W}}(u)-\frac{1-u}{1-t}\widetilde{\mathcal {W}}(t)\right| \right) . \end{aligned}$$

The continuous mapping theorem completes the proof

$$\begin{aligned} \mathcal {R}_n(\psi ,\gamma )\xrightarrow [n\rightarrow \infty ]{\mathscr {D}} \sup _{\gamma \le t\le 1-\gamma }\frac{\underset{0\le u\le t}{\sup }\big |\mathcal {W}(u)-u/t\mathcal {W}(t)\big |}{\underset{t\le u\le 1}{\sup }\big |\widetilde{\mathcal {W}}(u)-(1-u)/(1-t)\widetilde{\mathcal {W}}(t)\big |}. \end{aligned}$$

$\square$

Proof of Theorem 2

Let $k>\tau +1$ and $k=[\xi n]$ for some $\zeta<\xi <1-\gamma$. Note that $\tau =O(n)$ and $k=O(n)$ as $n\rightarrow \infty$. By the mean value theorem, we get

$$\begin{aligned} 0= & {} \sum _{i=1}^k\psi \left( Y_i-\widehat{\mu }_{1k}(\psi )\right) \nonumber \\= & {} \sum _{i=1}^k\psi \left( Y_i-\mu \right) +\left[ \sum _{i=1}^k\frac{\text{ d }}{\text{ d }\mu }\bigg .\psi \left( Y_i-\mu \right) \bigg |_{\mu =\mu ^*}\right] \left( \widehat{\mu }_{1k}(\psi )-\mu \right) , \end{aligned}$$

(11)

where $\mu ^*$ lies between $\mu$ and $\widehat{\mu }_{1k}(\psi )$. The first sum in (11) can be expanded using Lemma 4.3 by Hušková and Marušiaková (2012) and Assumptions A1–A4 as

$$\begin{aligned} \sum _{i=1}^k\psi \left( Y_i-\mu \right)= & {} \sum _{i=1}^k\psi \left( \varepsilon _i+\delta _n\mathcal {I} \{i>\tau \}\right) \nonumber \\= & {} \sum _{i=1}^k\psi (\varepsilon _i)+\sum _{i=1}^k\mathsf {E}\,\psi \left( \varepsilon _i+\delta _n\mathcal {I}\{i>\tau \}\right) \nonumber \\&+\,o_{\mathsf {P}\,}\left( k^{\theta -\nu +1}\right) \end{aligned}$$

(12)

as $k\rightarrow \infty$ for any $\theta \in [-1/2,0]$ and $\nu \in \left( 0,\eta /(3(2+\chi +\chi '))\right)$. The Taylor expansion of $\psi$ in the neighborhood of 0 with respect to Assumption A4 provides

$$\begin{aligned} \sum _{i=1}^k\mathsf {E}\,\psi \left( \varepsilon _i+\delta _n\mathcal {I}\{i>\tau \}\right)= & {} \sum _{i=\tau +1}^k\mathsf {E}\,\psi \left( \varepsilon _i+\delta _n\right) =k(1-\zeta /\xi )\delta _n\lambda '(0)\nonumber \\&+\,k(1-\zeta /\xi )o(\delta _n),\quad k\rightarrow \infty . \end{aligned}$$

(13)

The second sum in (11) can be rewritten using Lemma 4.4 by Hušková and Marušiaková (2012) with respect to the Lipschitz property from Assumption A4 as

$$\begin{aligned} \sum _{i=1}^k\frac{\text{ d }}{\text{ d }\mu }\bigg .\psi \left( Y_i-\mu \right) \bigg |_{\mu =\mu ^*}&=\sum _{i=1}^k\frac{\text{ d }}{\text{ d }d}\bigg .\mathsf {E}\,\psi \left( \varepsilon _i+\delta _n\mathcal {I}\{i>\tau \}-d\right) \bigg |_{d=0}\nonumber \\&\quad +\,O_{\mathsf {P}\,}\left( k^{1/2+(1/2+\theta )/(3+\chi )}\right) \nonumber \\&=-\sum _{i=1}^k\lambda '(\delta _n\mathcal {I}\{i>\tau \})=-k\left( \lambda '(0)+O(\delta _n)\right) \nonumber \\&\quad +\,O_{\mathsf {P}\,}\left( k^{1/2+(1/2+\theta )/(3+\chi )}\right) \end{aligned}$$

(14)

as $k\rightarrow \infty$ for any $\theta \in [-1/2,0]$. Combining (11)–(14), we end up with

$$\begin{aligned} \widehat{\mu }_{1k}(\psi )-\mu&=\frac{\sum _{i=1}^k\psi (\varepsilon _i)+k(1-\zeta /\xi )\delta _n\lambda '(0)+k(1-\zeta /\xi )o(\delta _n)+o_{\mathsf {P}\,}\left( k^{\theta -\nu +1}\right) }{k\left( \lambda '(0)+O(\delta _n)\right) +O_{\mathsf {P}\,}\left( k^{1/2+(1/2+\theta )/(3+\chi )}\right) }\nonumber \\&=\frac{1}{k\lambda '(0)}\sum _{i=1}^k\psi (\varepsilon _i)+(1-\zeta /\xi )\delta _n\nonumber \\&\quad +\,o_{\mathsf {P}\,}\left( k^{\theta -\nu }\right) , \end{aligned}$$

(15)

since $\delta _n=O(k^{\theta })$ for some $\theta \in \left( -\frac{1}{2},\frac{\eta }{3(2+\chi +\chi ')}-\frac{1}{2}\right) \subset [-1/2,0]$.

Consequently, applying Lemma 4.3 by Hušková and Marušiaková (2012) again, we obtain

$$\begin{aligned} \max _{1\le i\le k}k^{\nu -\theta -1}\Bigg |\sum _{j=1}^i\left( \psi \left( Y_j-\widehat{\mu }_{1k}(\psi )\right) -\psi (\varepsilon _j)-\mathsf {E}\,\psi \left( Y_j-\widehat{\mu }_{1k}(\psi )\right) \right) \Bigg | \xrightarrow [n\rightarrow \infty ]{\mathsf {P}\,}0. \end{aligned}$$

(16)

Assumption A1 allows us to apply law of large numbers for $\alpha$-mixing sequences (Chen and Wu 1989). Hence,

$$\begin{aligned} \sum _{j=1}^i\mathsf {E}\,(\psi \left( Y_j-\widehat{\mu }_{1k}(\psi )\right)&=\sum _{j=1}^i\mathsf {E}\,\psi \left( \varepsilon _j+\delta _n\mathcal {I}\{j>\tau \}+\mu -\widehat{\mu }_{1k}(\psi )\right) \nonumber \\&=\sum _{j=1}^i\frac{1}{k}\sum _{l=1}^k\psi \left( \varepsilon _l+\delta _n\mathcal {I}\{j>\tau ,l>\tau \}\right. \nonumber \\&\left. \quad +\,\mu -\widehat{\mu }_{1k}(\psi )\right) +o_{\mathsf {P}\,}(i),\quad k\rightarrow \infty \end{aligned}$$

(17)

uniformly for $i=1,\ldots ,k$. Furthermore, due to the mean value theorem and Assumption A4, we have

$$\begin{aligned}&\sum _{j=1}^i\frac{1}{k}\sum _{l=1}^k\psi \left( \varepsilon _l+\delta _n\mathcal {I}\{j>\tau ,l>\tau \}+\mu -\widehat{\mu }_{1k}(\psi )\right) \nonumber \\&\quad =\sum _{j=1}^i\frac{1}{k}\sum _{l=1}^k\psi \left( \varepsilon _l\right) +\sum _{j=1}^i\frac{1}{k}\sum _{l=1}^k\left[ \frac{\text{ d }}{\text{ d }e}\bigg .\psi \left( \varepsilon _l+e\right) \bigg |_{e=e^*}\right] \nonumber \\&\qquad \left( \mu -\widehat{\mu }_{1k}(\psi )+\delta _n\mathcal {I}\{j>\tau ,l>\tau \}\right) \nonumber \\&\quad =\frac{i}{k}\sum _{l=1}^k\psi \left( \varepsilon _l\right) -\frac{1}{k}\sum _{l=1}^k\left( \lambda '(0)+O(\delta _n)\right) \sum _{j=1}^i\nonumber \\&\qquad \left( \mu -\widehat{\mu }_{1k}(\psi )+\delta _n\mathcal {I}\{j>\tau ,l>\tau \}\right) , \end{aligned}$$

(18)

where $e^*$ is between 0 and $\mu -\widehat{\mu }_{1k}(\psi )+\mathcal {I}\{j>\tau ,l>\tau \}$. Plugging (15) into (18) yields

$$\begin{aligned}&-\frac{1}{k}\sum _{l=1}^k\left( \lambda '(0)+O(\delta _n)\right) \sum _{j=1}^i\left( \mu -\widehat{\mu }_{1k}(\psi )+\delta _n\mathcal {I}\{j>\tau ,l>\tau \}\right) \nonumber \\&\quad =\lambda '(0)\left( \frac{i}{k\lambda '(0)}\sum _{s=1}^k\psi (\varepsilon _s)+i(1-\zeta /\xi ) \delta _n\right. \nonumber \\&\left. \qquad -\,i(1-(\tau +1)/i)(1-\zeta /\xi )\delta _n+i o_{\mathsf {P}\,}\left( k^{\theta -\nu }\right) \right) \nonumber \\&\quad =\frac{i}{k}\sum _{s=1}^k\psi (\varepsilon _s)+\lambda '(0)\delta _n(1-\zeta /\xi )(\tau +1)\nonumber \\&\qquad +\,o_{\mathsf {P}\,}\left( k^{\theta -\nu +1}\right) ,\quad k\rightarrow \infty . \end{aligned}$$

(19)

Let us take into account (16) together with (17), (18), and (19). Thus, we obtain

$$\begin{aligned}&k^{\nu -\theta -1}\Bigg |\sum _{j=1}^{\tau +1}\psi \left( Y_j-\widehat{\mu }_{1k}(\psi )\right) \nonumber \\&\qquad -\left( \sum _{j=1}^{\tau +1}\psi (\varepsilon _j)-\frac{\tau +1}{k}\sum _{l=1}^k \psi (\varepsilon _l)-\lambda '(0)\delta _n (1-\zeta /\xi )(\tau +1)\right) \Bigg |\\&\quad \le \max _{1\le i\le k}k^{\nu -\theta -1}\Bigg |\sum _{j=1}^i\psi \left( Y_j-\widehat{\mu }_{1k}(\psi )\right) \nonumber \\&\qquad -\,\left( \sum _{j=1}^i\psi (\varepsilon _j)-\frac{i}{k}\sum _{l=1}^k\psi (\varepsilon _l)-\lambda '(0)\delta _n(1-\zeta /\xi )(\tau +1)\right) \Bigg | \xrightarrow []{\mathsf {P}\,}0. \end{aligned}$$

as $k\rightarrow \infty$. Let us choose $\nu =\theta +1/2$. Thus, $\theta \in \left( -\frac{1}{2},\frac{\eta }{3(2+\chi +\chi ')}-\frac{1}{2}\right)$ iff $\nu \in \left( 0,\frac{\eta }{3(2+\chi +\chi ')}\right)$. Since $\delta _n=O\left( k^{\theta }\right)$ as $k\rightarrow \infty$ and

$$\begin{aligned} \frac{1}{\sqrt{k}}\left| \sum _{j=1}^{\tau +1}\psi (\varepsilon _j)-\frac{\tau +1}{k}\sum _{l=1}^k\psi (\varepsilon _l)\right| =O_{\mathsf {P}\,}(1),\quad n\rightarrow \infty \end{aligned}$$

according to the proof of Theorem 1 (requiring Assumption A5), we get

$$\begin{aligned}&\max _{1\le i\le k}\frac{1}{\sqrt{k}}\left| \sum _{j=1}^i\psi \left( Y_j-\widehat{\mu }_{1k}(\psi )\right) \right| \ge \frac{1}{\sqrt{k}}\left| \sum _{j=1}^{\tau +1}\psi \left( Y_j-\widehat{\mu }_{1k}(\psi )\right) \right| \xrightarrow [n\rightarrow \infty ]{\mathsf {P}\,}\infty . \end{aligned}$$

Note that there is no change in the means of $Y_k,\ldots ,Y_n$. Again from the proof of Theorem 1, we have

$$\begin{aligned}&\max _{[\xi n]\le i\le n-1}\frac{1}{\sqrt{n}}\left| \sum _{j=i+1}^n\psi \left( Y_j-\widehat{\mu }_{2k}(\psi )\right) \right| \xrightarrow [n\rightarrow \infty ]{\mathscr {D}[\gamma ,1-\gamma ]}\sigma (\psi )\\&\quad \sup _{\xi \le u\le 1}\left| \widetilde{\mathcal {W}}(u)-\frac{1-u}{1-\xi }\widetilde{\mathcal {W}}(\xi )\right| , \end{aligned}$$

which completes the proof. $\square$

Proof of Theorem 3

Let us denote $\mathsf {P}\,^*(\cdot )\equiv \mathsf {P}\,(\cdot |Y_1,\ldots ,Y_n)$. Moreover, let us define $\beta _n=o_{\mathsf {P}\,,\mathsf {P}\,^*}(1), n\rightarrow \infty$ for some random sequence $\{\beta _n\}_{n\in \mathbb {N}}$ as follows

$$\begin{aligned} \forall \epsilon>0,\,\forall \phi >0\,:\lim _{n\rightarrow \infty }\mathsf {P}\,\left[ \mathsf {P}\,^*\left[ \left| \beta _n\right| \ge \epsilon \right] \ge \phi \right] =0. \end{aligned}$$

Lemma 4.3 and Lemma 4.4 by Hušková and Marušiaková (2012) together with the mean value theorem used in a similar way as in the proof of Theorem 2 provide

$$\begin{aligned}&\sqrt{(l-1)K+k}\left( m_{L,K}^{\varvec{U}}(l,k)-\mu \right) \\&\quad =\frac{1}{\lambda '(0)\sqrt{(l-1)K+k}}\left[ \sum _{r=1}^{l-1}\sum _{s=1}^{K}\psi \left( \varepsilon _{U_r+s}\right) +\sum _{s=1}^{k}\psi \left( \varepsilon _{U_l+s}\right) \right] \\&\qquad +\delta _n\frac{\sqrt{(l-1)K+k}}{n}\left[ \sum _{r=1}^{l-1}\sum _{s=1}^{K}\mathcal {I}\{U_r+s>\tau \}+\sum _{s=1}^{k}\mathcal {I}\{U_l+s>\tau \}\right] \\&\qquad +o_{\mathsf {P}\,,\mathsf {P}\,^*}(1),\quad L\rightarrow \infty . \end{aligned}$$

Consequently, applying Lemma 4.3 by Hušková and Marušiaková (2012) again similarly as in the proof of Theorem 2, we obtain

$$\begin{aligned}&\max _{\begin{array}{c} (p,q)\in \varPi _{l,k,L,K} \end{array}}\Bigg |\sum _{i=1}^{p-1}\sum _{j=1}^{K}\psi \left( Y_{U_i+j}-m_{L,K}^{\varvec{U}}(l,k)\right) +\sum _{j=1}^q\psi \left( Y_{U_p+j}-m_{L,K}^{\varvec{U}}(l,k)\right) \Bigg . \end{aligned}$$

(20)

$$\begin{aligned}&\quad -\Bigg [\sum _{i=1}^{p-1}\sum _{j=1}^{K}\psi \left( \varepsilon _{U_i+j}\right) +\sum _{j=1}^q\psi \left( \varepsilon _{U_p+j}\right) \nonumber \\&\qquad -\frac{(p-1)K+q}{(l-1)K+k}\left( \sum _{r=1}^{l-1}\sum _{s=1}^{K}\psi \left( \varepsilon _{U_r+s}\right) +\sum _{s=1}^{k}\psi \left( \varepsilon _{U_l+s}\right) \right) \Bigg .\end{aligned}$$

(21)

$$\begin{aligned}&\quad \Bigg .\Bigg . -\lambda '(0)\delta _n\Bigg \{\sum _{i=1}^{p-1}\sum _{j=1}^{K}\mathcal {I}\{U_i+j>\tau \}+\sum _{j=1}^{q}\mathcal {I}\{U_p+j>\tau \}\Bigg .\end{aligned}$$

(22)

$$\begin{aligned}&\quad -\Bigg .\left( \frac{(p-1)K+q}{(l-1)K+k}\sum _{r=1}^{l-1}\sum _{s=1}^{K}\mathcal {I}\{U_r+s>\tau \}+\sum _{s=1}^{k}\mathcal {I}\{U_l+s>\tau \}\right) \Bigg \}\Bigg ]\Bigg |\nonumber \\&\qquad =o_{\mathsf {P}\,,\mathsf {P}\,^*}(\sqrt{KL}),\quad L\rightarrow \infty . \end{aligned}$$

(23)

Note that (20) contains $S_{L,K}^{\varvec{U}}(p,q,l,k)$ and the expression in square brackets in (21)–(23) can be rewritten as

$$\begin{aligned}&\sum _{i=1}^{p-1}\sum _{j=1}^{K}\left( \psi \left( \varepsilon _{U_i+j}\right) -\lambda '(0)\delta _n\mathcal {I}\{U_i+j>\tau \}\right) \\&\quad +\sum _{j=1}^q\left( \psi \left( \varepsilon _{U_p+j}\right) -\lambda '(0)\delta _n\mathcal {I}\{U_p+j>\tau \}\right) \\&\quad -\frac{(p-1)K+q}{(l-1)K+k}\Bigg (\sum _{r=1}^{l-1}\sum _{s=1}^{K}\left( \psi \left( \varepsilon _{U_r+s}\right) -\lambda '(0)\delta _n\mathcal {I}\{U_r+s>\tau \}\right) \Bigg .\\&\quad +\Bigg .\sum _{s=1}^{k}\left( \psi \left( \varepsilon _{U_l+s}\right) -\lambda '(0)\delta _n\mathcal {I}\{U_l+s>\tau \}\right) \Bigg )=:T_{L,K}^{\varvec{U}}(p,q,l,k). \end{aligned}$$

Furthermore, we define

$$\begin{aligned}&\widetilde{T}_{L,K}^{\varvec{U}}(p,q,l,k):=\sum _{j=k+1}^{K}\left( \psi \left( \varepsilon _{U_l+j}\right) -\lambda '(0)\delta _n\mathcal {I}\{U_l+j>\tau \}\right) \mathcal {I}\{p\ge l+1\}\\&\quad +\sum _{i=l+1}^{p-1}\sum _{j=1}^{K}\left( \psi \left( \varepsilon _{U_i+j}\right) -\lambda '(0)\delta _n\mathcal {I}\{U_i+j>\tau \}\right) \mathcal {I}\{p\ge l+2\}\\&\quad +\sum _{j=1}^{q}\left( \psi \left( \varepsilon _{U_p+j}\right) -\lambda '(0)\delta _n\mathcal {I}\{U_p+j>\tau \}\right) \\&\quad -\frac{(p-l)K+q-k}{(L-l)K+K-k}\Bigg (\sum _{s=k+1}^{K}\left( \psi \left( \varepsilon _{U_l+s}\right) -\lambda '(0)\delta _n\mathcal {I}\{U_l+s>\tau \}\right) \Bigg .\\&\quad +\sum _{r=l+1}^{L}\sum _{s=1}^{K}\left( \psi \left( \varepsilon _{U_r+s}\right) -\lambda '(0)\delta _n\mathcal {I}\{U_r+s>\tau \}\right) \Bigg ). \end{aligned}$$

If $\{\varepsilon _i, i\in \mathbb {N}\}$ is an $\alpha$-mixing sequence, then $\{\psi (\varepsilon _i), i\in \mathbb {N}\}$ is also $\alpha$-mixing, but with smaller or equal mixing coefficients than $\{\varepsilon _i, i\in \mathbb {N}\}$ (Bradley 2005, Theorem 5.2). Proof of Theorem 3.6.2 by Kirch (2006) for $q(t)=1,\,t\in (0,1)$; $e(i)=\psi (\varepsilon _i)$; and $d=-\lambda '(0)\delta _n$ together with Remarks 3.5.4 and 3.5.5 from Kirch (2006) provide, conditionally on $\varepsilon _1,\ldots ,\varepsilon _n$,

$$\begin{aligned}&\left( \max _{\begin{array}{c} (p,q)\in \varPi _{l,k,L,K} \end{array}} \frac{\left| T_{L,K}^{\varvec{U}}(p,q,l,k)\right| }{\sigma (\psi )\sqrt{LK}}, \max _{\begin{array}{c} (p,q)\in \widetilde{\varPi }_{l,k,L,K} \end{array}} \frac{\left| \widetilde{T}_{L,K}^{\varvec{U}}(p,q,l,k)\right| }{\sigma (\psi )\sqrt{LK}}\right) \nonumber \\&\quad \xrightarrow [L\rightarrow \infty ]{\mathscr {D}^2[\gamma ,1-\gamma ]} \left( \sup _{0\le u\le t}\left| \mathcal {W}(u)-u/t\mathcal {W}(t)\right| , \sup _{t\le u\le 1}\left| \widetilde{\mathcal {W}}(u)-(1-u)/(1-t)\widetilde{\mathcal {W}}(t)\right| \right) ,\nonumber \\ \end{aligned}$$

(24)

in probability $\mathsf {P}\,$ along $\varepsilon _1,\ldots ,\varepsilon _n$. In contrast to Kirch (2006), we drop the assumption of random errors being a linear process (Kirch 2006, Remark 3.5.3), because this is only assumed in order to show that the original (not bootstrapped) statistic weakly converges under the null hypothesis. The assumptions of Theorem 3 and the null hypothesis (2) provide us the asymptotic distribution of $\mathcal {R}_n(\psi ,\gamma )$ from Theorem 1.

By the uniform stochastic closeness (20)–(23) in a $\mathsf {P}\,$-stochastic sense and (24), we get, conditionally on $Y_1,\ldots ,Y_n$,

$$\begin{aligned}&\frac{1}{\sqrt{LK}}\Bigg ( \max _{\begin{array}{c} (p,q)\in \varPi _{l,k,L,K} \end{array}}\left| S_{L,K}^{\varvec{U}}(p,q,l,k)\right| ,\max _{\begin{array}{c} (p,q)\in \widetilde{\varPi }_{l,k,L,K} \end{array}}\left| \widetilde{S}_{L,K}^{\varvec{U}}(p,q,l,k)\right| \Bigg )\\&\quad \xrightarrow [L\rightarrow \infty ]{\mathscr {D}^2[\gamma ,1-\gamma ]} \sigma (\psi )\left( \sup _{0\le u\le t}\left| \mathcal {W}(u)-u/t\mathcal {W}(t)\right| ,\right. \nonumber \\&\left. \qquad \sup _{t\le u\le 1}\left| \widetilde{\mathcal {W}}(u)-(1-u)/(1-t)\widetilde{\mathcal {W}}(t)\right| \right) , \end{aligned}$$

in probability $\mathsf {P}\,$ along $Y_1,\ldots ,Y_n$. Finally, the assertion of Theorem 3 is straightforward, since the considered bootstrap statistic is a continuous function of the above vector of statistics. $\square$