Testing in linear composite quantile regression models

Abstract

Composite quantile regression (CQR) can be more efficient and sometimes arbitrarily more efficient than least squares for non-normal random errors, and almost as efficient for normal random errors. Based on CQR, we propose a test method to deal with the testing problem of the parameter in the linear regression models. The critical values of the test statistic can be obtained by the random weighting method without estimating the nuisance parameters. A distinguished feature of the proposed method is that the approximation is valid even the null hypothesis is not true and power evaluation is possible under the local alternatives. Extensive simulations are reported, showing that the proposed method works well in practical settings. The proposed methods are also applied to a data set from a walking behavior survey.

Appendix

To prove main results in this paper, the following technical conditions are imposed.

A1. :

$d^2_n=\max _{1\le i \le n}\{X_i^TS_nX_i\}\rightarrow 0$ as $n\rightarrow \infty$.

A2. :

$S=\lim _{n\rightarrow \infty }S_n/n$ is a $p\times p$ positive definite matrix. For each p-vector $u$,

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{i=1}^{n}\int \limits _0^{u_0+X_i^Tu}\sqrt{n}[F(a+t/\sqrt{n})-F(a)]dt =\frac{1}{2}f(a)(u_0,u^T)\left[ \begin{array}{cc} 1&{} 0\\ 0 &{} S\\ \end{array}\right] \\&\quad (u_0,u^T)^T. \end{aligned}$$

A3. :

The random weights $\omega _{1},\ldots ,\omega _{n}$ are i.i.d. with $P(\omega _{1}\ge 0)$, $E(\omega _{1})=Var(\omega _{1})=1$, and the sequence $\{\omega _{i}\}$ and $\{Y_{i},X_{i}\}$ are independent.

Remark 3

Conditions A1 and A3 are standard conditions in the random weighting method, see Rao and Zhao (1992). And conditions A2 is commonly assumed in the quantile regression, see Koenker (2005).

Write

$$\begin{aligned} X_{ni}=S_n^{-1/2}X_i,~~\beta (n)=S_n^{1/2}(\beta -b),~~Y_{ni}=Y_i-X_i^Tb. \end{aligned}$$

The model (2.1) can be written as

$$\begin{aligned} Y_{ni}=X_{ni}^T\beta (n)+\varepsilon _i,~~i=1,\ldots ,n, \end{aligned}$$

(5.1)

Denote

$$\begin{aligned}&L_{n}(b_1,\ldots ,b_K, \beta (n))\\&\quad =\sum _{k=1}^{K}\sum _{i=1}^{n}\left[ \rho _{\tau _{k}} (Y_{ni}-b_k-X_{ni}^{T}\beta (n))-\rho _{\tau _{k}} (Y_{ni}-b_{\tau _k}-X_{ni}^{T}\beta _0(n))\right] , \end{aligned}$$

$$\begin{aligned} L_{n}^*(b_1,\ldots ,b_K, \beta (n))&= \sum _{k=1}^{K}\sum _{i=1}^{n}\omega _i\left[ \rho _{\tau _{k}} (Y_{ni}-b_k-X_{ni}^{T}\beta (n))-\rho _{\tau _{k}} (Y_{ni}-b_{\tau _k}\right. \\&\left. -X_{ni}^{T}\beta _0(n))\right] . \end{aligned}$$

where $\beta _0(n)=S_n^{1/2}(\beta _0-b)$ and $\beta _0$ is the true value.

Lemma 1

Under the conditions of Theorem 2, we have

$$\begin{aligned} \begin{aligned} \sqrt{n}(\hat{b}_k^*-b_{\tau _k})&=-\frac{1}{\sqrt{n}f(b_{\tau _k})}\sum _{i=1}^{n}\omega _i\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1),\\ \hat{\beta }^*(n)-\beta _0(n)&=-\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\sum _{i=1}^{n}\omega _iX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1), \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} L^*_{n}(\hat{b}^*_1,\ldots ,\hat{b}^*_K, \hat{\beta }^*(n))=&\sum _{i=1}^{n}\omega _iX_{ni}^T\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] (\hat{\beta }^*(n)-\beta _0(n))\\&+\frac{1}{2}\left( \sum _{k=1}^{K}f(b_{\tau _k})\right) (\hat{\beta }^*(n)-\beta _0(n))^T(\hat{\beta }^*(n)\\&-\beta _0(n))+A^*+o_p(1), \end{aligned} \end{aligned}$$

where $A^*=-\frac{1}{2n}\sum _{k=1}^{K}f^{-1}(b_{\tau _k})\left[ \sum _{i=1}^{n}\omega _i\{I(\varepsilon _i<b_{\tau _k})-\tau _k\}\right] ^2$.

Particularly, when $w\equiv 1$, we have

$$\begin{aligned} \begin{aligned}&\sqrt{n}(\hat{b}_k-b_{\tau _k})=-\frac{1}{\sqrt{n}f(b_{\tau _k})}\sum _{i=1}^{n}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1),\\&\hat{\beta }(n)-\beta _0(n)=-\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\sum _{i=1}^{n}X_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1), \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} L_{n}(\hat{b}_1,\ldots ,\hat{b}_K, \hat{\beta }(n))=&\sum _{i=1}^{n}X_{ni}^T\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] (\hat{\beta }(n)-\beta _0(n))\\&+\frac{1}{2}\left( \sum _{k=1}^{K}f(b_{\tau _k}) \right) (\hat{\beta }(n)\\&-\beta _0(n))^T(\hat{\beta }(n)-\beta _0(n))+A+o_p(1), \end{aligned} \end{aligned}$$

where $A=-\frac{1}{2n}\sum _{k=1}^{K}f^{-1}(b_{\tau _k})\left[ \sum _{i=1}^{n}\{I(\varepsilon _i<b_{\tau _k})-\tau _k\}\right] ^2$.

Proof

Let $\hat{\beta }^*(n)-\beta _0(n)=\mathbf{u}_{n}$ and $\sqrt{n}(\hat{b}_{k}-b_{\tau _{k}})=v_{n,k}$. Then $(v_{n,1},\ldots ,v_{n,q},\mathbf{u}_{n})$ is the minimizer of the following criterion:

$$\begin{aligned} \begin{aligned} L_{n}^*&=\sum _{k=1}^{K}\sum _{i=1}^{n}\omega _i[\rho _{\tau _{k}} (\varepsilon _{i}-b_{\tau _{k}}-[v_{k}/ \sqrt{n}+X_{ni}^{T}\mathbf{u}])-\rho _{\tau _{k}} (\varepsilon _{i}-b_{\tau _{k}})], \end{aligned} \end{aligned}$$

To apply the identity (Knight 1998)

$$\begin{aligned} \rho _{\tau }(x-y)-\rho _{\tau }(x)=y\{I(x<0)-\tau \}+\int \limits _{0}^{y}\{I(x\le z)-I(x\le 0)\}dz, \end{aligned}$$

Thus we rewrite $L^*_{n}$ as follows:

$$\begin{aligned} \begin{aligned} L^*_{n}=&\sum _{k=1}^{K}\sum _{i=1}^{n}\omega _i[v_{k}/ \sqrt{n}+X_{ni}^{T}\mathbf{u}][I(\varepsilon _{i}<b_{\tau _{k}})-\tau _{k}]\\&+\sum _{k=1}^{K}\sum _{i=1}^{n}\omega _i\int \limits _{0}^{[v_{k}/ \sqrt{n}+X_{ni}^{T}\mathbf{u}]} [I(\varepsilon _{i}\le b_{\tau _{k}}+t)-I(\varepsilon _{i}\le b_{\tau _{k}})]dt\\ \equiv&L^*_{1n}+L^*_{2n}. \end{aligned} \end{aligned}$$

Denote $L^*_{2n}=\sum _{k=1}^{K}L_{2n}^{*(k)}$, where $L_{2n}^{*(k)}=\sum _{i=1}^{n}\omega _i\int _{0}^{[v_{k}/ \sqrt{n}+X_{ni}^{T}\mathbf{u}]} [I(\varepsilon _{i}\le b_{\tau _{k}}+t)-I(\varepsilon _{i}\le b_{\tau _{k}})]dt$. By A3, noting that $\max _{1\le i\le n}\parallel X_{in}\parallel =d_n\rightarrow 0$ by A1, we have

$$\begin{aligned} E[L_{2n}^{*(k)}]&= \sum _{i=1}^{n}\int \limits _{0}^{[v_{k}/ \sqrt{n}+X_{ni}^{T}\mathbf{u}]} [F(b_{\tau _{k}}+t)-F( b_{\tau _{k}})]dt\\&= \frac{1}{n}\sum _{i=1}^{n}\int \limits _{0}^{[v_{k}+\sqrt{n}X_{ni}^{T}\mathbf{u}]}\sqrt{n}[F(b_{\tau _{k}}+t/\sqrt{n})-F( b_{\tau _{k}})]dt\\&\rightarrow \frac{1}{2}f(b_{\tau _k})(v_{k},\mathbf{u})\left[ \begin{array}{cc} 1&{} 0\\ 0 &{} I_p \end{array} \right] (v_{k},\mathbf{u}^T)^{T}.\\ Var[L_{2n}^{*(k)}]&\le 4E[L_{2n}^{*(k)}]\max _{1\le i\le n}|v_{k}/\sqrt{n}+X_{ni}^{T}\mathbf{u}|\rightarrow 0. \end{aligned}$$

Hence, $L_{2n}^{*(k)} \xrightarrow {p}\frac{1}{2}f(b_{\tau _k})(v_{k},\mathbf{u})\left[ \begin{array}{cc} 1&{} 0\\ 0 &{} I_p \end{array} \right] (v_{k},\mathbf{u}^T)^{T}.$ Thus it follows that

$$\begin{aligned} \begin{aligned} L^*_{n}&= \sum _{k=1}^{K}\sum _{i=1}^{n}\omega _i[v_{k}/ \sqrt{n}+X_{ni}^{T}\mathbf{u}][I(\varepsilon _{i}<b_{\tau _{k}})-\tau _{k}]\\&\quad +\frac{1}{2}\sum _{k=1}^{K}f(b_{\tau _k})(v_{k},\mathbf{u})\left[ \begin{array}{cc} 1&{} 0\\ 0 &{} I_p \end{array} \right] (v_{k},\mathbf{u}^T)^{T}+o_p(1). \end{aligned} \end{aligned}$$

Since $L^*_{n}-\sum _{k=1}^{K}\sum _{i=1}^{n}\omega _i[v_{k}/ \sqrt{n}+X_{ni}^{T}\mathbf{u}][I(\varepsilon _{i}<b_{\tau _{k}})-\tau _{k}]$ converges in probability to the convex function $\frac{1}{2}\sum _{k=1}^{K}f(b_{\tau _k})(v_{k},\mathbf{u})\left[ \begin{array}{cc} 1&{} 0\\ 0 &{} I_p \end{array} \right] (v_{k},\mathbf{u}^T)^{T}$, it follows from the convexity lemma (Pollard 1991) that, for any compact set, the quadratic approximation to $L^*_{n}$ holds uniformly for $(v_{1},\ldots ,v_{K},\mathbf{u})$ in any compact set, which leads to

$$\begin{aligned} \begin{aligned}&\sqrt{n}(\hat{b}_k^*-b_{\tau _k})=-\frac{1}{\sqrt{n}f(b_{\tau _k})}\sum _{i=1}^{n}\omega _i\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1),\\&\hat{\beta }^*(n)-\beta _0(n)=-\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\sum _{i=1}^{n}\omega _iX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1). \end{aligned} \end{aligned}$$

Thus, we can obtain

$$\begin{aligned} L^*_{n}(\hat{b}^*_1,\ldots ,\hat{b}^*_K, \hat{\beta }^*(n))&= \sum _{i=1}^{n}\omega _iX_{ni}^T\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] (\hat{\beta }^*(n)-\beta _0(n))\\&+\frac{1}{2}\left( \sum _{k=1}^{K}f(b_{\tau _k}) \right) (\hat{\beta }^*(n)-\beta _0(n))^T(\hat{\beta }^*(n)\\&-\beta _0(n))+A^*+o_p(1). \end{aligned}$$

The Lemma is proved.

Now we proceed to prove the theorems.

Proof of Theorem 1

Suppose $0<q<p$ and let $K$ be a $p\times (p-q)$ matrix of the rank $(p-q)$ such that $H^TK=0$ and $K^T\omega _{n}=0$. Without loss of generality, $H_{0}$ and $H_{2,n}$ can be written as

$$\begin{aligned}&H_{0}: \mathbf{\beta }-b=K\gamma ,~for~some~ \gamma \in \mathbf{R}^{p-q},~H_{2,n}:\mathbf{\beta }-b=K\gamma +\omega _{n},\\&\quad ~for~some~ \gamma \in \mathbf{R}^{p-q}. \end{aligned}$$

Write

$$\begin{aligned} H_{n}=S_n^{-1/2}H(H^{T}S_n^{-1}H)^{-1/2},~~~~K_{n}=S_n^{1/2}K(K^{T}S_nK)^{-1/2}, \end{aligned}$$

thus,

$$\begin{aligned} H_{n}^{T}H_{n}=I_{q},~~K_{n}^{T}K_{n}=I_{p-q},~~H_{n}^{T}K_{n}=0,~~H_{n}H_{n}^{T}+K_{n}K_{n}^{T}=I_{p}. \end{aligned}$$

When H$_0$ is true, model (A.1) can be written as

$$\begin{aligned} Y_{ni}=X_{ni}^TK_n\gamma _0(n)+\varepsilon _i,~~i=1,\ldots ,n, \end{aligned}$$

where $\gamma _0(n)=(K^{T}S_nK)^{1/2}\gamma$. Then, replacing $X_{ni}$ in (A.1) by $K_n^TX_{ni}$, and using a similar argument as that of Lemma 1, we have

$$\begin{aligned} \tilde{b}_k-b_{\tau _k}&= -\frac{1}{nf(b_{\tau _k})}\sum _{i=1}^{n}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1),\\ \hat{\gamma }(n)-\gamma _0(n)&= -\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\sum _{i=1}^{n}K_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1), \end{aligned}$$

$$\begin{aligned} \begin{aligned} L_{n}(\tilde{b}_1,\ldots ,\tilde{b}_K, \tilde{\beta }(n))=&\sum _{i=1}^{n}X_{ni}^TK_n\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] (\hat{\gamma }(n)-\gamma _0(n))\\&+\frac{1}{2}\left( \sum _{k=1}^{K}f(b_{\tau _k}) \right) (\hat{\gamma }(n)-\gamma _0(n))^TK_n^TK_n(\hat{\gamma }(n)\\&-\gamma _0(n))+A+o_p(1),\\ =&-\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i{=}1}^{n}K_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k}){-}\tau _k\right] \right\| ^2\\&+A+o_p(1), \end{aligned} \end{aligned}$$

where $\hat{\gamma }(n)$ is the CQR estimate of $\gamma _0(n)$ and $\tilde{\beta }(n)=S_n^{1/2}(\tilde{\beta }-b)=K_n\hat{\gamma }(n)$.

Hence, under the null hypotheses, we have

$$\begin{aligned} M_{n}&= L_{n}(\tilde{b}_1,\ldots ,\tilde{b}_K, \tilde{\beta }(n))-L_{n}(\hat{b}_1,\ldots ,\hat{b}_K, \hat{\beta }(n))\\&= -\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}K_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2\\&+\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}X_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2+o_{p}(1)\\&= \frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}H_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2+o_{p}(1) \end{aligned}$$

When $H_{2,n}$ is true, the model (A.1) can be written as

$$\begin{aligned} Y_{ni}=X_{ni}^T[K_n\gamma _2(n)+H_n\delta (n)]+\varepsilon _i,~~i=1,\ldots ,n, \end{aligned}$$

where $\gamma _2(n)=(K^{T}S_nK)^{1/2}\gamma +K_n^TS_n^{1/2}\omega _n$ and $\delta (n)=H_n^TS_n^{1/2}\omega _n$. Then by Lemma 1, we have

$$\begin{aligned} \tilde{b}_k-b_{\tau _k}&= -\frac{1}{nf(b_{\tau _k})}\sum _{i=1}^{n}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1),\\ \hat{\gamma }_2(n)-\gamma _2(n)&= -\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\sum _{i=1}^{n}K_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] +o_p(1), \end{aligned}$$

$$\begin{aligned}&L_{n}(\tilde{b}_1,\ldots ,\tilde{b}_K, \tilde{\beta }(n))\\&\quad =\sum _{i=1}^{n}X_{ni}^T\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] (K_n\hat{\gamma }_2(n)-K_n\gamma _{2}(n)-H_n\delta (n))\\&\qquad +\frac{1}{2}\left( \sum _{k=1}^{K}f(b_{\tau _k}) \right) (K_n\hat{\gamma }_2(n)-K_n\gamma _{2}(n)\\&\qquad -H_n\delta (n))^T(K_n\hat{\gamma }_2(n)-K_n\gamma _{2}(n)-H_n\delta (n))+A+o_p(1),\\&\quad =-\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}K_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2\\&\qquad -\sum _{i=1}^{n}X_{ni}^TH_n\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \delta (n)\\&\qquad +\frac{1}{2}\sum _{k=1}^{K}f(b_{\tau _k})\Vert \delta (n)\Vert ^2 +A+o_p(1). \end{aligned}$$

Hence, under the local alternative hypotheses, we have

$$\begin{aligned}&\!\!\!\!\! M_{n}=L_{n}(\tilde{b}_1,\ldots ,\tilde{b}_K, \tilde{\beta }(n))-L_{n}(\hat{b}_1,\ldots ,\hat{b}_K, \hat{\beta }(n))\\&\quad =-\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}K_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2\\&\qquad +\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}X_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2\\&\qquad -\sum _{i=1}^{n}X_{ni}^TH_n\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \delta (n) +\frac{1}{2}\sum _{k=1}^{K}f(b_{\tau _k})\Vert \delta (n)\Vert ^2+o_{p}(1)\\&\quad =\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}H_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2\\&\qquad -\sum _{i=1}^{n}X_{ni}^TH_n\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \delta (n) +\frac{1}{2}\sum _{k=1}^{K}f(b_{\tau _k})\Vert \delta (n)\Vert ^2+o_{p}(1)\\&\quad =\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}H_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right. \\&\qquad \left. -\sum _{k=1}^{K}f(b_{\tau _k})\delta (n)\right\| ^2+o_{p}(1)\\&\quad =\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}H_n^TX_{ni}\sum _{k=1}^{K}\left[ \tau _k-I(\varepsilon _i<b_{\tau _k})\right] \right. \\&\qquad \left. +\sum _{k=1}^{K}f(b_{\tau _k})\delta (n)\right\| ^2+o_{p}(1). \end{aligned}$$

The theorem is thus proved.

Proof of Theorem 2

By Lemma 2, we can obtain

$$\begin{aligned}&L^*_{n}(\hat{b}^*_1,\ldots ,\hat{b}^*_K, \hat{\beta }^*(n))-L^*_{n}(\hat{b}_1,\ldots ,\hat{b}_K, \hat{\beta }(n))\\&\quad = \sum _{i=1}^{n}\omega _iX_{ni}^T\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] (\hat{\beta }^*(n)-\hat{\beta }(n))\\&\qquad +\frac{1}{2}\left( \sum _{k=1}^{K}f(b_{\tau _k}) \right) \left[ (\hat{\beta }^*(n)-\beta _0(n))^T(\hat{\beta }^*(n)\right. \\&\qquad \left. -\beta _0(n))-(\hat{\beta }(n)-\beta _0(n))^T(\hat{\beta }(n)-\beta _0(n))\right] +o_p(1)\\&\quad =-\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}(\omega _i-1)X_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2+o_{p}(1). \end{aligned}$$

Similarly, under the null and local alternative hypotheses, we can obtain

$$\begin{aligned}&L^*_{n}(\tilde{b}^*_1,\ldots ,\tilde{b}^*_K, \tilde{\beta }^*(n))-L^*_{n}(\tilde{b}_1,\ldots ,\tilde{b}_K, \tilde{\beta }(n))\\&\quad =-\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}(\omega _i-1)K_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2+o_{p}(1). \end{aligned}$$

where $\hat{\beta }^*(n)=S_n(\hat{\beta }^*-b)$ and $\tilde{\beta }^*(n)=S_n(\tilde{\beta }^*-b)$. Therefore, we can obtain

$$\begin{aligned} \begin{aligned} M_n^*=&\left[ L^*_{n}(\hat{b}^*_1,\ldots ,\hat{b}^*_K, \hat{\beta }^*(n))\right. \\&\left. \!\!-L^*_{n}(\hat{b}_1,\ldots ,\hat{b}_K, \hat{\beta }(n))\right] -\left[ L^*_{n}(\tilde{b}^*_1,\ldots ,\tilde{b}^*_K, \tilde{\beta }^*(n))-L^*_{n}(\tilde{b}_1,\ldots ,\tilde{b}_K, \tilde{\beta }(n))\right] \\ =&-\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}(\omega _i-1)X_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2\\&+\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}(\omega _i-1)K_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2+o_{p}(1)\\ =&\frac{1}{2}\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-1}\left\| \sum _{i=1}^{n}(\omega _i-1)H_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right] \right\| ^2+o_{p}(1). \end{aligned} \end{aligned}$$

By the checking the Lindeberg condition, we know that the conditional distribution of $\sum _{i=1}^{n}(\omega _i-1)H_n^TX_{ni}\sum _{k=1}^{K}\left[ I(\varepsilon _i<b_{\tau _k})-\tau _k\right]$ given $Y_1,\ldots ,Y_n$ converge to $N\left( \mathbf{0}, A\left[ \sum _{k=1}^{K}f(b_{\tau _k})\right] ^{-2}\cdot I_q\right)$. Therefore, the conditional distribution of $M_n^*$ given $Y_1,\ldots ,Y_n$ converges to $\chi ^2_q/\left[ 2A^{-1}\sum _{k=1}^{K}f(b_{\tau _k})\right]$. The proof of Theorem 2 is completed.