A constructive hypothesis test for the single-index models with two groups

Abstract

Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.

Appendix

1.1 Proofs of Theorems 1 and 3

Lemma 1

Suppose that $\varvec{X}_{i}$, $i=1, \ldots , n$ are i.i.d. random vectors. Let $m(\varvec{x})$ be a continuous function and its derivatives up to second order are bounded, satisfying $E[m^{2}(\varvec{X})]<\infty$. $E[m(\varvec{X})|\varvec{\beta }^{\tau }\varvec{X}=u]$ has a continuous bounded second derivative on u. Let K(u) be a bounded positive function with a bounded support satisfying the Lipschitz condition: there exists a neighborhood of the origin, say $\Upsilon$, and a constant $c>0$ such that for any $\epsilon \in \Upsilon$: $|K(u+\epsilon )-K(u)|<c|\epsilon |$. Given that $h=n^{-d}$ for some $d<1$, we have, for $s_{0}>0$, and $j=0,1,2$,

$$\begin{aligned}&\displaystyle {\sup _{(x, {\varvec{\beta }})\in \mathcal {X} \times \Delta }} \Bigg |\frac{1}{n}\sum _{i=1}^{n}K_{h}(\varvec{\beta }^{\tau }\varvec{X}_{i}-\varvec{\beta }^{\tau }x)\left( \frac{\varvec{\beta }^{\tau }\varvec{X}_{i}-\varvec{\beta }^{\tau }x}{h} \right) ^{j}m(\varvec{X}_{i})\\&\qquad -\,f_{{\varvec{\beta }}_{0}^{\tau }{\varvec{X}}}(\varvec{\beta }_{0}^{\tau }\varvec{x}) E\big [m(\varvec{X})|\varvec{\beta }_{0}^{\tau }\varvec{X}=\varvec{\beta }_{0}^{\tau }x\big ] \mu _{K,j}-hS(\varvec{\beta }_{0}^{\tau }\varvec{x})\mu _{K,j+1}\Bigg |\\&\quad =\,O_{P}(c_{n}), \end{aligned}$$

where $\Delta = \{\varvec{\beta }\in \Theta , \Vert \varvec{\beta }-\varvec{\beta }_{0}\Vert \le Cn^{-1/2}\}$ for some positive constant C, $\Theta =\{\varvec{\beta }, \Vert \varvec{\beta }\Vert =1, \beta _{1}>0\}$, $\mu _{K,l}=\int t^{l}K(t)dt$, $S(\varvec{\beta }_{0}^{\tau }\varvec{x})=\frac{d }{du}\left\{ f_{{\varvec{\beta }}_{0}^{\tau }{\varvec{X}}}(u) E\big [m(\varvec{X})|\varvec{\beta }_{0}^{\tau }\varvec{X}=u\big ]\right\} |_{u={\varvec{\beta }}_{0}^{\tau }{\varvec{x}}}$, and $c_{n}=\left\{ \dfrac{(\log n)^{1+s_{0}}}{nh}\right\} ^{1/2}+h^{2}$.

Proof

This proof can be completed by a similar argument of Lemma A.4 in Wang et al. (2010). See also the Lemma A6.1 in Xia (2006). $\square$

Proofs of Theorems 1 and 3

We present the proof of Theorem 3. The proof of Theorem 1 is similar and we omit the details. We define $c_{n_{s}}=\left\{ \dfrac{(\log n_{s})^{1+s_{0}}}{n_{s}h_{s}}\right\} ^{1/2}+h_{s}^{2}$ for $s=1, 2$ for simplicity in the following.

Proof

Note that $\mathcal {W}_{n_{1}n_{2}}\left( \hat{\varvec{\beta }}_{\mathcal {H}_{0}}^{(1)}\right) =\mathbf{0}$. Taylor expansion entails that

$$\begin{aligned}&-\,\frac{1}{\sqrt{n_{1}+n_{2}}}\mathcal {W}_{n_{1}n_{2}}\left( {\varvec{\beta }}_{0}^{(1)}\right) \nonumber \\&\quad =\,\left[ \frac{1}{n_{1}+n_{2}}\frac{\partial \mathcal {W}_{n_{1}n_{2}}\left( {\varvec{\beta }}^{(1)}\right) }{\partial \varvec{\beta }^{(1)}}\Big |_{\varvec{\beta }^{(1)}=\tilde{\varvec{\beta }}^{(1)}_{0}}\right] \left[ \sqrt{n_{1}+n_{2}}\left( \hat{\varvec{\beta }}_{\mathcal {H}_{0}}^{(1)}-\varvec{\beta }_{0}^{(1)}\right) \right] ,\qquad \end{aligned}$$

(18)

where $\tilde{\varvec{\beta }}^{(1)}_{0}$ is between $\hat{\varvec{\beta }}_{\mathcal {H}_{0}}^{(1)}$ and $\varvec{\beta }_{0}^{(1)}$.

Step 1 :

In the following, we define $N=n_{1}+n_{2}$ for simplicity. In this step, we deal with ${{N}^{-1/2}}\mathcal {W}_{n_{1}n_{2}}\left( {\varvec{\beta }}_{0}^{(1)}\right)$. Using Lemma 1 and the detailed proofs of Lemma A.4 in Zhang et al. (2014), we have $\hat{g}_{s}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{si}, \varvec{\beta }_{0})=g_{s}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{si})+O_{P}(c_{ns})$, $\hat{V}_{s}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{si}, \varvec{\beta }_{0})=V_{s, {\varvec{\beta }}_{0}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{si})+O_{P}(c_{ns})$, for $s=1,2$. Moreover,

$$\begin{aligned}&S_{n_{1},l_{1}1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i},\varvec{\beta }_{0})\nonumber \\&\quad =\, \frac{1}{n_{1}}\sum _{j=1}^{n_{1}}K_{h_{1}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j}-\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}) (\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j}-\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})^{l_{1}} \sigma _{1}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j})\epsilon _{1j}^{2}\nonumber \\&\qquad +\,\frac{2}{n_{1}}\sum _{j=1}^{n_{1}}K_{h_{1}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j}-\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}) (\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j}-\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})^{l_{1}} \nonumber \\&\qquad \times \,\left[ g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j})-\hat{g}_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j}, \varvec{\beta }_{0})\right] \sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j})\epsilon _{1j}\nonumber \\&\qquad +\,\frac{1}{n_{1}}\sum _{j=1}^{n_{1}}K_{h_{1}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j}-\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}) (\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j}-\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})^{l_{1}}\nonumber \\&\qquad \times \,\left[ g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j})-\hat{g}_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1j}, \varvec{\beta }_{0})\right] ^{2}\nonumber \\&\quad =\,h_{1}^{l_{1}}f_{{\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}) \sigma _{1}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})\mu _{Kl_{1}}+O_{P}(h_{1}^{l_{1}}c_{n1}+h_{1}^{l_{1}}c_{n1}^{2}), \end{aligned}$$

(19)

for $l_{1}=0,1,2$. Using (19), we obtain $\hat{\sigma }_{1}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i},\varvec{\beta }_{0})=\sigma _{1}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})+O_{P}(c_{n1})$. Similarly, $\hat{\sigma }_{2}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i},\varvec{\beta }_{0})=\sigma _{2}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i})+O_{P}(c_{n2})$.

Let $G^{{\varvec{x}}}_{1,w}(u,\varvec{\beta })= E\left[ Y^{w}_{1}\{\varvec{X}_{1}-\varvec{x}\}|\varvec{\beta }^{\tau }\varvec{X}_{1}=u\right] f_{{\varvec{\beta }}^{\tau }{\varvec{X}}}(u)$, $K'_{h_{1}}(u)=\frac{1}{h_{1}}K'(u/h_{1})$. Using condition (C3), we have

$$\begin{aligned}&E\left[ \frac{\partial }{\partial \varvec{\beta }}T_{n_{1},l_{1}l_{2}}(\varvec{\beta }^{\tau }\varvec{x},\varvec{\beta })\right] \nonumber \\&\quad =\,\frac{1}{n_{1}}\sum _{i=1}^{n_{1}}E\left[ K'_{h_{1}} (\varvec{\beta }^{\tau }\varvec{X}_{1i}-\varvec{\beta }^{\tau }\varvec{x})J_{{\varvec{\beta }}}^{\tau }\left( \frac{\varvec{X}_{1i}-\varvec{x}}{h_{1}}\right) (\varvec{\beta }^{\tau }\varvec{X}_{1i}-\varvec{\beta }^{\tau }\varvec{x})^{l_{1}} Y_{1i}^{l_{2}}\right] \nonumber \\&\qquad +\,\frac{1}{n_{1}}\sum _{i=1}^{n_{1}}E\left[ K_{h_{1}}(\varvec{\beta }^{\tau }\varvec{X}_{1i}-\varvec{\beta }^{\tau }\varvec{x}) J_{{\varvec{\beta }}}^{\tau }\left( {\varvec{X}_{1i}-\varvec{x}}\right) l_{1}(\varvec{\beta }^{\tau }\varvec{X}_{1i}-\varvec{\beta }^{\tau }\varvec{x})^{l_{1}-1}I\{l_{1}\ge 1\} Y_{1i}^{l_{2}}\right] \nonumber \\&\quad =\,-\sum _{v=0}^{2}\frac{l_{1}+v}{v!}J_{{\varvec{\beta }}}^{\tau }G^{{\varvec{x}}(v)}_{1,l_{2}}(\varvec{\beta }^{\tau }\varvec{x},\varvec{\beta })h_{1}^{l_{1}-1+v} \mu _{K,l_{1}-1+v}I\{l_{1}+v\ge 1\}\nonumber \\&\qquad ~~~+\,\sum _{v=0}^{3}\frac{l_{1}}{v!}J_{{\varvec{\beta }}}^{\tau }G^{{\varvec{x}}(v)}_{1,l_{2}}(\varvec{\beta }^{\tau }\varvec{x},\varvec{\beta })h_{1}^{l_{1}-1+v}\mu _{K,l_{1}-1+v}I\{l_{1}\ge 1\} +O(h_{1}^{l_{1}+2}), \end{aligned}$$

(20)

where $G^{{\varvec{x}}(v)}_{1,l_{2}}(u,\varvec{\beta })=\frac{\partial ^{v}}{\partial u^{v}}G^{{\varvec{x}}}_{1,l_{2}}(u,\varvec{\beta })$, and $I\{u\}$ is the indicator function. Similar to the proof of Theorem 3.1 in Fan and Gijbels (1996) and Lemma A.5 in Zhang et al. (2014), together with (20) and Lemma 1, we can have

$$\begin{aligned}&\frac{\partial \hat{g}_{1}(\varvec{\beta }^{\tau }\varvec{X}_{1i}, \varvec{\beta })}{\partial \varvec{\beta }^{(1)}}\Big |_{\varvec{\beta }^{(1)}=\tilde{\varvec{\beta }}^{(1)}_{0}}\nonumber \\&\quad =\,J^{\tau }_{{\varvec{\beta }}_{0}} \left[ \varvec{X}_{1i}-V_{1,{\varvec{\beta }}_{0}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})\right] g_{1}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}) +O_{P}\left( h_{1}^{2}+\sqrt{\frac{(\log n_{1})^{1+s_{0}}}{n_{1}h_{1}^{3}}}\right) .\nonumber \\ \end{aligned}$$

(21)

Under the null hypothesis $\mathcal {H}_{0}$,

$$\begin{aligned}&\frac{\partial \hat{g}_{2}(\varvec{\beta }^{\tau }\varvec{X}_{2i}, \varvec{\beta })}{\partial \varvec{\beta }^{(1)}}\Big |_{\varvec{\beta }^{(1)}=\tilde{\varvec{\beta }}^{(1)}_{0}}\nonumber \\&\quad =\,J^{\tau }_{{\varvec{\beta }}_{0}} \left[ \varvec{X}_{2i}-V_{2,{\varvec{\beta }}_{0}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i})\right] g_{2}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i}) +O_{P}\left( h_{2}^{2}+\sqrt{\frac{(\log n_{2})^{1+s_{0}}}{n_{2}h_{2}^{3}}}\right) .\nonumber \\ \end{aligned}$$

(22)

Define that $\mathcal {Q}_{n_{1}}(u,\varvec{\beta }_{0})=\frac{1}{n_{1}h_{1}^{2}}T_{n_{1},20}(u, \varvec{\beta }_{0})\frac{1}{n_{1}}T_{n_{1},00}(u, \varvec{\beta }_{0})-\frac{1}{n_{1}^{2}h_{1}^{2}}T_{n_{1},10}^{2}(u, \varvec{\beta }_{0})$ and $\mathcal {L}_{n_{1}}(u,\varvec{\beta }_{0})=\frac{1}{n^{2}_{1}h_{1}^{2}}T_{n_{1},20}(u, \varvec{\beta }_{0})T_{n_{1},01}(u, \varvec{\beta }_{0})-\frac{1}{n^{2}_{1}h^{2}_{1}}T_{n_{1},10}(u, \varvec{\beta }_{0})T_{n_{1},11}(u, \varvec{\beta }_{0})$. Then, $\hat{g}_{1}(u, \varvec{\beta }_{0})=\frac{\mathcal {L}_{n_{1}}(u,{\varvec{\beta }}_{0})}{\mathcal {Q}_{n_{1}}(u,{\varvec{\beta }}_{0})}$ and $\hat{g}_{1}'(u, \varvec{\beta }_{0})=\frac{\partial \mathcal {L}_{n_{1}}(u,{\varvec{\beta }}_{0})/\partial u}{\mathcal {Q}_{n_{1}}(u,{\varvec{\beta }}_{0})}-\frac{\mathcal {L}_{n_{1}}(u,{\varvec{\beta }}_{0})\partial \mathcal {Q}_{n_{1}}(u,{\varvec{\beta }}_{0})/\partial u}{\mathcal {Q}^{2}_{n_{1}}(u,{\varvec{\beta }}_{0})}$. Following the proof of Lemma A.5 in Zhang et al. (2014), together with Lemma 1 and (20), we have $\hat{g}_{1}'(u, \varvec{\beta }_{0})=g_{1}'(u)+O_{P}\left( h_{1}^{2}+\sqrt{\frac{(\log n_{1})^{1+s_{0}}}{n_{1}h_{1}^{3}}}\right)$ and $\hat{g}_{1}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}, \varvec{\beta }_{0})=g_{1}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})+O_{P}\left( h_{1}^{2}+\sqrt{\frac{(\log n_{1})^{1+s_{0}}}{n_{1}h_{1}^{3}}}\right)$. Similarly, $\hat{g}_{2}'(u, \varvec{\beta }_{0})=g_{2}'(u)+O_{P}\left( h_{2}^{2}+\sqrt{\frac{(\log n_{2})^{1+s_{0}}}{n_{2}h_{2}^{3}}}\right)$, $\hat{g}_{2}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i}, \varvec{\beta }_{0})=g_{2}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i})+O_{P}\left( h_{2}^{2}+\sqrt{\frac{(\log n_{2})^{1+s_{0}}}{n_{2}h_{2}^{3}}}\right)$. Using the asymptotic results (21) and (22) and the condition of that $\frac{n_{1}}{n_{1}+n_{2}}=\frac{n_{1}}{N}\rightarrow \lambda \in (0, 1)$, as $\max \left\{ \frac{(\log n_{1})^{2+2s_{0}}}{n_{1}h_{1}^{2}}, \frac{(\log n_{2})^{2+2s_{0}}}{n_{2}h_{2}^{2}}\right\} \rightarrow 0$, and also $\max \{n_{1}h_{1}^{8},n_{2}h_{2}^{8}\}\rightarrow 0$, we have

$$\begin{aligned}&{({n_{1}+n_{2})}^{-1/2}}\mathcal {W}_{n_{1}n_{2}}\left( {\varvec{\beta }}_{0}^{(1)}\right) \nonumber \\&\quad =\,\sqrt{\frac{n_{1}}{n_{1}+n_{2}}}n_{1}^{-1/2} \sum _{i=1}^{n_{1}}J_{{\varvec{\beta }}_{0}}^{\tau }\frac{\hat{g}_{1}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}, \varvec{\beta }_{0})}{\hat{\sigma }_{1}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}, \varvec{\beta }_{0})}\left[ \varvec{X}_{1i}-\hat{V}_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}, \varvec{\beta }_{0})\right] \nonumber \\&\qquad \times \left[ Y_{1i}-\hat{g}_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}, \varvec{\beta }_{0})\right] \nonumber \\&\qquad +\,\sqrt{\frac{n_{2}}{n_{1}+n_{2}}}n_{2}^{-1/2} \sum _{i=1}^{n_{2}}J_{{\varvec{\beta }}_{0}}^{\tau }\frac{\hat{g}_{2}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i}, \varvec{\beta }_{0})}{\hat{\sigma }_{2}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i}, \varvec{\beta }_{0})}\left[ \varvec{X}_{2i}-\hat{V}_{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i}, \varvec{\beta }_{0})\right] \nonumber \\&\qquad \times \left[ Y_{2i}-\hat{g}_{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i}, \varvec{\beta }_{0})\right] \nonumber \\&\quad =\sqrt{\frac{n_{1}}{n_{1}+n_{2}}} n_{1}^{-1/2} \sum _{i=1}^{n_{1}}J_{{\varvec{\beta }}_{0}}^{\tau }{g}_{1}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})\left[ \varvec{X}_{1i}-{V}_{1, {\varvec{\beta }}_{0}} (\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})\right] {\sigma }_{1}^{-1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})\epsilon _{1i}\nonumber \\&\qquad +\,\sqrt{\frac{n_{2}}{n_{1}+n_{2}}} n_{2}^{-1/2} \sum _{i=1}^{n_{2}}J_{{\varvec{\beta }}_{0}}^{\tau }{g}_{2}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i})\left[ \varvec{X}_{2i}-{V}_{2, {\varvec{\beta }}_{0}} (\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i})\right] {\sigma }_{2}^{-1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i})\epsilon _{2i}\nonumber \\&\qquad +\,o_{P}(1), \end{aligned}$$

(23)

where $V_{2, {\varvec{\beta }}_{0}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2})=E[\varvec{X}_{2}|\varvec{\beta }^{\tau }_{0}\varvec{X}_{2}]$.

Step 2 :

In this sub-step, we deal with $\frac{1}{n_{1}+n_{2}}\frac{\partial \mathcal {W}_{n_{1}n_{2}}\left( {\varvec{\beta }}^{(1)}\right) }{\partial {\varvec{\beta }}^{(1)}}\big |_{{\varvec{\beta }}^{(1)}=\tilde{{\varvec{\beta }}}^{(1)}_{0}}$. Define

$$\begin{aligned}&\mathcal {S}_{n_{1}n_{2}}(\tilde{\varvec{\beta }}^{(1)}_{0})\mathop {=}\limits ^\mathrm{def}\frac{1}{n_{1}+n_{2}}\sum _{s=1}^{2}\sum _{i=1}^{n_{s}} \left[ Y_{si}-\hat{g}_{s}(\tilde{\varvec{\beta }}_{0}^{\tau }\varvec{X}_{si}, \tilde{\varvec{\beta }}_{0})\right] \\&\times \frac{\partial }{\partial \varvec{\beta }^{(1)}}\left\{ J_{{\varvec{\beta }}}^{\tau }\hat{g}_{s}'(\varvec{\beta }^{\tau }\varvec{X}_{si}, \varvec{\beta })\left[ \varvec{X}_{si}-\hat{V}_{s}(\varvec{\beta }^{\tau }\varvec{X}_{si}, \varvec{\beta })\right] \hat{\sigma }_{s}^{-2}(\varvec{\beta }^{\tau }\varvec{X}_{si}, \varvec{\beta })\right\} \Big |_{\varvec{\beta }^{(1)}=\tilde{\varvec{\beta }}^{(1)}_{0}}, \end{aligned}$$

and

$$\begin{aligned}&\mathcal {L}_{n_{1}n_{2}}(\tilde{\varvec{\beta }}^{(1)}_{0})\\&\mathop {=}\limits ^\mathrm{def}\frac{1}{n_{1}+n_{2}}\sum _{s=1}^{2}\sum _{i=1}^{n_{s}}\Big \{J_{\tilde{{\varvec{\beta }}}_{0}}^{\tau } \hat{g}_{s}'(\tilde{\varvec{\beta }}_{0}^{\tau }\varvec{X}_{si}, \tilde{\varvec{\beta }}_{0})\left[ \varvec{X}_{si}-\hat{V}_{s}(\tilde{\varvec{\beta }}_{0}^{\tau }\varvec{X}_{si}, \tilde{\varvec{\beta }}_{0})\right] \\&\quad \times \hat{\sigma }_{s}^{-2}(\tilde{\varvec{\beta }}_{0}^{\tau }\varvec{X}_{si}, \tilde{\varvec{\beta }}_{0})\Big \}\frac{\partial \hat{g}_{s}(\varvec{\beta }^{\tau }\varvec{X}_{si}, \varvec{\beta })}{\partial \varvec{\beta }^{(1)}}\Bigg |_{\varvec{\beta }^{(1)}=\tilde{\varvec{\beta }}^{(1)}_{0}}. \end{aligned}$$

Then,

$$\begin{aligned} \frac{1}{n_{1}+n_{2}}\frac{\partial \mathcal {W}_{n_{1}n_{2}}\left( {\varvec{\beta }}^{(1)}\right) }{\partial \varvec{\beta }^{(1)}} \Bigg |_{\varvec{\beta }^{(1)}=\tilde{\varvec{\beta }}^{(1)}_{0}} =\mathcal {S}_{n_{1}n_{2}}(\tilde{\varvec{\beta }}^{(1)}_{0})+\mathcal {L}_{n_{1}n_{2}}(\tilde{\varvec{\beta }}^{(1)}_{0}), \end{aligned}$$

(24)

where $\tilde{\varvec{\beta }}_{0}=\left( \sqrt{1-\tilde{\varvec{\beta }}_{0}^{(1)\tau }\tilde{\varvec{\beta }}_{0}^{(1)}}, \tilde{\varvec{\beta }}_{0}^{(1)\tau }\right) ^{\tau }$. Note that $\tilde{\varvec{\beta }}^{(1)}_{0}$ is between $\hat{\varvec{\beta }}_{\mathcal {H}_{0}}^{(1)}$ and $\varvec{\beta }_{0}^{(1)}$. By using (18), we have $\hat{\varvec{\beta }}_{\mathcal {H}_{0}}^{(1)}=\varvec{\beta }_{0}^{(1)}+O_P((n_{1}+n_{2})^{-1/2})$. Note that $\tilde{\varvec{\beta }}^{(1)}_{0}\mathop {\longrightarrow }\limits ^{P}\varvec{\beta }_{0}^{(1)}$, $\tilde{\varvec{\gamma }}^{(1)}_{0}\mathop {\longrightarrow }\limits ^{P}\varvec{\beta }_{0}^{(1)}$ and $\tilde{\varvec{\beta }}^{}_{0}\mathop {\longrightarrow }\limits ^{P}\varvec{\gamma }_{0}^{}$, $\tilde{\varvec{\gamma }}^{}_{0}\mathop {\longrightarrow }\limits ^{P}\varvec{\gamma }_{0}^{}$. Together with (21)–(22) and condition of that $\frac{n_{1}}{n_{1}+n_{2}}\rightarrow \lambda \in (0, 1)$, we have

$$\begin{aligned}&\mathcal {L}_{n_{1}n_{2}}(\tilde{\varvec{\beta }}^{(1)}_{0})\mathop {\longrightarrow }\limits ^{P} \lambda J_{{{\varvec{\beta }}}_{0}}^{\tau }E\left[ \frac{g_{1}^{'2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1})}{ {\sigma }_{1}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1})}\left[ \varvec{X}_{1}-V_{1,{\varvec{\beta }}_{0}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1})\right] ^{\otimes 2}\right] J_{{{\varvec{\beta }}}_{0}}\nonumber \\&\quad +\, (1-\lambda ) J_{{{\varvec{\beta }}}_{0}}^{\tau }E\left[ \frac{g^{'2}_{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2})}{{\sigma }_{2}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2})} \left[ \varvec{X}_{2}-V_{2, {\varvec{\beta }}_{0}}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2})\right] ^{\otimes 2}\right] J_{{{\varvec{\beta }}}_{0}}. \end{aligned}$$

(25)

Moreover, a direct calculation for $\mathcal {S}_{n_{1}n_{2}}(\tilde{\varvec{\beta }}^{(1)}_{0})$ and Lemma 1 entail that $\mathcal {S}_{n_{1}n_{2}}(\tilde{\varvec{\beta }}^{(1)}_{0})=o_{P}(1)$. Together with (23) and (25), we complete the proof of Theorem 2. $\square$

1.2 Proof of Theorem 3

Proof

From the proof of Theorem 3, we can have that

(26)

$$\begin{aligned}&\sqrt{n_{2}}\left( \hat{\varvec{\gamma }}^{(1)}_{0}-{\varvec{\gamma }}^{(1)}_{0}\right) \nonumber \\&\quad =\,\varvec{\varOmega }_{2}^{-1} n_{2}^{-1/2} \sum _{i=1}^{n_{2}}J_{{\varvec{\gamma }}_{0}}^{\tau }{g}_{2}'(\varvec{\gamma }_{0}^{\tau }\varvec{X}_{2i})\left[ \varvec{X}_{2i}-{V}_{2, {\varvec{\gamma }}_{0}} (\varvec{\gamma }_{0}^{\tau }\varvec{X}_{2i})\right] {\sigma }_{2}^{-1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{2i})\epsilon _{2i}+o_{P}(1).\nonumber \\ \end{aligned}$$

(27)

Under the null hypothesis $\mathcal {H}_{0}: \varvec{\beta }_{0}=\varvec{\gamma }_{0}$, we can have

$$\begin{aligned}&\sqrt{n_{1}+n_{2}}\left( \hat{\varvec{\beta }}^{(1)}_{0}-\hat{\varvec{\gamma }}^{(1)}_{0}\right) \\&\quad =\,\left( \frac{\sqrt{n_{1}}}{\sqrt{\lambda }}\left( \hat{\varvec{\beta }}_{0}^{(1)}-\varvec{\beta }_{0}^{(1)}\right) - \frac{\sqrt{n_{2}}}{\sqrt{1-\lambda }}\left( \hat{\varvec{\gamma }}_{0}^{(1)}-\varvec{\beta }_{0}^{(1)}\right) \right) +o_{P}(1)\\&\quad \xrightarrow [\mathcal {H}^{*}_{0}]{\mathcal {L}}N\left( \mathbf{0}_{p-1}, \frac{1}{\lambda }\varvec{\varOmega }_{1}^{-1}+ \frac{1}{1-\lambda }\varvec{\varOmega }_{2}^{-1}\right) . \end{aligned}$$

Moreover,

$$\begin{aligned}&(n_{1}+n_{2})\widehat{\varvec{A}}\\&\quad =\,\frac{n_{1}+n_{2}}{n_{1}}\left[ J^{\tau }_{\hat{{\varvec{\beta }}}_{0}} \frac{1}{n_{1}}\sum _{i=1}^{n_{1}}\frac{ \hat{g}_{1}^{'2}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{X}_{1i}, \hat{\varvec{\beta }}_{0})}{\hat{\sigma }_{1}^{2}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{X}_{1i}, \hat{\varvec{\beta }}_{0}),}\left[ \varvec{X}_{1i}-\hat{V}_{1}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{X}_{1i}, \hat{\varvec{\beta }}_{0})\right] ^{\otimes 2}J_{\hat{{\varvec{\beta }}}_{0}}\right] ^{-1}\\&\qquad ~~~+\,\frac{n_{1}+n_{2}}{n_{2}}\left[ J^{\tau }_{\hat{{\varvec{\gamma }}}_{0}}\frac{1}{n_{2}} \sum _{i=1}^{n_{2}}\frac{ \hat{g}_{2}^{'2}(\hat{\varvec{\gamma }}_{0}^{\tau }\varvec{X}_{2i}, \hat{\varvec{\gamma }}_{0})}{\hat{\sigma }_{2}^{2}(\hat{\varvec{\gamma }}_{0}^{\tau }\varvec{X}_{2i}, \hat{\varvec{\gamma }}_{0}),}\left[ \varvec{X}_{2i}-\hat{V}_{2}(\hat{\varvec{\gamma }}_{0}^{\tau }\varvec{X}_{2i}, \hat{\varvec{\gamma }}_{0})\right] ^{\otimes 2}J_{\hat{{\varvec{\gamma }}}_{0}}\right] ^{-1}\\&\quad \mathop {\longrightarrow }\limits ^{P}\frac{1}{\lambda }\varvec{\varOmega }_{1}^{-1}+ \frac{1}{1-\lambda }\varvec{\varOmega }_{2}^{-1}. \end{aligned}$$

Then, the Slutsky Theorem and continuous mapping theorem entail that

$$\begin{aligned}&\mathcal {T}_{n_{1}n_{2}}\\&\quad =\,\left[ \sqrt{n_{1}+n_{2}}\left( \hat{\varvec{\beta }}^{(1)}_{0}-\hat{\varvec{\gamma }}^{(1)}_{0}\right) \right] ^{\tau } \left( (n_{1}+n_{2})\widehat{\varvec{A}}\right) ^{-1}\left[ \sqrt{n_{1}+n_{2}}\left( \hat{\varvec{\beta }}^{(1)}_{0}-\hat{\varvec{\gamma }}^{(1)}_{0}\right) \right] \\&\qquad \times \xrightarrow [\mathcal {H}^{*}_{0}]{\mathcal {L}}\chi ^{2}_{p-1}. \end{aligned}$$

We complete the proof of Theorem 3. $\square$

1.3 Proof of Theorem 4

Lemma 2

Suppose that conditions (C1)–(C5) hold. Let $F_{\hat{\epsilon }_{s}}(t|\mathscr {Q}_{n_{s}})$ be the distribution function of $\hat{\epsilon }_{s}=\frac{Y_{s}-\hat{g}_{s}(\hat{\omega }_{s,0}^{\tau }{\varvec{X}}_{s}, \hat{\omega }_{s,0})}{\hat{\sigma }_{s}(\hat{\omega }_{s,0}^{\tau }{\varvec{X}}_{s},\hat{\omega }_{s,0})}$ conditional on the data $\mathscr {Q}_{n_{s}}=\{\varvec{X}_{si}, Y_{si}\}_{i=1}^{n_{s}}$ (i.e., considering $\hat{g}_{s}\left( \hat{\omega }_{s,0}^{\tau }\varvec{x}_{s}, \hat{\omega }_{s,0}\right) , \hat{\sigma }_{s}\left( \hat{\omega }_{s,0}^{\tau }\varvec{x}_{s},\hat{\omega }_{s,0}\right)$ as fixed functions on $\varvec{x}_{s}$) for $s=1,2$ respectively. Here, $\hat{\omega }_{1,0}=\hat{\varvec{\beta }}_{0}$ and $\hat{\omega }_{2,0}=\hat{\varvec{\gamma }}_{0}$. Then, we have

$$\begin{aligned}&\sup _{t\in \mathbb {R}}\left| n_{1}^{-1}\sum _{i=1}^{n_{1}}\left[ I\left\{ \hat{\epsilon }_{1i}\le t\right\} -I\{\epsilon _{1i}\le t\}- F_{\hat{\epsilon }_{1}}(t|\mathscr {Q}_{n_{1}})+F_{\epsilon _{1}}(t)\right] \right| \nonumber \\&\quad =\,o_{P}\left( n_{1}^{-1/2}\right) , \end{aligned}$$

(28)

$$\begin{aligned}&\sup _{t\in \mathbb {R}}\left| n_{2}^{-1}\sum _{i=1}^{n_{2}}\left[ I\left\{ \hat{\epsilon }_{2i}\le t\right\} -I\{\epsilon _{2i}\le t\}- F_{\hat{\epsilon }_{2}}(t|\mathscr {Q}_{n_{2}})+F_{\epsilon _{2}}(t)\right] \right| \nonumber \\&\quad =\,o_{P}\left( n_{2}^{-1/2}\right) . \end{aligned}$$

(29)

Proof

In the following, we only prove (28), the proof of (29) is similar and we omit the details. Let

$$\begin{aligned} \mathscr {O}= & {} \Big \{I\{\epsilon _{1}\le tf_{2}(\varvec{X}_{1})+f_{1}(\varvec{X}_{1})\}-I\{\epsilon _{1}\le t\}- P(\epsilon _{1}\le tf_{2}(\varvec{X}_{1})+f_{1}(\varvec{X}_{1}))\nonumber \\&\quad ~~~~+\,P(\epsilon _{1}\le t); t\in \mathbb {R}, f_{1}, f_{2}\in M_{1}^{1+\delta }(\mathfrak {R}_{c}^{p}) \Big \}, \end{aligned}$$

where $M_{1}^{1+\delta }( \mathfrak {R}_{c}^{p})$ is the class of all differential functions f(u) defined on the domain $\mathfrak {R}_{c}^{p}$ of $\varvec{x}_{1}$ and $\Vert f\Vert _{1+\delta }\le 1$. Here $\mathfrak {R}_{c}^{p}$ is a compact set of $\mathbb {R}^{p}$ and

$$\begin{aligned}&\Vert f\Vert _{1+\delta }\\&\quad =\sup _{{\varvec{x}}_{1}\in \mathfrak {R}_{c}^{p}}\left| f(\varvec{x}_{1})\right| +\sum _{l=1}^{p}\sup _{{\varvec{x}}_{1}\in \mathfrak {R}_{c}^{p}}\left| \frac{\partial f({\varvec{x}}_{1})}{\partial x_{1l}}\right| +\sup _{{\varvec{x}}_{1,1}, {\varvec{x}}_{1,2}\in \mathfrak {R}_{c}^{p}}\frac{|\partial f( {\varvec{x}}_{1,1})-\partial f( {\varvec{x}}_{1,2})|}{\Vert {\varvec{x}}_{1,1}-{\varvec{x}}_{1,2}\Vert ^{\delta }}. \end{aligned}$$

Using Lemma 1 and $\Vert \hat{\varvec{\beta }}_{0}-\varvec{\beta }_{0}\Vert =O_{P}(n_{1}^{-1/2})$, and similar to the proofs of (21) and (22), we have that

$$\begin{aligned} \hat{g}_{1}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}, \hat{\varvec{\beta }}_{0})= & {} g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})+O_{P}\left( n_{1}^{-1/2} +c_{n_{1}}\right) , \end{aligned}$$

(30)

$$\begin{aligned} \hat{\sigma }_{1}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}, \hat{\varvec{\beta }}_{0})= & {} \sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1}) =O_{P}\left( n_{1}^{-1/2}+c_{n_{1}}\right) , \end{aligned}$$

(31)

uniformly in $\varvec{x}_{1}\in \mathfrak {R}_{c}^{p}$. Let $A_{n_{1}}(\varvec{x}_{1})=\frac{\hat{g}_{1}(\hat{{\varvec{\beta }}}_{0}^{\tau }{\varvec{x}}_{1}, \hat{{\varvec{\beta }}}_{0})-{g}_{1}({{\varvec{\beta }}}_{0}^{\tau } {\varvec{x}}_{1})}{\sigma _{1}({{\varvec{\beta }}}_{0}^{\tau } {\varvec{x}})}$, $B_{n_{1}}(\varvec{x})=\frac{\hat{\sigma }_{1}(\hat{{\varvec{\beta }}}_{0}^{\tau }{\varvec{x}}_{1}, \hat{{\varvec{\beta }}}_{0})}{\sigma _{1}({{\varvec{\beta }}}_{0}^{\tau }{\varvec{x}}_{1})}$. So, (30) and (31) entail $P\left( A_{n_{1}}\in M_{1}^{1+\delta }(\mathfrak {R}_{c}^{p}) \right) \rightarrow 1$, $P\left( B_{n_{1}}\in M_{1}^{1+\delta }(\mathfrak {R}_{c}^{p})\right) \rightarrow 1$ as $n_{1}\rightarrow \infty$, $h_{1}\rightarrow 0$ and $\frac{n_{1}h_{1}}{(\log n_{1})^{1+s}}\rightarrow \infty$.

By directly using the Corollary 2.7.2 of van der Vaart and Wellner (1996), the bracketing number $N_{[~]}\left( \upsilon ^{2}, M_{1}^{1+\delta }(\mathfrak {R}_{c}^{p}), L_{2}(P)\right)$ can be at most $\exp \left( c_{0}\upsilon ^{-\frac{2p}{1+\delta }}\right)$ for some positive constant $c_{0}$, According to the proof of Lemma 1 in Appendix B of Akritas and Van Keilegom (2001), and then the class $\mathscr {O}$ defined above is a Donsker class, i.e., we have that ${\displaystyle \int }_{0}^{\infty }\sqrt{N_{[~]}(\bar{\upsilon }, \mathscr {O}, L_{2}(P))}d\bar{\upsilon }<\infty$. Then, the proof of (28) is complete. $\square$

Proof of Theorem 4

We can have that

$$\begin{aligned}&\hat{F}_{\epsilon _{1}}(t)-F_{\epsilon _{1}}(t)\nonumber \\&\quad =\frac{1}{n_{1}}\sum _{i=1}^{n_{1}}I\left\{ {\epsilon }_{1i}\le t\right\} -F_{\epsilon _{1}}(t)+\left( F_{\hat{\epsilon }_{1}}(t|\mathscr {Q}_{n_{1}})-F_{\epsilon _{1}}(t)\right) +R_{n_{1},1}(t), \end{aligned}$$

(32)

where $R_{n_{1},1}(t)=o_{P}(n_{1}^{-1/2})$ uniformly in $t\in \mathbb {R}$ by using Lemma 2. Taylor expansion entails that

$$\begin{aligned}&F_{\hat{\epsilon }_{1}}(t|\mathscr {Q}_{n_{1}})-F_{\epsilon _{1}}(t)\nonumber \\&\quad =\,\int \left[ F_{\epsilon _{1}}(t+t[B_{n_{1}}(\varvec{x}_{1})-1]+A_{n_{1}}(\varvec{x}_{1})) -F_{\epsilon _{1}}(t)\right] dF_{{\varvec{X}}_{1}}(\varvec{x}_{1})\nonumber \\&\quad =\,f_{\epsilon _{1}}(t)t\int [B_{n_{1}}(\varvec{x}_{1})-1]dF_{{\varvec{X}}_{1}}(\varvec{x}_{1})+f_{\epsilon _{1}}(t)\int A_{n_{1}}(\varvec{x}_{1})dF_{{\varvec{X}}_{1}}(\varvec{x}_{1})\nonumber \\&\qquad +\,\int f'_{\epsilon _{1}}(t+v^{*}_{n_{1}}(t,\varvec{x}_{1}))\left\{ t[B_{n_{1}}(\varvec{x}_{1})-1]+A_{n_{1}}(\varvec{x}_{1}) \right\} ^{2}dF_{{\varvec{X}}_{1}}(\varvec{x}_{1})\nonumber \\&\quad =\,R_{n_{1},2}(t)+R_{n_{1},3}(t)+R_{n_{1},4}(t), \end{aligned}$$

(33)

where $v_{n_{1}}^{*}(t,\varvec{x}_{1})$ is between 0 and $t[B_{n_{1}}(\varvec{x}_{1})-1]+A_{n_{1}}(\varvec{x}_{1})$. Note that

$$\begin{aligned}&A_{n_{1}}(\varvec{x}_{1})\nonumber \\&\quad =\,\frac{\hat{g}_{1}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}, \varvec{\beta }_{0})- \hat{g}_{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1}, \varvec{\beta }_{0})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}+ \frac{\hat{g}_{1}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}, \varvec{\beta }_{0}) -g(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}. \end{aligned}$$

(34)

Recall the definition of $\hat{g}_{1}(u,\varvec{\beta }_{0})$ and using Lemma 1,

$$\begin{aligned}&\hat{g}_{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1},\varvec{\beta }_{0})-g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})\nonumber \\&\quad =\,\frac{1}{n_{1}f_{{\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1}}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})} \sum _{i=1}^{n_{1}}K_{h_{1}}\left( {\varvec{\beta }}_{0}^{\tau }\varvec{X}_{1i}-{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}\right) \sigma _{1}({\varvec{\beta }}_{0}^{\tau }\varvec{X}_{1i})\epsilon _{1i}+O_{P}(c_{n_{1}}).\qquad \end{aligned}$$

(35)

Similar to (21), we can also have

$$\begin{aligned}&\hat{g}_{1}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}, \varvec{\beta }_{0})- \hat{g}_{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1}, \varvec{\beta }_{0}) \nonumber \\&\quad =\, \left[ \varvec{x}_{1}-V_{1,{\varvec{\beta }}_{0}}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})\right] ^{\tau }g_{1}'(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})\left( \hat{\varvec{\beta }}_{0}-\varvec{\beta }_{0}\right) +O_{P}\left( h_{1}^{2}+\sqrt{\frac{(\log n_{1})^{1+s}}{n_{1}h_{1}^{3}}}\right) .\nonumber \\ \end{aligned}$$

(36)

Together with (34), (35) and (36), we have

$$\begin{aligned} R_{n_{1},3}(t)=f_{\epsilon _{1}}(t)\int A_{n_{1}}(\varvec{x}_{1})dF_{{\varvec{X}}_{1}}(\varvec{x}_{1})=\frac{f_{\epsilon _{1}}(t)}{n_{1}}\sum _{i=1}^{n_{1}}\epsilon _{1i}+o_{P}(n_{1}^{-1/2}). \end{aligned}$$

(37)

Similarly,

$$\begin{aligned}&\hat{\sigma }^{2}_{1}\left( \hat{{\varvec{\beta }}}_{0}^{\tau }{\varvec{x}}_{1}, \hat{{\varvec{\beta }}}_{0}\right) -\sigma _{1}^{2}({\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1})\nonumber \\&\quad =\frac{1}{f^{}_{{\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1}}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})n_{1}} \sum _{i=1}^{n_{1}}K_{h_{1}}\left( {\varvec{\beta }}_{0}^{\tau }\varvec{X}_{1i}-{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}\right) \sigma _{1}^{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})(\epsilon _{1i}^{2}-1)\nonumber \\&\quad ~~~+\,2\left[ \varvec{x}_{1}-V_{1,{\varvec{\beta }}_{0}}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})\right] ^{\tau }\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1}) \sigma _{1}'(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1}) \left( \hat{\varvec{\beta }}_{0}-\varvec{\beta }_{0}\right) +o_{P}(n_{1}^{-1/2}).\qquad \qquad \end{aligned}$$

(38)

Then, Taylor expansion for $\sqrt{\hat{\sigma }^{2}_{1}\left( \hat{{\varvec{\beta }}}_{0}^{\tau }{\varvec{x}}, \hat{{\varvec{\beta }}}_{0}\right) }-\sqrt{\sigma ^{2}_{1}({\varvec{\beta }}_{0}^{\tau }\varvec{x})}$ and asymptotic expression (38) entail that

$$\begin{aligned} R_{n_{1},2}(t)= & {} tf_{\epsilon _{1}}(t)\int [B_{n_{1}}(\varvec{x}_{1})-1]dF_{{\varvec{X}}_{1}}(\varvec{x}_{1})\nonumber \\= & {} tf_{\epsilon _{1}}(t)\frac{1}{2n_{1}}\sum _{i=1}^{n_{1}}(\epsilon _{1i}^{2}-1)+o_{P}(n_{1}^{-1/2}). \end{aligned}$$

(39)

Moreover, (34), (39) and Condition (C5) entail that $R_{n_{1},4}(t)=o_{P}(n_{1}^{-1/2})$ uniformly in t. Together with (32), (26) and (37)–(39), we complete the proof of Theorem 2.

$\square$

1.4 Proof of Theorems 5 and 6

Proof

Recalling the definition of $F^{*}_{\widetilde{\mathcal {H}}_{0},\epsilon _{1}}(t)=E\left[ F_{\epsilon _{1}}\left( t+\frac{ g_{2}({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})-g_{1}({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})}{\sigma ({\varvec{\beta }}_{0}^{\tau }{\varvec{X}})}\right) \right]$.

$$\begin{aligned}&\hat{F}_{\widetilde{\mathcal {H}}_{0},\epsilon _{1}}(t)-F^{*}_{\widetilde{\mathcal {H}}_{0},\epsilon }(t)= \frac{1}{n_{1}}\sum _{i=1}^{n_{1}}I\{\hat{\epsilon }_{\widetilde{\mathcal {H}}_{0},i}\le t\}-F^{*}_{\widetilde{\mathcal {H}}_{0}, \epsilon _{1}}(t)\nonumber \\&\quad =\,\frac{1}{n_{1}}\sum _{i=1}^{n_{1}}I\left\{ {\epsilon }_{1i}+\frac{g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}) -g_{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})}\le t\right\} -F^{*}_{\widetilde{\mathcal {H}}_{0},\epsilon _{1}}(t)\nonumber \\&\qquad + \,\left[ F_{\widetilde{\mathcal {H}}_{0},\hat{\epsilon }_{1}}(t|\mathscr {V}_{n_{1}n_{2}}) -F^{*}_{\widetilde{\mathcal {H}}_{0}, \epsilon _{1}}(t)\right] +S_{n_{1},1}(t), \end{aligned}$$

(40)

where $F_{\widetilde{\mathcal {H}}_{0}, \hat{\epsilon }_{1}}(t|\mathscr {V}_{n_{1}n_{2}})$ be the distribution function of $\hat{\epsilon }_{\widetilde{\mathcal {H}}_{0},1}=\frac{Y-\hat{g}_{2} (\hat{{\varvec{\beta }}}_{0}^{\tau } {\varvec{X}},\hat{{\varvec{\gamma }}}_{0})}{\hat{\sigma }_{1}(\hat{{\varvec{\beta }}}_{0}^{\tau }{\varvec{X}},\hat{{\varvec{\beta }}}_{0})}$ conditional on the data $\mathscr {V}_{n_{1}n_{2}}=\{\varvec{X}_{1i}, Y_{1i},\varvec{X}_{2j}, Y_{2j}, 1\le i\le n_{1}, 1\le j \le n_{2} \}$, and similar to the analysis of Lemma 2, we have $\displaystyle {\sup _{t\in \mathbb {R}}}|S_{n_{1},1}(t)|=o_{P}(n_{1}^{-1/2})$. Taylor expansion entails that

$$\begin{aligned}&F_{\widetilde{\mathcal {H}}_{0},\hat{\epsilon }_{1}}(t|\mathscr {V}_{n_{1}n_{2}}) -F^{*}_{\widetilde{\mathcal {H}}_{0}, \epsilon _{1}}(t)\nonumber \\&\quad =\,\int F_{\epsilon _{1}}\Bigg (t+\frac{g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1}) -g_{2}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}+[B_{n_{1}}(\varvec{x}_{1})-1]t\nonumber \\&\qquad +\,\frac{\hat{g}_{2}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}, \hat{\varvec{\gamma }}_{0}) -g_{2}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}\Bigg )dF_{{\varvec{X}}_{1}}(\varvec{x}_{1})-F^{*}_{\widetilde{\mathcal {H}}_{0}, \epsilon _{1}}(t)\nonumber \\&\quad =\,t\int f_{\epsilon _{1}}\left( t+\frac{g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1}) -g_{2}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}\right) [B_{n_{1}}(\varvec{x}_{1})-1]dF_{{\varvec{X}}_{1}}(\varvec{x}_{1})\nonumber \\&\qquad +\,\int f_{\epsilon _{1}}\left( t+\frac{g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1}) -g_{2}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}\right) \frac{\hat{g}_{2}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}, \hat{\varvec{\gamma }}_{0}) -g_{2}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})}dF_{{\varvec{X}}_{1}}(\varvec{x}_{1})\nonumber \\&\qquad +\,R_{n_{1}n_{2}}(t). \end{aligned}$$

(41)

Similar to the analysis of (21), we can have that

$$\begin{aligned}&\hat{g}_{2}(\hat{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}, \varvec{\gamma }_{0})- \hat{g}_{2}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1}, \varvec{\gamma }_{0}) \nonumber \\&\quad =g_{2}'(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})\varvec{x}_{1}^{\tau }\left( \hat{\varvec{\beta }}_{0}-\varvec{\beta }_{0}\right) +O_{P}\left( h_{2}^{2}+\sqrt{\frac{(\log n_{2})^{1+s_{0}}}{n_{2}h_{2}^{3}}}\right) . \end{aligned}$$

(42)

Recall the definition of $\hat{g}_{2}(u,\varvec{\gamma }_{0})$ and using Lemma 1,

$$\begin{aligned}&\hat{g}_{2}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1},\varvec{\gamma }_{0})-g_{2}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})\nonumber \\&\quad =\frac{1}{n_{2}f_{{\varvec{\gamma }}_{0}^{\tau }{\varvec{X}}_{2}}(\varvec{\beta }_{0}^{\tau }\varvec{x}_{1})} \sum _{i=1}^{n_{2}}K_{h_{2}}\left( {\varvec{\gamma }}_{0}^{\tau }\varvec{X}_{2i}-{\varvec{\beta }}_{0}^{\tau }\varvec{x}_{1}\right) \sigma _{2}({\varvec{\gamma }}_{0}^{\tau }\varvec{X}_{2i})\epsilon _{2i} +O_{P}(c_{n_{2}}).\qquad \end{aligned}$$

(43)

We can also show that $R_{n_{1}n_{2}}(t)$ defined in (41) is $o_{P}(n^{-1/2}_{1}+n_{2}^{-1/2})$ uniformly in $t\in \mathbb {R}$. Together with (39), (42) and (43), we have

$$\begin{aligned}&F_{\widetilde{\mathcal {H}}_{0},\hat{\epsilon }_{1}}(t|\mathscr {V}_{n_{1}n_{2}}) -F^{*}_{\widetilde{\mathcal {H}}_{0}, \epsilon _{1}}(t)\\&\quad =\frac{t}{2n_{1}}\sum _{i=1}^{n_{1}}(\epsilon _{1i}^{2}-1)f_{\epsilon _{1}}\left( t+ \frac{g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i}) -g_{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})}\right) \\&\qquad +\,E\left[ f_{\epsilon _{1}}\left( t+ \frac{g_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1}) -g_{2}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1})}\right) \frac{g'_{2}(\varvec{\beta }_{0}^{\tau }\varvec{X})}{\sigma _{1}(\varvec{\beta }_{0}^{\tau }\varvec{X})}V_{1, {\varvec{\beta }}_{0}}(\varvec{\beta }_{0}^{\tau }\varvec{X})\right] ^{\tau }\left( \hat{\varvec{\beta }}_{0}-\varvec{\beta }_{0}\right) \\&\qquad +\,\frac{1}{n_{2}}\sum _{i=1}^{n_{2}}f_{\epsilon _{1}}\left( t+ \frac{g_{1}(\varvec{\gamma }_{0}^{\tau }\varvec{X}_{2i}) -g_{2}(\varvec{\gamma }_{0}^{\tau }\varvec{X}_{2i})}{\sigma _{1}(\varvec{\gamma }_{0}^{\tau }\varvec{X}_{2i})}\right) \frac{f_{{\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1}} (\varvec{\gamma }_{0}^{\tau }\varvec{X}_{2i})}{f_{{\varvec{\gamma }}_{0}^{\tau }{\varvec{X}}_{2}}(\varvec{\gamma }_{0}^{\tau }\varvec{X}_{2i}) }\frac{\sigma _{2}(\varvec{\gamma }_{0}^{\tau }\varvec{X}_{2i})}{\sigma _{1}(\varvec{\gamma }_{0}^{\tau }\varvec{X}_{2i})}\epsilon _{2i}\\&\qquad +\,o_{P}(n^{-1/2}). \end{aligned}$$

Recalling the definitions of $D_{1}(u)$ and $\rho _{f,\sigma }(u)$, we complete the proof of Theorem 5. Moreover, the proof of Theorem 6 is completed by following the asymptotic result of Theorem 5 and recalling that $D_{1}(u)\equiv D_{2}(u)\equiv 0$ under the null hypothesis, we omit the details. $\square$

1.5 Proof of Theorem 7

Proof

By using the detailed proof of Theorem 1 in Stute et al. (2008), the class of functions

$$\begin{aligned} \ell _{t}(\epsilon _{1}, \varvec{x}_{1})= & {} I\left\{ \epsilon _{1}\le t-\frac{1}{\sqrt{n_{1}+n_{2}}}\frac{\mu ({\varvec{\beta }}_{0}^{\tau }{\varvec{x}}_{1})}{\sigma _{1}({\varvec{\beta }}_{0}^{\tau }{\varvec{x}}_{1})}\right\} -I\left\{ \epsilon _{1}\le t\right\} \\&\quad ~-\,F_{\epsilon _{1}}\left( t-\frac{1}{\sqrt{n_{1}+n_{2}}} \frac{\mu ({\varvec{\beta }}_{0}^{\tau }{\varvec{x}}_{1})}{\sigma _{1}({\varvec{\beta }}_{0}^{\tau }{\varvec{x}}_{1})}\right) +F_{\epsilon _{1}}(t) \end{aligned}$$

is a Vapnik-Chervonenkis class with envelop function 4 (Pollard 1984, Ch. 2). Then, we can have that

$$\begin{aligned}&n_{1}^{-1/2}\Bigg |\sum _{i=1}^{n_{1}}\Bigg [I\left\{ \epsilon _{1i}\le t-\frac{1}{\sqrt{n_{1}+n_{2}}}\frac{\mu ({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1i})}{\sigma _{1}({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1i})}\right\} -I\left\{ \epsilon _{1i}\le t\right\} \Bigg ]\nonumber \\&\quad ~~~~~~~-\,E\left[ F_{\epsilon _{1}}\left( t-\frac{1}{\sqrt{n_{1}+n_{2}}} \frac{\mu ({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})}{\sigma _{1}({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})}\right) \right] +F_{\epsilon _{1}}(t)\Bigg |=o_{P}\big (n_{1}^{-1/2}\big ). \end{aligned}$$

(44)

Moreover, Taylor expansion entails that

$$\begin{aligned}&E\left[ F_{\epsilon _{1}}\left( t-\frac{1}{\sqrt{n_{1}+n_{2}}} \frac{\mu ({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})}{\sigma _{1}({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})}\right) \right] -F_{\epsilon _{1}}(t)\nonumber \\&\quad =\,-\frac{1}{\sqrt{n_{1}+n_{2}}}f_{\epsilon _{1}}(t)E\left[ \frac{\mu ({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})}{\sigma _{1}({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})}\right] +o\big (n_{1}^{-1/2}+n_{2}^{-1/2}\big ). \end{aligned}$$

(45)

If the local alternative hypothesis $\mathcal {H}_{1n_{1}n_{2}}$ is true, together with (44) and (45), we have

$$\begin{aligned}&\hat{F}_{\widetilde{\mathcal {H}}_{0},\epsilon _{1}}(t)-\hat{F}_{\epsilon _{1}}(t)=-\frac{1}{\sqrt{n_{1}+n_{2}}}f_{\epsilon _{1}}(t)E\left[ \frac{\mu ({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})}{\sigma _{1}({\varvec{\beta }}_{0}^{\tau }{\varvec{X}}_{1})}\right] \\&\quad +\,f_{\epsilon _{1}}(t)\mathcal {N}_{1}^{\tau }J_{{{\varvec{\beta }}}_{0}}\varvec{\varOmega }_{1}^{-1}\frac{1}{n_{1}} \sum _{i=1}^{n_{1}}J_{{\varvec{\beta }}_{0}}^{\tau }\frac{{g}_{1}'(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})}{{\sigma }_{1}(\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})}\left[ \varvec{X}_{1i}-{V}_{1, {\varvec{\beta }}_{0}} (\varvec{\beta }_{0}^{\tau }\varvec{X}_{1i})\right] \epsilon _{1i}\\&\quad +\,f_{\epsilon _{1}}(t)\frac{1}{n_{2}} \sum _{i=1}^{n_{2}}\rho _{f,\sigma }(Y{\varvec{\gamma }}_{0}^{\tau }{\varvec{X}}_{2i})\epsilon _{2i} +f_{\epsilon _{1}}(t)\frac{1}{n_{1}}\sum _{i=1}^{n_{1}}\epsilon _{1i}\\&\quad +\,o_{P}(n_{1}^{-1/2}+n_{2}^{-1/2}). \end{aligned}$$

We can also obtain a similar expression for $\hat{F}_{\widetilde{\mathcal {H}}_{0},\epsilon _{2}}(t)-\hat{F}_{\epsilon _{2}}(t)$ and we omit the details. Using the continuous mapping theorem, we complete the proof of Theorem 7. $\square$