Identification and estimation in quantile varying-coefficient models with unknown link function

Abstract

In this paper, we consider the estimation problem of quantile varying-coefficient models when the link function is unspecified, which significantly expands the existing works on varying-coefficient models with unspecified link function focusing only on mean regression. We provide new identification conditions which are weaker than existing ones. Under these identification conditions, we use polynomial splines to estimate both the varying coefficients and the link functions and establish the convergence rate of the estimator. Our simulation studies and a real data application illustrate the finite sample performance of the estimators.

Appendix: Technical proofs

1.1 Proof of Proposition 1

Suppose we have $g(\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u))=h(\mathbf{x}^{\mathrm{T}}{\varvec{\alpha }}(u))$ for $(\mathbf{x},u)\in R^{p+1}$ in the support of $(\mathbf{X},U)$, with h and ${\varvec{\alpha }}(u)$ also satisfying (C2) and (C3). Setting $u=0$, since $\beta _1(0)>0$ and $\Vert {\varvec{\beta }}(0)\Vert =1$, by the identification of single-index models (Lin and Kulasekera 2007), we know that ${\varvec{\beta }}(0)={\varvec{\alpha }}(0)$ and $g=h$ on $\mathcal{S}_0$. Let $u_0=\inf \{u\in [0,1]:{\varvec{\beta }}(u)\ne {\varvec{\alpha }}(u)\}$. By continuity we have ${\varvec{\beta }}(u_0)={\varvec{\alpha }}(u_0)$. If $u_0=1$, the identification is achieved. Thus we assume $u_0<1$ below.

Let $\inf _u\Vert {\varvec{\beta }}(u)\Vert =\epsilon >0$. since continuity implies uniform continuity, there exists $\delta >0$ such that $\Vert {\varvec{\beta }}(u_1)-{\varvec{\beta }}(u_2)\Vert \le \epsilon$ and $\Vert {\varvec{\alpha }}(u_1)-{\varvec{\alpha }}(u_2)\Vert \le \epsilon$ whenever $|u_1-u_2|\le \delta$.

By the assumption that $\mathbf{X}$ has a convex support containing the origin, $\mathcal{S}_u$ is an interval containing zero for any u. Since g is nonconstant on $\mathcal{S}_{u_0}$, there exists some $M>0$ such that g is nonconstant on $\mathcal{S}_{u_0}^M:=\mathcal{S}_{u_0}\cap \{\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u_0): \Vert \mathbf{x}\Vert \le M\}$ (if g is constant on $\mathcal{S}_{u_0}^M$ for all $M>0$ then g is also a constant on $\cup _{M} \mathcal{S}_{u_0}^M$ leading to a contradiction). Similarly, we can find a constant $a\in (0,1)$ such that g is not a constant on $a\mathcal{S}_{u_0}^M$. For other values of u, we also denote $\mathcal{S}_u^M=\mathcal{S}_u\cap \{\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u): \Vert \mathbf{x}\Vert \le M\}$.

Let $L_u<0$ and $U_u>0$ be the left and right boundary point of $\mathcal{S}_u^M$, respectively. Assumption (C1) also implies that $\inf _u \min \{-L_u,U_u\}$ is bounded below by a positive constant, say $\epsilon '$. Since $\mathcal{S}_{u_0}^M$ is an interval containing zero, $a\mathcal{S}_{u_0}^M$ is a proper subset of $\mathcal{S}_{u_0}^M$. More concretely, we have, for any $\mathbf{x}\in \mathcal{X}\cap [-M,M]$,

$$\begin{aligned} a\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u_0)\le U_{u_0}-(1-a)\epsilon ' \end{aligned}$$

(A.1)

and

$$\begin{aligned} a\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u_0)\ge L_{u_0}+(1-a)\epsilon '. \end{aligned}$$

(A.2)

In fact, for example, the first inequality above can be seen by the trivial inequality $U_{u_0}-a\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u_0)\ge (1-a)U_{u_0}$.

Now we claim that there exists a constant $\delta '>0$ such that $u_0\le u\le u_0+\delta '$ implies

$$\begin{aligned} a\mathcal{S}_{u_0}^M\subseteq \mathcal{S}_u^M\cap \mathcal{S}_{u_0}^M. \end{aligned}$$

In fact, we obviously have $a\mathcal{S}_{u_0}^M\subseteq \mathcal{S}_{u_0}^M$ since $a<1$. To see $a\mathcal{S}_{u_0}^M\subseteq \mathcal{S}_{u}^M$, we only need to show that for all $\mathbf{x}\in \mathcal{X}\cap \{\mathbf{x}:\Vert \mathbf{x}\Vert \le M\}$,

$$\begin{aligned} a\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u_0)\le \max \{\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u): \mathbf{x}\in \mathcal {X}\cap \{\mathbf{x}:\Vert \mathbf{x}\Vert \le M\}\} \end{aligned}$$

and

$$\begin{aligned} a\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u_0)\ge \min \{\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u): \mathbf{x}\in \mathcal {X}\cap \{\mathbf{x}:\Vert \mathbf{x}\Vert \le M\}\}, \end{aligned}$$

which are easily seen to be implied by (A.1) and (A.2), respectively, using that $\Vert \mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u_0)-\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u)\Vert \le M \Vert {\varvec{\beta }}(u_0)-{\varvec{\beta }}(u)\Vert$ and the continuity of ${\varvec{\beta }}(u)$.

Summarizing the above, we can find $\delta ''>0$ such that for $u=u_0+\delta ''$, we have $a\mathcal{S}_{u_0}^M\subseteq \mathcal{S}_u^M\cap \mathcal{S}_{u_0}^M$ and $\Vert {\varvec{\beta }}(u)-{\varvec{\beta }}(u_0)\Vert \le \epsilon$, $\Vert {\varvec{\alpha }}(u)-{\varvec{\alpha }}(u_0)\Vert \le \epsilon$. In the following, we fixed u to be $u_0+\delta ''$.

Differentiating both sides of $g(\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u))=h(\mathbf{x}^{\mathrm{T}}{\varvec{\alpha }}(u))$ we get $g'(\mathbf{x}^{\mathrm{T}}{\varvec{\beta }}(u)){\varvec{\beta }}(u)=h'(\mathbf{x}^{\mathrm{T}}{\varvec{\alpha }}(u)){\varvec{\alpha }}(u)$ and thus ${\varvec{\beta }}(u)\propto {\varvec{\alpha }}(u)$. For this fixed u, simply write ${\varvec{\beta }}={\varvec{\beta }}(u)$, ${\varvec{\alpha }}={\varvec{\alpha }}(u)$ and thus ${\varvec{\beta }}=c{\varvec{\alpha }}$ for some $c\in R$. We assume $|c|\le 1$ (otherwise we can simply switch ${\varvec{\beta }}$ and ${\varvec{\alpha }}$). If $c=1$, we have ${\varvec{\beta }}={\varvec{\alpha }}$ which contradicts the definition of $u_0$. If $c=-1$, we have ${\varvec{\beta }}=-{\varvec{\alpha }}$. and thus $\Vert {\varvec{\beta }}(u)-{\varvec{\beta }}(u_0)\Vert ^2+\Vert {\varvec{\alpha }}(u)-{\varvec{\alpha }}(u_0)\Vert ^2=\Vert {\varvec{\beta }}(u)-{\varvec{\beta }}(u_0)\Vert ^2+\Vert {\varvec{\beta }}(u)+{\varvec{\beta }}(u_0)\Vert ^2=2\Vert {\varvec{\beta }}(u)\Vert ^2+2\Vert {\varvec{\beta }}(u_0)\Vert ^2\ge 4\epsilon ^2$, which contradict that $\Vert {\varvec{\beta }}(u)-{\varvec{\beta }}(u_0)\Vert \le \epsilon$ and $\Vert {\varvec{\alpha }}(u)-{\varvec{\alpha }}(u_0)\Vert \le \epsilon$. Finally, if $|c|<1$, we have $g(\mathbf{x}^{\mathrm{T}}{\varvec{\beta }})=h(c \mathbf{x}^{\mathrm{T}}{\varvec{\beta }})$ and by the identification of single-index models, we have $g(x)=h(cx)$ for $x\in \mathcal{S}_u$. Since $g=h$ on $\mathcal{S}_{u_0}$, we have $h(cx)=h(x)$ for $x\in \mathcal{S}_u\cap \mathcal{S}_{u_0}$. Note $\mathcal{S}_u\cap \mathcal{S}_{u_0}$ is an interval containing zero. This implies $h(x)=h(cx)=h(c^2x)=\ldots \rightarrow h(0)$ and h is a constant on $\mathcal{S}_u\cap \mathcal{S}_{u_0}\supseteq \mathcal{S}_u^M\cap \mathcal{S}_{u_0}^M$, which is in turn a superset of $a\mathcal{S}_{u_0}^M$, leading to contradiction (since we argued above that $g=h$ is nonconstant on $a\mathcal{S}_{u_0}^M$).

$\square$

1.2 Proofs for convergence rates

Let $F(\cdot |\mathbf{X},U)$ be the conditional cdf of e given the covariates. We also write the true conditional quantile $g(\mathbf{X}_i^{\mathrm{T}}{\varvec{\beta }}(U_i))$ as $m_i$. In the proofs, C denotes a generic positive constant which may assume different values even on the same line.

Lemma 1

Let $r_n=\sqrt{K/n}+K^{-d}$. Define $\mathbf{Z}_i=\{\mathbf{A}^{-1}\mathbf{B}(U_i)\}\otimes \mathbf{X}_i$.

$$\begin{aligned}&\sup _{\Vert {{\phi }}-{{\phi }}_0\Vert +\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert \le Cr_n} \sum _{i=1}^n\rho _\tau \{Y_i-\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}\}-\sum _{i=1}^n\rho _\tau \{Y_i-\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}\}\\&\quad +\sum _{i=1}^n \{\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}-\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\{\tau -I(e_i\le 0)\}\\&\quad -E\sum _{i=1}^n\rho _\tau \{Y_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}\}+E\sum _{i=1}^n\rho _\tau \{Y_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}=o_p(nr_n^2), \end{aligned}$$

where the expectations are over $Y_i$ conditional on $\mathbf{X}_i$ and $U_i$ (all expectations below are also such conditional expectations).

Proof

As in He and Shi (1994), in the proof we consider median regression with $\tau =1/2,\;\rho _\tau (u)=|u|/2$ and the general case can be shown in the same way. Let $\mathcal{N}=\{({{\phi }}_{(1)},{\varvec{\theta }}_{(1)}),\ldots ,({{\phi }}_{(N)},{\varvec{\theta }}_{(N)})\}$ be a $\delta _n$ covering of $\{({{\phi }},{\varvec{\theta }}): \Vert {{\phi }}-{{\phi }}_0\Vert +\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert \le Cr_n\}$, with size bounded by $N\le (Cr_n/\delta _n)^{CK}$ and thus $\mathrm{log} N\le C K\mathrm{log} n$ if we choose $\delta _n\sim n^{-a}$ for some $a>0$ (we will choose a to be large enough).

Let $M_{ni}({{\phi }},{\varvec{\theta }})=\frac{1}{2}|Y_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}|-\frac{1}{2} |Y_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0|+ \{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\{1/2-I(e_i\le 0)\}$, and $M_n({{\phi }},{\varvec{\theta }})=\sum _{i=1}^nM_{ni}({{\phi }},{\varvec{\theta }})$. Using the Lipschitz property of |u|, and that for any $({{\phi }},{\varvec{\theta }})$ there exists $({{\phi }}_{(l)},{\varvec{\theta }}_{(l)})$ such that $\Vert {{\phi }}-{{\phi }}_{(l)}\Vert ^2+\Vert {\varvec{\theta }}-{\varvec{\theta }}_{(l)}\Vert ^2\le \delta _n^2$, we have

$$\begin{aligned}&M_n({{\phi }},{\varvec{\theta }})-EM_n({{\phi }},{\varvec{\theta }})-M_n({{\phi }}_{(l)},{\varvec{\theta }}_{(l)})+EM_n({{\phi }}_{(l)},{\varvec{\theta }}_{(l)})\\&\quad \le C\sum _{i=1}^n |\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}-\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_{(l)}){\varvec{\theta }}_{(l)} |, \end{aligned}$$

which can obviously be made smaller than $nr_n^2$ by the Lipschitz property of the spline functions, by setting $\delta _n\sim n^{-a}$ for a large enough.

Furthermore, by simple algebra

$$\begin{aligned} |M_{ni}({{\phi }},{\varvec{\theta }})|= & {} \bigg | \frac{1}{2}|Y_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}|-\frac{1}{2}|Y_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0|\\&+\, \{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\{1/2-I(e_i\le 0)\}\bigg |\\= & {} \bigg | \frac{1}{2}|e_i+m_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}|-\frac{1}{2}|e_i+m_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0|\\&+ \, \{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\{1/2-I(e_i\le 0)\}\bigg |\\\le & {} | \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0|\cdot \\&I(|e_i|\le | \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0) {\varvec{\theta }}_0|+|m_i-\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0|). \end{aligned}$$

Thus

$$\begin{aligned} |M_{ni}({{\phi }},{\varvec{\theta }})|\le & {} | \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0|\\\le & {} C|\mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}^*){\varvec{\theta }}\mathbf{X}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_0)|+|\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)({\varvec{\theta }}-{\varvec{\theta }}_0)|\\\le & {} C |\mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}^*){\varvec{\theta }}_0\mathbf{X}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_0)|+|\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)({\varvec{\theta }}-{\varvec{\theta }}_0)|\\&+|\mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}^*)({\varvec{\theta }}-{\varvec{\theta }}_0)\mathbf{X}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_0)|\\\le & {} C(r_n+\sqrt{K}r_n+K^{3/2}r_n^2)\\\le & {} C\sqrt{K}r_n\\=: & {} A, \end{aligned}$$

where ${{\phi }}^*$ lies between ${{\phi }}$ and ${{\phi }}_0$ and we used that $\Vert \mathbf{B}(x)\Vert \le C\sqrt{K}$ and $\Vert \mathbf{B}^{(1)}(x)\Vert \le CK^{3/2}$ at any fixed point $x\in [0,1]$.

Furthermore, we have

$$\begin{aligned} E|M_{ni}({{\phi }},{\varvec{\theta }})|^2\le & {} C(\sqrt{K}r_n) E |\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}-\mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0|^2\nonumber \\\le & {} C(\sqrt{K}r_n)(r_n^2)=:D^2. \end{aligned}$$

(A.3)

Using Bernstein’s inequality, together with union bound, we have

$$\begin{aligned} P(\sup _{({{\phi }},{\varvec{\theta }})\in \mathcal {N}} |M_n({{\phi }},{\varvec{\theta }})-EM_n({{\phi }},{\varvec{\theta }})|>a) \le C\exp (-\frac{a^2}{aA+nD^2}-CK\mathrm{log} (n)). \end{aligned}$$

The right-hand side converges to zero with

$a=O\left( \max \{K^{3/2}r_n\mathrm{log} (n), \sqrt{nK^{3/2}r_n^3\mathrm{log} (n)}\}\right) =o(nr_n^2)$. $\square$

Lemma 2

Suppose $\Vert {{\phi }}-{{\phi }}_0\Vert +\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert =Lr_n$ for sufficiently large $L>0$.

$$\begin{aligned}&\inf _{\Vert {{\phi }}-{{\phi }}_0\Vert +\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert = Lr_n}\sum _iE\rho _\tau \{e_i+m_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}\}\\&\quad -\,\sum _iE\rho _\tau \{e_i+m_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\ge L^2 Cnr_n^2, \end{aligned}$$

with probability approaching one.

Proof

Using the Knight’s identity $\rho _\tau (x-y)-\rho _\tau (x)=-y\{\tau -I(x\le 0)\}+\int _0^y \{I(x\le t)-I(x\le 0)\}dt$, we have that

$$\begin{aligned}&E\sum _{i=1}^n \rho _\tau \{e_i+m_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}\}-E\sum _{i=1}^n\rho _\tau \{e_i+m_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\\&\quad =\sum _i\int _{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0-m_i}^{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}_{j}-m_i} F(t|\mathbf{X}_i,U_i)-F(0|\mathbf{X}_i,U_i)dt\\&\quad \ge C\sum _i f(0|\mathbf{X}_i,U_i)\left[ \{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}^2\right. \\&\qquad \left. +\, 2\{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0-m_i\}\right] \end{aligned}$$

We have, by Taylor’s expansion,

$$\begin{aligned}&\sum _i\{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}^2\\&\quad \ge \sum _i\left\{ \mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_{0}){\varvec{\theta }}_0\mathbf{Z}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_0)+ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)({\varvec{\theta }}-{\varvec{\theta }}_0)\right\} ^2+o_p(nr_n^2)\\&\quad \ge CL^2nr_n^2, \end{aligned}$$

with probability approaching one. The final lower bound above is nontrivial and we put its proof in Lemma 3. On the other hand, using Taylor’s expansion, we can easily get similar upper bound

$$\begin{aligned} \sum _i\{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}^2 \le CL^2nr_n^2, \end{aligned}$$

and by the approximation property of splines,

$$\begin{aligned} \sum _i\{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0-m_i\}^2\le CnK^{-2d}. \end{aligned}$$

(A.4)

Combining various bounds above, we get

$$\begin{aligned} E\sum _{i=1}^n \rho _\tau \{e_i+m_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}\}-E\rho _\tau \{e_i+m_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\ge CL^2nr_n^2, \end{aligned}$$

if L is large enough. $\square$

Lemma 3

Under the setup of Lemma 2, we have

$$\begin{aligned}&\inf _{\Vert {{\phi }}-{{\phi }}_0\Vert +\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert = Lr_n} \frac{1}{n}\sum _i\left\{ \mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_{0}){\varvec{\theta }}_0\mathbf{Z}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_0)\right. \nonumber \\&\qquad \left. +\, \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)({\varvec{\theta }}-{\varvec{\theta }}_0)\right\} ^2\ge CL^2r_n^2, \end{aligned}$$

(A.5)

with probability approaching one.

Proof

We use $\mathrm{vec}^{-1}$ to denote the inverse mapping of ${{\phi }}=\mathrm{vec}({\varvec{\Phi }})$. Let ${\varvec{\Phi }}=\mathrm{vec}^{-1}({{\phi }})$, ${\varvec{\Phi }}_0=\mathrm{vec}^{-1}({{\phi }}_0)$, ${\varvec{\beta }}(u)={\varvec{\Phi }}\mathbf{A}^{-1}\mathbf{B}(u)$ and $\widetilde{\varvec{\beta }}(u)={\varvec{\Phi }}_0\mathbf{A}^{-1}\mathbf{B}(u)$. Obviously $\mathbf{Z}_i({{\phi }}-{{\phi }}_0)=\mathbf{X}_i^{\mathrm{T}}\{{\varvec{\beta }}(U_i)-\widetilde{\varvec{\beta }}(U_i)\} =\mathbf{X}_i^{\mathrm{T}}\mathbf{J}(U_i)\{{\varvec{\beta }}^{(-r)}(U_i)-\widetilde{\varvec{\beta }}^{(-r)}(U_i)\}+o_p(r_n^2)$ and we have $\Vert {\varvec{\beta }}^{(-r)}-\widetilde{\varvec{\beta }}^{(-r)}\Vert \ge CLr_n$. Then the right-hand side of (A.5) can be written as

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^n \left( ({\varvec{\beta }}^{(-r)}(U_i)-\widetilde{\varvec{\beta }}^{(-r)}(U_i)) ^{\mathrm{T}},{\varvec{\theta }}^{\mathrm{T}}-{\varvec{\theta }}_0^{\mathrm{T}}\right) \left( \begin{array}{c} \mathbf{J}(U_i)\mathbf{X}_i(\mathbf{B}^{(1)\mathrm{T}}(\mathbf{X}_i^{\mathrm{T}}\widetilde{\varvec{\beta }}(U_i)) {\varvec{\theta }}_0\\ \mathbf{B}(\mathbf{X}_i^{\mathrm{T}}\widetilde{\varvec{\beta }}(U_i)) \end{array}\right) ^{\otimes 2}\\&\quad \left( \begin{array}{c} {\varvec{\beta }}^{(-r)}(U_i)-\widetilde{\varvec{\beta }}^{(-r)}(U_i)\\ {\varvec{\theta }}-{\varvec{\theta }}_0 \end{array}\right) +o_p(r_n^2). \end{aligned}$$

By law of large number, we only need to show the lower bound for

$$\begin{aligned}&E\left[ \left( ({\varvec{\beta }}^{(-r)}(U_i)-\widetilde{\varvec{\beta }}^{(-r)}(U_i))^{\mathrm{T}},{\varvec{\theta }}^{\mathrm{T}}-{\varvec{\theta }}_0^{\mathrm{T}}\right) \left( \begin{array}{c} \mathbf{J}(U_i)\mathbf{X}_i(\mathbf{B}^{(1)\mathrm{T}}(\mathbf{X}_i^{\mathrm{T}}\widetilde{\varvec{\beta }}(U_i)) {\varvec{\theta }}_0\\ \mathbf{B}(\mathbf{X}_i^{\mathrm{T}}\widetilde{\varvec{\beta }}(U_i)) \end{array}\right) ^{\otimes 2}\right. \\&\quad \left. \left( \begin{array}{c} {\varvec{\beta }}^{(-r)}(U_i)-\widetilde{\varvec{\beta }}^{(-r)}(U_i)\\ {\varvec{\theta }}-{\varvec{\theta }}_0 \end{array}\right) \right] , \end{aligned}$$

which is obviously lower bounded by $E\Vert {\varvec{\beta }}^{(-r)}(U)-\widetilde{\varvec{\beta }}^{(-r)}(U)\Vert ^2+\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert ^2\ge CL^2r_n^2$ if we can show that the eigenvalues of

$$\begin{aligned} E\left[ \left( \begin{array}{c} \mathbf{J}(U)\mathbf{X}\mathbf{B}^{(1)\mathrm{T}}(\mathbf{X}^{\mathrm{T}}\widetilde{\varvec{\beta }}(U)) {\varvec{\theta }}_0\\ \mathbf{B}(\mathbf{X}^{\mathrm{T}}\widetilde{\varvec{\beta }}(U)) \end{array}\right) ^{\otimes 2}\big | U=u \right] \end{aligned}$$

are bounded away from zero for all $u\in [0,1]$.

Since $|\mathbf{B}^{(1)\mathrm{T}}(\mathbf{X}^{\mathrm{T}}\widetilde{\varvec{\beta }}(u)) {\varvec{\theta }}_0-g^{(1)}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(u))|\le CK^{-d+1}$, we only need to show that the eigenvalues of

$$\begin{aligned} E\left[ \left( \begin{array}{c} \mathbf{J}(U)\mathbf{X}g^{(1)}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U))\\ \mathbf{B}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U)) \end{array}\right) ^{\otimes 2} \big |U=u\right] \end{aligned}$$

(A.6)

are bounded away from zero.

Under condition (A4), let $\mathbf{L}_0$ be the $p\times K$ matrix of spline coefficients with $\Vert E\{\mathbf{X}|\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U)\}-\mathbf{L}_0\mathbf{B}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U))\Vert \le C K^{-d'}$.

Pre-multiplying (A.6) by

$$\begin{aligned} \left( \begin{array}{cc} \mathbf{I}&{} -\mathbf{J}(U)\mathbf{L}_0 \\ \mathbf{0}&{} \mathbf{I}\end{array}\right) \end{aligned}$$

(A.7)

and post-multiplying (A.6) by its transpose we get the matrix

$$\begin{aligned} E\left[ \left( \begin{array}{c} \mathbf{J}(U)\mathbf{X}g^{(1)}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U))-\mathbf{J}(U)\mathbf{L}_0 \mathbf{B}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U))\\ \mathbf{B}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U)) \end{array}\right) ^{\otimes 2}\big |U=u\right] . \end{aligned}$$

(A.8)

It is easy to see that singular values of (A.7) are bounded and bounded away from zero, and thus we only need to show that the eigenvalues of (A.8) are bounded away from zero. Since $\Vert E\{\mathbf{X}|\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U)\}-\mathbf{L}_0\mathbf{B}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U))\Vert \le C K^{-d'}$, we can replace $\mathbf{L}_0\mathbf{B}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U))$ with $E\{\mathbf{X}|\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U)\}$ in (A.8), and we only need to consider

$$\begin{aligned} E\left[ \left( \begin{array}{c} \mathbf{J}(U)\left[ \mathbf{X}- E\{\mathbf{X}|\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U)\}\right] g^{(1)}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U))\\ \mathbf{B}(\mathbf{X}^{\mathrm{T}}{\varvec{\beta }}_0(U)) \end{array}\right) ^{\otimes 2}\big |U=u\right] . \end{aligned}$$

(A.9)

It is easy to see that (A.9) is block-diagonal and the eigenvalues of both blocks are bounded and bounded away from zero, by the property of splines and condition (A5). $\square$

The following lemma deals with one of the terms in the statement of Lemma 1. For models with single-index structure, its proof is more complicated than additive or varying-coefficient models (due to that for the latter models the parametric part and nonparametric part are added up to give the regression function).

Lemma 4

$$\begin{aligned}&\sup _{\Vert {{\phi }}-{{\phi }}_0\Vert {+}\Vert {\varvec{\theta }}{-}{\varvec{\theta }}_0\Vert =Lr_n} \sum _i\left\{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}{-} \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\right\} \{\tau -I(e_i\le 0)\}{=}L\cdot O_p(nr_n^2). \end{aligned}$$

Proof

For simplicity of notation, let $\epsilon _i=\tau -I(e_i\le 0)$. We have

$$\begin{aligned}&\sum _i\{ \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\epsilon _i\nonumber \\&\quad =\sum _i \mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\mathbf{Z}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_{0})\epsilon _i \end{aligned}$$

(A.10)

$$\begin{aligned}&\qquad +\sum _i\mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)({\varvec{\theta }}-{\varvec{\theta }}_0)\mathbf{Z}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_{0})\epsilon _i \end{aligned}$$

(A.11)

$$\begin{aligned}&\qquad +\sum _i\{\mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}^*)-\mathbf{B}^{(1)T}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)\}{\varvec{\theta }}_0\mathbf{Z}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_{0})\epsilon _i \end{aligned}$$

(A.12)

$$\begin{aligned}&\qquad +\sum _i\{\mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}^*)-\mathbf{B}^{(1)T}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)\}({\varvec{\theta }}-{\varvec{\theta }}_0)\mathbf{Z}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_{0})\epsilon _i \end{aligned}$$

(A.13)

$$\begin{aligned}&\qquad +\sum _i \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)({\varvec{\theta }}-{\varvec{\theta }}_0)\epsilon _i. \end{aligned}$$

(A.14)

The term (A.10) obviously has order $L\cdot O_p(\sqrt{n}r_n)$. For (A.14), we have that $\Vert \mathbf{B}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)\epsilon _i\Vert ^2=O_p(\sum _i\Vert \mathbf{B}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)\Vert ^2)=O_p(nK)$ and thus (A.14) is $O_p(\sqrt{nK}\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert )=L\cdot O_p(\sqrt{nK}r_n)$.

For the term (A.11), since $\Vert \sum _i\mathbf{B}^{(1)}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)\epsilon _i\Vert ^2=O_p(\sum _i\Vert \mathbf{B}^{(1)}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)\Vert ^2)=O_p(nK^3)$ we have

$$\begin{aligned}&\sum _i\sum _j \mathbf{B}^{(1)\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)({\varvec{\theta }}-{\varvec{\theta }}_0)\mathbf{Z}_i^{\mathrm{T}}({{\phi }}-{{\phi }}_{0})\epsilon _i=O_p(\sqrt{n}K^{3/2}r_n^2)=o_p(nr_n^2). \end{aligned}$$

With further Taylor expansion $\mathbf{B}^{(1)}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}^*)-\mathbf{B}^{(1)}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0)=\mathbf{B}^{(2)}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}^{**})\mathbf{Z}_i^{\mathrm{T}}({{\phi }}^*-{{\phi }}_0)$, (A.12) and (A.13) are also of order $o_p(nr_n^2)$ and the proof is complete. Note that all these bounds are trivially uniform over $\Vert {{\phi }}-{{\phi }}_0\Vert +\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert = Lr_n$ since expressions ${{\phi }}-{{\phi }}_0$ and ${\varvec{\theta }}-{\varvec{\theta }}_0$ can be put in front of the sum. $\square$

Proof of Theorem 1

Combining Lemmas 1, 2 and 4, we get for sufficiently large $L>0$,

$$\begin{aligned} P\left[ \inf _{\Vert {{\phi }}-{{\phi }}_0\Vert +\Vert {\varvec{\theta }}-{\varvec{\theta }}_0\Vert =Lr_n}\sum _{i=1}^n\rho _\tau \{Y_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}){\varvec{\theta }}\}>\sum _{i=1}^n\rho _\tau \{Y_i- \mathbf{B}^{\mathrm{T}}(\mathbf{Z}_i^{\mathrm{T}}{{\phi }}_0){\varvec{\theta }}_0\}\right] \rightarrow 1, \end{aligned}$$

and thus there is a local minimizer of $(\widehat{{\phi }},\widehat{\varvec{\theta }})$ with $\Vert \widehat{{\phi }}-{{\phi }}_0\Vert +\Vert \widehat{\varvec{\theta }}-{\varvec{\theta }}_0\Vert =O_p(r_n)$. $\square$