Rank estimation for the function-on-scalar model

Abstract

Rank regression method has been widely pursued for robust inference in statistical models. Unfortunately, there does not exist related literature for the function-on-scalar model, which is the focus of this paper. We study the robust estimation based on rank regression and B-spline approximations for the function-on-scalar model and further establish the theoretical properties of the proposed method under regularity conditions. Extensive simulation studies and two real data applications are given to illustrate the merits of the proposed approach. Numerical results show that the proposed method is competitive with existing robust estimation procedures.

Appendix: Technical proofs

We denote the cdf and pdf of $\varepsilon _{ijk}=\varepsilon _{ik}-\varepsilon _{jk}$ as $G_k$ and $g_k$ respectively. By simple algebra operation we can get $g_k(\nu )=\int f_k(t)f_k(t-\nu )dt$. By the Corollary 6.21 of Schumaker (1981), there exists a vector ${\varvec{\lambda }}_0=({\varvec{\lambda }}^T_{01},\ldots ,{\varvec{\lambda }}^T_{0p})^T$ such that $\Vert {\varvec{B}}^T(s){\varvec{\lambda }}_{0\ell }-\beta _{0\ell }(s)\Vert =O_{p}({J^{-r}_n})$ for $\ell =1,\ldots ,p$. In the proofs, C denotes a generic constant that might assume different values at different places.

Proof of Theorem 1

Let $\delta _n=n^{-r/(2r+1)}$, ${\varvec{u}}_n=\delta ^{-1}_n({\varvec{\lambda }}-{\varvec{\lambda }}_0)$ with ${\varvec{u}}_{n\ell }=\delta ^{-1}_n({\varvec{\lambda }}_\ell -{\varvec{\lambda }}_{0\ell })$. We want to show that for any given $\eta >0$, there exists a large constant C such that, for large n we have

$$\begin{aligned} P\left\{ \inf _{\Vert {\varvec{u}}_n\Vert =C} {\mathcal {L}}_{n}({\varvec{\lambda }}_{0}+\delta _{n} {\varvec{u}}_{n})>\mathcal { L}_{n}({\varvec{\lambda }}_{0})\right\} \ge 1-\eta . \end{aligned}$$

(9)

This implies that, with probability tending to one, there is local minimum $\hat{{\varvec{\lambda }}}$ in the ball $\{{\varvec{\lambda }}_0+\delta _{n}{\varvec{u}}_{n},\Vert {\varvec{u}}_{n}\Vert \le C\}$ such that $\Vert \hat{{\varvec{\lambda }}}-{\varvec{\lambda }}_{0}\Vert =O_{p}(\delta _{n})$.

Let $D_n={\mathcal {L}}_{n}({\varvec{\lambda }}_{0}+\delta _{n} {\varvec{u}}_{n})-{\mathcal {L}}_{n}({\varvec{\lambda }}_{0})$, then

$$\begin{aligned} D_n=\frac{1}{2nm}\sum _{k=1}^m\sum _{i,j}\left\{ |\varepsilon _{ijk}-R_{ijk}-\delta _n{\varvec{\Pi }}^T_{ijk}{\varvec{u}}_{n}|-|\varepsilon _{ijk}-R_{ijk}|\right\} \triangleq&I, \end{aligned}$$

(10)

where $R_{ijk}={\varvec{\Pi }}^T_{ijk}{\varvec{\lambda }}_{0}-{\varvec{Z}}^T_i{\varvec{\beta }}_0(s_k)+{\varvec{Z}}^T_j{\varvec{\beta }}_0(s_k)$ with ${\varvec{\Pi }}_{ijk}={\varvec{\Pi }}_{ik}-{\varvec{\Pi }}_{jk}$.

By knight’s identity (Knight 1998),

$$\begin{aligned} (|r-z|-|r|) / 2=-z\left( \frac{1}{2}-I(r<0)\right) +\int _{0}^{z}[I(r \le t)-I(r \le 0)] dt, \end{aligned}$$

we have

$$\begin{aligned} I=&-\frac{1}{nm}\sum _{k=1}^m\sum _{i, j} \delta _{n}{\varvec{\Pi }}_{i jk}^{T}{\varvec{u}}_{n}\left[ \frac{1}{2}-I(\varepsilon _{i jk}-R_{ijk}<0)\right] \nonumber \\&+\frac{1}{nm}\sum _{k=1}^m\sum _{i, j} \int _{0}^{\delta _{n} {\varvec{\Pi }}_{i jk}^{T}{\varvec{u}}_{n}}[I(\varepsilon _{i jk} \le t+R_{ i jk})-I(\varepsilon _{i jk} \le R_{ i jk})] d t\nonumber \\ \triangleq&\frac{1}{m}\sum _{k=1}^mI_1+\frac{1}{m}\sum _{k=1}^mI_2, \end{aligned}$$

(11)

For $I_1$, we have

$$\begin{aligned} I_{1}&=-\frac{1}{n} \sum _{i, j} \delta _{n}{\varvec{\Pi }}_{i jk}^{T}{\varvec{u}}_{n}\left[ \frac{1}{2}-I(\varepsilon _{i jk}-R_{ijk}<0)\right] \nonumber \\&=-\frac{1}{n} \sum _{i, j} \delta _{n} {\varvec{\Pi }}_{i jk}^{T}{\varvec{u}}_{n}\left[ \frac{1}{2}-I(\varepsilon _{i jk}<0)\right] \nonumber \\&\quad -\frac{1}{n} \sum _{i, j} \delta _{n}{\varvec{\Pi }}_{i jk}^{T}{\varvec{u}}_{n}[I(\varepsilon _{i jk}<0)-I(\varepsilon _{i jk}-R_{ i jk}<0)] \nonumber \\&=-\frac{1}{n} \delta _{n}\sum _{i=1}^{n} {\varvec{\Pi }}_{ik}^{T}\{2 R(\varepsilon _{ik})-(n+1)\} {\varvec{u}}_{n}\nonumber \\&\quad -\frac{1}{n}\sum _{i, j} \delta _{n}{\varvec{\Pi }}_{i jk}^{T}[I(\varepsilon _{i jk}<0)-I(\varepsilon _{i jk}-R_{ ijk}<0)] {\varvec{u}}_{n} \nonumber \\&\triangleq I_{11}+I_{12}, \end{aligned}$$

(12)

where $R(\varepsilon _{ik})$ is the rank statistic of $\varepsilon _{ik}$, $I_{11}= W^T_n{\varvec{u}}_{n}$ with $W_n=-\frac{1}{n} \delta _{n}\sum _{i=1}^{n} {\varvec{\Pi }}_{ik}^{T}\{2 R(\varepsilon _{ik})-(n+1)\}$. By the independence between ${\varvec{Z}}_i$ and $\varepsilon _{ik}$ and condition (C1), ${\text {E}}(W_n)=0$, ${\text {Cov}}(W_n)=\frac{1}{n^2}\delta ^2_{n}{\varvec{\Pi }}^T_{k} {\text {Cov}}(\zeta _k){\varvec{\Pi }}_{k}$ for ${\varvec{\Pi }}_{k}=(\Pi _{1k},\ldots ,\Pi _{nk})^T$ and $\zeta _k=(2 R(\varepsilon _{1k})-(n+1), \ldots , 2 R(\varepsilon _{nk})-(n+1))^{T}$. According to the Theorem 1 of Leng (2010), the diagonal terms of $\frac{1}{n^2} {\text {Cov}}(\zeta )$ are

$$\begin{aligned} \frac{1}{n^2}{\text {Var}}(\zeta _{ik})&=\frac{1}{n^2} \sum _{i=1}^{n}\{2 i-(n+1)\}^{2} \frac{1}{n}=\frac{4(n+1)^{2}}{n^{3}} \\&\quad \sum _{i=1}^{n}\left( \frac{i}{(n+1)}-\frac{1}{2}\right) ^{2} \rightarrow 4 \int \left(t-\frac{1}{2}\right)^{2} d t=1 / 3, \end{aligned}$$

and its off-diagonal terms are

$$\begin{aligned} \frac{1}{n^2} {\text {Cov}}(\zeta _{ik}, \zeta _{jk})&=\frac{1}{n^2}\sum _{i=1}^{n} \sum _{j \ne i}\{2 i-(n+1)\}\{2 j-(n+1)\} \frac{1}{n(n-1)}\\&=-\frac{4(n+1)^{2}}{n^{2}(n-1)} \int (t-\frac{1}{2})^{2} d t \rightarrow 0. \end{aligned}$$

Thus, we have ${\text {E}}(I_{11})=0$ and ${\text {Var}}(I_{11})=\delta _{n}^{2} {\varvec{u}}^{T}_{n} {\varvec{\Pi }}^T_{k} n^{-2} {\text {Cov}}(\zeta ) {\varvec{\Pi }}_{k} {\varvec{u}}_{n} \rightarrow \frac{1}{3} \delta _{n}^{2} {\varvec{u}}^{T}_{n} {\varvec{\Pi }}^T_{k}{\varvec{\Pi }}_{k} {\varvec{u}}_{n}$.

Applying the Markov inequality, we have

$$\begin{aligned} P(|I_{11}|&\ge n \delta _{n}^{2}\Vert {\varvec{u}}_{n}\Vert ) \le \frac{{\text {Var}}(I_{11})}{\Vert {\varvec{u}}_{n}\Vert ^{2} n^{2} \delta _{n}^{4}} \rightarrow \frac{1}{3\Vert {\varvec{u}}_{n}\Vert ^{2} n^{2} \delta _{n}^{4}} \delta _{n}^{2} {\varvec{u}}^T_{n} {\varvec{\Pi }}^T_{k}{\varvec{\Pi }}_{k} {\varvec{u}}_{n} \\&\le \frac{\delta _{n}^{2}\Vert {\varvec{u}}_{n}\Vert ^{2} \lambda _{\max }({\varvec{\Pi }}^T_{k}{\varvec{\Pi }}_{k})}{3\Vert {\varvec{u}}_{n}\Vert ^{2} n^{2} \delta _{n}^{4}} \rightarrow 0, \end{aligned}$$

where $\lambda _{\max }(M)$ denotes the maximum eigenvalue of a positive definite matrix M. Hence, $I_{11}=o_{p}(n \delta _{n}^{2})\Vert {\varvec{u}}_{n}\Vert$.

On the other hand, based on the independence between ${\varvec{Z}}_i$ and $\varepsilon _{ik}$ and condition (C1), it follows that ${\text {E}}(I_{12})=0$. By the similar arguments of Theorem 1(a) of Noh et al. (2012), we have

$$\begin{aligned} {\text {E}}(I_{12}^{2})&={\text {E}}\bigg (n^{-1}\sum _{i, j} \delta _{n}{\varvec{\Pi }}_{i jk}^{T}[I(\varepsilon _{i jk}<0)-I(\varepsilon _{i jk}-R_{i jk}<0)] {\varvec{u}}_{n}\bigg )^{2} \\&=n^{-2}\delta _{n}^{2}{\text {E}}\bigg (\sum _{i, j}{\varvec{\Pi }}_{i jk}^{T}{\varvec{u}}_{n}[I(\varepsilon _{i jk}<0)-I(\varepsilon _{i jk}-R_{i jk}<0)]\bigg )^{2} \\&\le n^{-2} \delta _{n}^{2} n^{4}\bigg \{{\text {E}}[I(\varepsilon _{12k}<0)-I(\varepsilon _{12k}-R_{ 12k}<0)]\bigg \}^{2} {\text {E}}({\varvec{u}}^T_{n} {\varvec{\Pi }}_{12k} {\varvec{\Pi }}_{12k}^{T} {\varvec{u}}_{n}) \\&=n^{-2} \delta _{n}^{2} n^{4}[G_k(0)-G_k(R_{12k})]^{2} {\text {E}}({\varvec{u}}^T_{n} {\varvec{\Pi }}_{12k} {\varvec{\Pi }}_{12k}^{T} {\varvec{u}}_{n})\\&=-n^{2} \delta _{n}^{2}\{g_k(0) R_{12k}(1+o(1))\}^{2}{\varvec{u}}^T_{n} {\text {E}}({\varvec{\Pi }}_{12k} {\varvec{\Pi }}_{12k}^{T} ){\varvec{u}}_{n}\\&=O(n^{2} \delta _{n}^{4})\Vert {\varvec{u}}_{n}\Vert ^{2}. \end{aligned}$$

Therefore, we have $I_{12}=O_p(n \delta _{n}^{2})\Vert {\varvec{u}}_{n}\Vert$.

Next, use condition (C1) and the Markov inequality, taking the similar arguments as in Guo et al. (2017), it can be shown that

$$\begin{aligned} P\bigg (\max _{i, j}(\delta _{n}|{\varvec{\Pi }}_{i jk}^{T} {\varvec{u}}_{n}|)>\eta \bigg )&=P\bigg (\bigcup _{i, j=1}^{n}(\delta _{n}|{\varvec{\Pi }}_{i jk}^{T} {\varvec{u}}_{n}|>\eta )\bigg ) \\&\le n^{2} P\left( \delta _{n}\left| {\varvec{\Pi }}_{12k}^{T} {\varvec{u}}_{n}\right| >\eta \right) \\&\le n^{2} \frac{\delta _{n}^{6}}{\eta ^{6}} {\text {E}}\left( {\varvec{\Pi }}_{12k}^{T} {\varvec{u}}_{n}\right) ^{6} \\&\le n^{2} \frac{\delta _{n}^{6}}{\eta ^{6}}\Vert {\varvec{u}}_{n}\Vert ^{6} {\text {E}}\left\| {\varvec{\Pi }}_{12k}\right\| ^{6} \\&\le n^{2} \frac{\delta _{n}^{6}}{\eta ^{6}} C^{6} M^{6} \rightarrow 0. \end{aligned}$$

Thus,

$$\begin{aligned} \max _{i, j}\left( \delta _{n}\left| {\varvec{\Pi }}_{i jk}^{T} {\varvec{u}}_{n}\right| \right) =o_p(1). \end{aligned}$$

(13)

By conditions (C1) and (C4), and the Lebesgue’s dominated convergence theorem, we have

$$\begin{aligned} {\text {E}}(I_{2})&={\text {E}}\left\{ {\text {E}}\left[ n^{-1} \sum _{i,j} \int _{0}^{\delta _{n} {\varvec{\Pi }}_{ijk}^{T} {\varvec{u}}_{n}}(I(\varepsilon _{i jk} \le t+R_{i jk})-I(\varepsilon _{i jk} \le R_{ ijk})) d t\right] \right\} \\&={\text {E}}\left[ n^{-1} \sum _{i, j} \int _{0}^{\delta _{n}{\varvec{\Pi }}_{ijk}^{T} {\varvec{u}}_{n}}\left\{ G_k\left( t+R_{i jk}\right) -G_k\left( R_{i jk}\right) \right\} d t\right] \\&=n^{-1} \sum _{i, j} E\left[ \int _{0}^{\delta _{n}{\varvec{\Pi }}_{ijk}^{T} {\varvec{u}}_{n}} g_k(R_{ i jk}) t(1+o(1)) d t\right] \\&=\frac{1}{2n} g_k(0)(1+o(1)) \delta _{n}^{2}{\varvec{u}}^T_{n}{\text {E}}(\sum _{i, j} {\varvec{\Pi }}_{ijk} {\varvec{\Pi }}^T_{ijk}) {\varvec{u}}_{n} \end{aligned}$$

Here we use the (13) in the third step.

By noting that

$$\begin{aligned}&{\text {Var}}\left( I_{2 i jk}\right) \\&= n^{-2} {\text {E}}\left\{ \int _{0}^{\delta _{n} {\varvec{\Pi }}_{ijk}^{T} {\varvec{u}}_{n}}\bigg [I(\varepsilon _{i jk} \le t+R_{i jk})-I(\varepsilon _{i jk} \le R_{i jk})-(G_k(t+R_{i jk}) -G_k(R_{i jk}))\bigg ] d t\right\} ^{2} \\&\le n^{-2} E\left\{ \left| \int _{0}^{\delta _{n} {\varvec{\Pi }}_{ijk}^{T} {\varvec{u}}_{n}}\bigg [I(\varepsilon _{i jk} \le t+R_{i jk})-I(\varepsilon _{i jk} \le R_{i jk}) -(G_k(t+R_{i jk})-G_k(R_{ i jk}))\bigg ] d t\right| \right\} \\&\quad \times 2 \delta _{n}\left| {\varvec{\Pi }}_{ijk}^{T} {\varvec{u}}_{n}\right| \le 4 n^{-2} \delta _{n} \max \left| {\varvec{\Pi }}_{ijk}^{T} {\varvec{u}}_{n}\right| {\text {E}}\left( I_{2 i jk}\right) . \end{aligned}$$

Again by (13), we have

$$\begin{aligned} {\text {Var}}\left( I_{2}\right) \le n^{2} \sum _{i j} {\text {Var}}\left( I_{2 i jk}\right) \le 4 \delta _{n} \max \left| {\varvec{\Pi }}_{ijk}^{T} {\varvec{u}}_{n}\right| {\text {E}}\left( I_{2}\right) =o\left( n \delta _{n}^{2}\right) \Vert {\varvec{u}}_{n}\Vert ^2, \end{aligned}$$

Therefor, we have $I_2=\frac{1}{2n} g_k(0)\delta _{n}^{2}{\varvec{u}}^T_{n}{\text {E}}(\sum _{i, j} {\varvec{\Pi }}_{ijk} {\varvec{\Pi }}^T_{ijk}) {\varvec{u}}_{n}+o_p(\sqrt{n \delta _{n}^{2}})\Vert {\varvec{u}}_{n}\Vert$. It follows from the above analysis that the $D_n$ in (10) is dominated by the positive quadratic term of $I_2$ as long as $\Vert {\varvec{u}}_{n}\Vert$ is large enough. Hence (9) holds. In particular, this implies that $\Vert {\hat{\beta }}_{\ell }(s)-\beta _{0\ell }(s)\Vert =O_{p}(n^{-r /(2 r+1)})$ for $\ell =1,2,\ldots ,p$. We complete the proof of Theorem 1.