Robust estimation of single index models with responses missing at random

Abstract

A single-index regression model is considered, where some responses in the model are assumed to be missing at random. Local linear rank-based estimators of the single-index direction and the unknown link function are proposed. Asymptotic properties of the estimators are established under mild regularity conditions. Monte Carlo simulation experiments show that the proposed estimators are more efficient than their least squares counterparts especially when the data are derived from contaminated or heavy-tailed model error distributions. When the errors follow a normal distribution, the least squares index direction estimator tends to be more efficient than the rank-based index direction estimator; however, the least squares link function estimator remains less efficient than the rank-based link function estimator. A real data example is analyzed and cross-validation studies show that the proposed procedure provides better prediction than the least squares method when the responses contain outliers and are missing at random.

Appendix

This section contains necessary lemmas needed in the proofs of the main results stated in this manuscript.

Lemma 1

Under assumptions $(I_2)-(I_6),$ we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{u\in {\mathscr {F}}}|{\widehat{g}}^{(r)}(u)-g^{(r)}(u)|\rightarrow 0\quad a.s.\; \text{ as } n\rightarrow \infty ,\quad r=0,1,2. \end{aligned}$$

This lemma whose proof can be constructed along the lines as given in Delecroix et al. (2006) ensures the uniform strong consistency of ${\widehat{g}}$ with its derivatives. For sake of brevity the proof is omitted here.

Lemma 2

Under assumptions $(I_{1})-(I_{7})$ and for ${\mathbf {x}}\in \varGamma ,$ we have

• (a) $\displaystyle \lim \nolimits _{n\rightarrow \infty }\sup \nolimits _{{\varvec{\beta }}\in {\mathscr {B}}} \left\| {\widetilde{S}}_{jn}({\varvec{\beta }})-S_{jn}({\varvec{\beta }})\right\| =0\;a.s.$ and $\displaystyle \frac{1}{n}\sum \nolimits _{i=1}^{n}w_{i}\{g^{\prime }({\mathbf {x}}_{i}^{\tau } {\varvec{\beta }}_{0})\}^{2}{\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }\varphi ^{2}(F_{ij}(\xi _{ij})) \rightarrow {\mathbf {V}}_{j}\;a.s.$

• (b) $\displaystyle \lim \nolimits _{n\rightarrow \infty }\sup \nolimits _{{\varvec{\beta }}\in {\mathscr {B}}} \left\| {\widetilde{S}}_{jn}({\varvec{\beta }})-{\widetilde{T}}_{jn}({\varvec{\beta }}) \right\| =0\;a.s.\quad \text{ and }\quad \sqrt{n}{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})\ \xrightarrow {{\mathscr {D}}} N(\mathbf {0},{\mathbf {V}}_{j})\quad \text{ as } n\rightarrow \infty .$

The first result in this lemma implies that $\sqrt{n}\big ({\widetilde{S}}_{jn}({\varvec{\beta }}_0)- {\widetilde{T}}_{jn}({\varvec{\beta }}_0)\big )\ \xrightarrow {P} \ 0.$ Thus, $\sqrt{n}{\widetilde{S}}_{jn}({\varvec{\beta }}_0)$ and $\sqrt{n}{\widetilde{T}}_{jn}({\varvec{\beta }}_0)$ have the same asymptotic distribution (Hettmansperger and McKean 2011). While the first result of this theorem holds for ${\varvec{\beta }}\in {\mathscr {B}},$ one could just restrict ourselves in ${\mathscr {B}}_{n}$ for the remaining part of the paper instead of considering ${\mathscr {B}}.$ As a consequence of Lemma 2, with probability 1, ${\widetilde{S}}_{n}({\varvec{\beta }})={\widetilde{T}}_{jn}({\varvec{\beta }})+o(1/\sqrt{n}).$ A Taylor expansion of $T_{jn}({\varvec{\beta }})$ around ${\varvec{\beta }}_0$ gives

$$\begin{aligned} {\widetilde{T}}_{jn}({\varvec{\beta }})={\widetilde{T}}_{jn}({\varvec{\beta }}_{0}) +({\varvec{\beta }}-{\varvec{\beta }}_{0})^{\tau }\nabla _{{\varvec{\beta }}} {\widetilde{T}}_{jn}({\varvec{\beta }}_{0}) +\frac{1}{2}({\varvec{\beta }}-{\varvec{\beta }}_{0})^{\tau }\nabla ^{2}_{{\varvec{\beta }}} {\widetilde{T}}_{jn}(\varvec{\zeta })({\varvec{\beta }}-{\varvec{\beta }}_{0}), \end{aligned}$$

where $\varvec{\zeta }$ belongs in the line segment joining ${\varvec{\beta }}_0$ and ${\varvec{\beta }}.$ Thus, with probability 1,

$$\begin{aligned} {\widetilde{S}}_{jn}({\varvec{\beta }})={\widetilde{T}}_{jn}({\varvec{\beta }}_{0}) +({\varvec{\beta }}-{\varvec{\beta }}_{0})^{\tau }\nabla _{{\varvec{\beta }}} {\widetilde{T}}_{jn}({\varvec{\beta }}_{0}) +\frac{1}{2}({\varvec{\beta }}-{\varvec{\beta }}_{0})^{\tau }\nabla ^{2}_{{\varvec{\beta }}} {\widetilde{T}}_{jn}(\varvec{\zeta })({\varvec{\beta }}-{\varvec{\beta }}_{0})+o(1/\sqrt{n}). \end{aligned}$$

$\widehat{{\varvec{\beta }}}_{n}^{j}$ being a solution of ${\widetilde{S}}_{jn}({\varvec{\beta }})=0,$ we have

$$\begin{aligned} 0= & {} {\widetilde{S}}_{jn}(\widehat{{\varvec{\beta }}}_{n}^{j})={\widetilde{T}}_{jn} ({\varvec{\beta }}_{0})+\nabla _{{\varvec{\beta }}} {\widetilde{T}}_{jn}({\varvec{\beta }}_{0})\cdot (\widehat{{\varvec{\beta }}}_{n}^{j} -{\varvec{\beta }}_{0})+\frac{1}{2}(\widehat{{\varvec{\beta }}}_{n}^{j} -{\varvec{\beta }}_{0})^{\tau }\nonumber \\&\quad \cdot \nabla ^{2}_{{\varvec{\beta }}}{\widetilde{T}}_{jn} (\varvec{\zeta }_{n}^{j})\cdot (\widehat{{\varvec{\beta }}}_{n}^{j}-{\varvec{\beta }}_{0})+o(1/\sqrt{n}), \end{aligned}$$

(1)

where $\varvec{\zeta }_{n}^{j}=\lambda {\varvec{\beta }}_{0}+(1-\lambda )\widehat{{\varvec{\beta }}}_{n}^{j}.$

Lemma 3

Under assumptions $(I_{1})-(I_7)$ and for any ${\mathbf {x}}\in \varGamma ,$ the following hold:

• (a) $\nabla _{{\varvec{\beta }}} {\widetilde{T}}_{jn}({\varvec{\beta }}_{0}) \rightarrow {\mathbf {W}}_{j} \ a.s.$

• (b) $\nabla ^{2}_{{\varvec{\beta }}}{\widetilde{T}}_{jn}(\varvec{\zeta }_{n}^{j})$ is almost surely bounded.

Proof of Lemma 2

Let $a_{ijn}=R(\xi _{ij}({\varvec{\beta }}))/(n+1)$ and let c be any arbitrary positive constant. Then, $\displaystyle S_{jn}({\varvec{\beta }})=\frac{1}{n}\sum \nolimits _{i=1}^{n}w_{i}{\mathbf {x}}_{i} {\widehat{g}}^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }})\varphi (a_{ijn}).$

(a) For any ${\mathbf {x}}\in \varGamma$ and ${\varvec{\beta }}\in {\mathscr {B}},$ the triangle inequality gives

$$\begin{aligned} \Vert {\widetilde{S}}_{jn}({\varvec{\beta }})-S_{jn}({\varvec{\beta }})\Vert\le & {} \frac{1}{n}\sum _{i=1}^{n}w_{i}\Vert {\mathbf {x}}_{i}\Vert |\varphi (a_{ijn})|| {\widehat{g}}^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }})-g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }})|. \end{aligned}$$

Since $\varphi$ is bounded, there exists a positive constant c such that $|\varphi (t)|\le c,$ for any $t\in (0,1).$ Also, since $w_{i}$ are positive and $\sum _{i=1}^{n}w_{i}=1,$ $w_{i}\le 1,$ for any i. This together with Swartz inequality imply that

$$\begin{aligned} \sup _{{\varvec{\beta }}\in {\mathscr {B}}}\Vert {\widetilde{S}}_{jn}({\varvec{\beta }})- S_{jn}({\varvec{\beta }})\Vert\le & {} c\left[ \frac{1}{n}\sum _{i=1}^{n}\Vert {\mathbf {x}}_{i}\Vert ^{2}\right] ^{1/2}\left[ \sup _{{\varvec{\beta }}\in {\mathscr {B}}}\max _{1\le i\le n}|{\widehat{g}}^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }})-g^{\prime } ({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }})|^{2}\right] ^{1/2}. \end{aligned}$$

By Lemma 1, $\displaystyle \lim \nolimits _{n\rightarrow \infty }\sup \nolimits _{{\varvec{\beta }}\in {\mathscr {B}}}\max _{1\le i\le n}|{\widehat{g}}^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }})-g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }})|^{2}= 0\;a.s.$ and by SLLN, $\displaystyle \frac{1}{n}\sum \nolimits _{i=1}^{n}\Vert {\mathbf {x}}_{i}\Vert ^{2}$ converges almost surely to $E[\Vert {\mathbf {x}}\Vert ^{2}]<\infty \;a.s.,$ by assumption $(I_6).$ Hence,

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{{\varvec{\beta }}\in {\mathscr {B}}}\Vert {\widetilde{S}}_{jn} ({\varvec{\beta }})-S_{jn}({\varvec{\beta }})\Vert =0\;a.s. \end{aligned}$$

(b) Also,

$$\begin{aligned}{\widetilde{S}}_{jn}({\varvec{\beta }})-{\widetilde{T}}_{jn}({\varvec{\beta }}) =\frac{1}{n}\sum _{i=1}^{n}w_{i}{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau } {\varvec{\beta }})\left[ \varphi (a_{ijn})-\varphi (F_{ij}(\xi _{ij}({\varvec{\beta }})))\right] . \end{aligned}$$

Recall that following Hájek and Šidák (1967), $a_{ijn}\rightarrow F_{ij}(\xi _{ij}({\varvec{\beta }}))\; a.s.$ as $n\rightarrow \infty ,$ for all ${\varvec{\beta }}\in {\mathscr {B}}$ and for all i,  $j=1,2.$ So, by the generalized continuous mapping theorem (Whitt 2002),

$$\begin{aligned}\max _{1\le i\le n}\left| \varphi (a_{ijn}) -\varphi (F_{ij}(\xi _{ij}({\varvec{\beta }})))\right| \rightarrow 0\; a.s. \quad \hbox { as}\ n\rightarrow \infty . \end{aligned}$$

From the boundedness of $g^{\prime }(t)$ and the fact that $w_{i}\le 1,$ for all i,  we have

$$\begin{aligned}&\sup _{{\varvec{\beta }}\in {\mathscr {B}}}\Vert {\widetilde{S}}_{jn}({\varvec{\beta }})- {\widetilde{T}}_{jn}({\varvec{\beta }})\Vert \le \left[ \frac{1}{n}\sum _{i=1}^{n} \{\Vert {\mathbf {x}}_{i}\Vert J({\mathbf {x}}_{i})\}^{2}\right] ^{1/2}\\&\quad \left[ \sup _{{\varvec{\beta }} \in {\mathscr {B}}}\max _{1\le i\le n}\left| \varphi (a_{ijn}) -\varphi (F_{ij}(\xi _{ij}({\varvec{\beta }})))\right| ^{2}\right] ^{1/2}, \end{aligned}$$

by assumption $(I_3)$ and Cauchy–Schwarz inequality. Thus,

$$\begin{aligned}\lim _{n\rightarrow \infty }\sup _{{\varvec{\beta }}\in {\mathscr {B}}} \Vert {\widetilde{S}}_{jn}({\varvec{\beta }})-{\widetilde{T}}_{jn}({\varvec{\beta }})\Vert =0\;a.s., \end{aligned}$$

since the SLLN ensures that $\displaystyle \frac{1}{n}\sum \nolimits _{i=1}^{n}\{\Vert {\mathbf {x}}_{i}\Vert J({\mathbf {x}}_{i})\}^{2}\rightarrow E(\{\Vert {\mathbf {x}}\Vert J({\mathbf {x}})\}^{2})<\infty \; a.s.,$ by assumption $(I_3)$ and $(I_6)$ and,

$$\begin{aligned} \sup _{{\varvec{\beta }}\in {\mathscr {B}}}\max _{1\le i\le n}\left| \varphi (a_{ijn})-\varphi (F_{ij}(\xi _{ij}({\varvec{\beta }})))\right| ^{2}\rightarrow 0\;a.s.\quad \text{ as } n\rightarrow \infty . \end{aligned}$$

Next,

$$\begin{aligned} E[{\widetilde{S}}_{jn}({\varvec{\beta }}_{0})-{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})] =\frac{1}{n}\sum _{i=1}^{n}E\left[ w_{i}{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau } {\varvec{\beta }}_{0})\left( \varphi \left( \frac{R(\xi _{j}({\varvec{\beta }}_{0}))}{n+1}\right) -\varphi \left( F_{j}(\xi _{j}({\varvec{\beta }}_{0}))\right) \right) \right] , \end{aligned}$$

from which Cauchy–Schwarz inequality gives

$$\begin{aligned} \Vert E[{\widetilde{S}}_{jn}({\varvec{\beta }}_{0})-{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})]\Vert\le & {} \left[ \frac{1}{n}\sum _{i=1}^{n}E\{(w_{i}{\mathbf {x}}_{i}g^{\prime } ({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0}) )^{2}\}\right] ^{1/2}\\&\left[ E\left( \left| \varphi \left( \frac{R(\xi _{j} ({\varvec{\beta }}_{0}))}{n+1}\right) -\varphi \left( F_{j}(\xi _{j} ({\varvec{\beta }}_{0}))\right) \right| ^{2}\right) \right] ^{1/2}. \end{aligned}$$

From the discussion above,

$$\begin{aligned}\left| \varphi \left( \frac{R(\xi _{j}({\varvec{\beta }}_{0}))}{n+1}\right) -\varphi \left( F_{j}(\xi _{j}({\varvec{\beta }}_{0}))\right) \right| ^{2} \rightarrow 0\; a.s.\quad \text{ as } n\rightarrow \infty . \end{aligned}$$

This together with the generalized dominated convergence theorem gives

$$\begin{aligned} E\left( \left| \varphi \left( \frac{R(\xi _{j}({\varvec{\beta }}_{0}))}{n+1} \right) -\varphi \left( F_{j}(\xi _{j}({\varvec{\beta }}_{0}))\right) \right| ^{2}\right) \rightarrow 0\;\hbox { as}\ n\rightarrow \infty . \end{aligned}$$

Thus, $E[{\widetilde{S}}_{jn}({\varvec{\beta }}_{0})-{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})]\rightarrow \;0$ as $n\rightarrow \infty .$ By Chebychev’s inequality, for any $\epsilon >0$, we have

$$\begin{aligned} P\left( \sqrt{n}\big ({\widetilde{S}}_{jn}({\varvec{\beta }}_0)-{\widetilde{T}}_{jn} ({\varvec{\beta }}_0)\big )>\epsilon \right) \le \epsilon ^{-2}E\Big [n\big ({\widetilde{S}}_{jn}({\varvec{\beta }}_0) -{\widetilde{T}}_{jn}({\varvec{\beta }}_0)\big )^{2}\Big ]. \end{aligned}$$

To complete the proof, it sufficies to show that $E\big [n\big ({\widetilde{S}}_{jn}({\varvec{\beta }}_0)-{\widetilde{T}}_{jn }({\varvec{\beta }}_0)\big )^{2}\big ]\rightarrow 0$ as $n\rightarrow \infty$. Following Hájek et al. (1999, p. 192),

$$\begin{aligned}&E\left[ n\big ({\widetilde{S}}_{jn}({\varvec{\beta }}_{0})-{\widetilde{T}}_{jn} ({\varvec{\beta }}_{0})\big )^{2}\right] \\&\quad =\frac{1}{n}E \left[ \left\{ \sum _{i=1}^{n}w_{i}{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau } {\varvec{\beta }}_{0})\left( \varphi \left( \frac{R(\xi _{ij} ({\varvec{\beta }}_{0}))}{n+1}\right) -\varphi \left( F_{j}(\xi _{ij} ({\varvec{\beta }}_{0}))\right) \right) \right\} ^{2}\right] \\&\quad \le \frac{n}{n-1}\left\{ \frac{1}{n}\sum _{i=1}^{n}E[(w_{i}{\mathbf {x}}_{i} g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0}))^{2}]\right\} E \left[ \left( \varphi \left( \frac{R(\xi _{j})}{n+1}\right) -\varphi \left( F_{j}(\xi _{j})\right) \right) ^{2}\right] . \end{aligned}$$

From the SLLN, it is not hard to see that

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}(w_{i}{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau } {\varvec{\beta }}_{0}))^{2}\le \frac{1}{n}\sum _{i=1}^{n}\Vert {\mathbf {x}}_{i}\Vert ^{2}J^{2}({\mathbf {x}}_{i})\rightarrow E[\Vert {\mathbf {x}}\Vert ^{2}J^{2}({\mathbf {x}})]<\infty \;a.s., \end{aligned}$$

by assumption $(I_3)$ and $(I_6).$ This implies that the right hand side the above inequality converges to zero. Thus,

$$\begin{aligned} E\left[ n\big ({\widetilde{S}}_{jn}({\varvec{\beta }}_{0})-{\widetilde{T}}_{jn} ({\varvec{\beta }}_{0})\big )^{2}\right] \rightarrow 0\quad \text{ as } n\rightarrow \infty \end{aligned}$$

and therefore,

$$\begin{aligned} \lim _{n\rightarrow \infty }P\left( \sqrt{n}\big ({\widetilde{S}}_{jn} ({\varvec{\beta }}_{0})-{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})\big )>\epsilon \right) =0. \end{aligned}$$

Now, by assumption $(I_7),$ we have $E\{{\widetilde{S}}_{jn}({\varvec{\beta }}_{0})\}\rightarrow 0$ as $n\rightarrow \infty .$ Since

$$\begin{aligned} E\{{\widetilde{S}}_{jn}({\varvec{\beta }}_{0})-{\widetilde{T}}_{jn}({\varvec{\beta }}_{0}))\}\rightarrow 0, \end{aligned}$$

we have $E\{{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})\}\rightarrow 0$ as $n\rightarrow \infty .$ Recall that ${\widetilde{T}}_{jn}({\varvec{\beta }}_{0})=\frac{1}{n}\sum _{i=1}^{n}w_{i}{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0})\varphi (F_{j}(\xi _{j}))=\frac{1}{n}\sum _{i=1}^{n}V_{ij},$ where $V_{ij}=w_{i}{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0})\varphi [F_{j}(\xi _{j})].$ To obtain the asymptotic distribution of ${\widetilde{T}}_{jn}({\varvec{\beta }}_0),$ we employ the Cramér–Wold device (Serfling 1980). To this end, set

$$\begin{aligned} U_{jn}=n^{-1/2}\sum _{i=1}^{n}{\mathbf {a}}^{T}w_{i}{\mathbf {x}}_{i}\{g^{\prime } ({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0})\}\varphi [F_{j}(\xi _{j})] ={\mathbf {a}}^{\tau }{\widetilde{T}}_{n}({\varvec{\beta }}_{0}), \end{aligned}$$

where ${\mathbf {a}}\in {\mathbb {R}}^q.$ Since $E\{{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})\}\rightarrow 0$ and $E(U_{jn})={\mathbf {a}}^{T} E\{T_{jn}({\varvec{\beta }}_0)\},$ we have $E(U_{jn})\rightarrow 0.$ From the fact that $({\mathbf {x}}_{i},y_{i},\varepsilon _{i})$, $i=1,\ldots ,n,$ is a random sample, $({\mathbf {x}}_{i},z_{ij}, \xi _{ij})$ is also a random sample. Conditional on ${\mathbf {x}}_{i},$ $\{V_{nj}\}_{n\ge 1}$ is a sequence of conditional independent random variables with identical distribution (Rao 2009).

$$\begin{aligned} \text{ Var }(U_{jn})= & {} \frac{1}{n}\sum _{i=1}^{n}E\left[ \left( w_{i} {\mathbf {a}}^{\tau }{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0})\right) ^{2} E\left\{ \varphi ^{2}(F_{j}(\xi _{j}))|{\mathbf {x}}_{i}\right\} \right] \ \rightarrow \ {\mathbf {a}}^{\tau }{\mathbf {V}}_{j}{\mathbf {a}}\quad \text{ as } n\rightarrow \infty . \end{aligned}$$

Note that ${\widetilde{T}}_{jn}({\varvec{\beta }}_{0})$ is the sum of identically distributed conditionally independent functions of random variables. Hence, the limiting distribution is established by verifying the conditional Lindeberg–Feller condition for the applicability of the conditional Central Limit Theorem. To do this, let $\zeta _{jn}=\sqrt{n}{\widetilde{T}}_{jn}({\varvec{\beta }}_{0}))$ and set $\sigma _{jn}^{2}=Var(\zeta _{jn}).$ We need to show that for any $\mu >0,$

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{\sigma _{nj}^{2}}\sum _{i=1}^{n}E\left[ \frac{1}{n} \left( w_{i}{\mathbf {a}}^{\tau }{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0}) \right) ^{2}E\{\varphi ^{2}[F_{j}(\xi _{j})|{\mathbf {x}}_{i}]\}\right] \\&\quad \times I\left( \Big |\frac{1}{\sqrt{n}}\big (w_{i}{\mathbf {a}}^{\tau }{\mathbf {x}}_{i}g^{\prime } ({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0})\big )\varphi [F_{j}(\xi _{j})]\Big |>\mu \sigma _{nj}\right) =0. \end{aligned}$$

To see this, note that since $w_{i}\le 1,$ for all i,  we have

$$\begin{aligned} \frac{1}{\sqrt{n}}|w_{i}{\mathbf {a}}^{\tau }{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau } {\varvec{\beta }}_{0})|\le n^{-1/2}\Vert {\mathbf {a}}\Vert \Vert {\mathbf {x}}_{i}\Vert J({\mathbf {x}}_{i})\rightarrow 0\;a.s. \quad \text{ as } n\rightarrow \infty , \end{aligned}$$

since $\Vert {\mathbf {x}}\Vert J({\mathbf {x}})$ is almost surely finite, as $E(\Vert {\mathbf {x}}\Vert J({\mathbf {x}}))<\infty ,$ by $(I_3)$ and $(I_6).$ Set

$$\begin{aligned} \lambda _{n}=\displaystyle \big [\max _{1\le i\le n}n^{-1}|w_{i}{\mathbf {a}}^{\tau }{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0})|\Big ]^{1/2}. \end{aligned}$$

Then $\lambda _{n}\rightarrow 0$ as $n\rightarrow \infty ,$ and, is independent of i. Since $\sigma ^{2}_{jn}$ converges to a positive quantity, the ratio $\sigma _{jn}/\lambda _{n}\rightarrow \infty$ as $n\rightarrow \infty .$ Now, it is easy to see that

$$\begin{aligned}&\sum _{i=1}^{n}E\Big [\frac{1}{n}(w_{i}{\mathbf {a}}^{\tau }{\mathbf {x}}_{i}g^{\prime } ({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0}))^{2}E\{\varphi ^{2}[F_{j}(\xi _{ij})] |{\mathbf {x}}_{i}\}I\\&\quad \Big (\Big |\frac{1}{\sqrt{n}}(w_{i}{\mathbf {a}}^{\tau }{\mathbf {x}}_{i}g^{\prime } ({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0})\varphi [F_{j}(\xi _{ij})]\Big |>\mu \sigma _{jn}\Big )\Big ] \end{aligned}$$

is less than or equal to

$$\begin{aligned} E\left\{ E\Big [\varphi ^{2}[F_{j}(\xi _{j})]I\Big (\big |\varphi [F_{j}(\xi _{j})] \big |>\mu \sigma _{jn}/\lambda _{n}\Big )\Big |{\mathbf {x}}\Big ]\right\} \times \frac{1}{n}\sum _{i=1}^{n}E\left[ (w_{i}{\mathbf {a}}^{\tau }{\mathbf {x}}_{i}g^{\prime } ({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0}))^{2}\right] . \end{aligned}$$

In this expression, the second term $\displaystyle \lim _{n \rightarrow \infty } \frac{1}{n}\sum \nolimits _{i=1}^{n}E\left[ (w_{i}{\mathbf {x}}_{i}g^{\prime }({\mathbf {x}}_{i}^{\tau } {\varvec{\beta }}_{0}))^{2}\right] \le E(\Vert {\mathbf {x}}\Vert ^{2}J^{2}({\mathbf {x}})) < \infty .$ From the boundedness of $\varphi$ and applying the Dominated Convergence Theorem to the first term, we have

$$\begin{aligned} E\left\{ E\Big [\varphi ^{2}[F_{j}(\xi _{j})]I\Big (\big |\varphi [F_{j}(\xi _{j})]\big |>\mu \sigma _{jn}/\lambda _{n}\Big )\Big |{\mathbf {x}}\Big ]\right\} \rightarrow 0\quad as\quad n\rightarrow \infty . \end{aligned}$$

This established the conditional Lindeberg–Feller condition and thus, the conditional Central Limit Theorem (Rao 2009) gives $\sqrt{n}U_{jn}({\varvec{\beta }}_0)\ \xrightarrow {{\mathscr {D}}} \ N(\mathbf {0},{\mathbf {a}}^{\tau }{\mathbf {V}}_{j}{\mathbf {a}})$ as $n\rightarrow \infty .$ Thus, by the Cramér–Wold device (Serfling 1980), $\sqrt{n}{\widetilde{T}}_{jn}({\varvec{\beta }}_0)\ \xrightarrow {{\mathscr {D}}} \ N(\mathbf {0},{\mathbf {V}}_{j})$ as $n\rightarrow \infty .$ $\square$

Proof of Lemma 3

Differentiating ${\widetilde{T}}_{jn}({\varvec{\beta }})$ with respect to ${\varvec{\beta }},$ we have

$$\begin{aligned} \nabla _{{\varvec{\beta }}}{\widetilde{T}}_{jn}({\varvec{\beta }})= & {} -\frac{1}{n}\sum _{i=1}^{n} w_{i}{\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }[g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }})]^{2} f_{ij}(\xi _{ij}({\varvec{\beta }}))\varphi ^{\prime }(F_{ij}(\xi _{ij}({\varvec{\beta }})))\\&\quad +\frac{1}{n}\sum _{i=1}^{n}w_{i}{\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }g^{\prime \prime } ({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }})\varphi (F_{ij}(\xi _{ij}({\varvec{\beta }}))). \end{aligned}$$

At ${\varvec{\beta }}={\varvec{\beta }}_{0},$ we have

$$\begin{aligned} \nabla _{{\varvec{\beta }}}{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})= & {} -\frac{1}{n}\sum _{i=1}^{n}w_{i}{\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }[g^{\prime }({\mathbf {x}}_{i}^{\tau } {\varvec{\beta }}_{0})]^{2}f_{j}(\xi _{j}))\varphi ^{\prime }(F_{j}(\xi _{j}))\\&\quad +\frac{1}{n}\sum _{i=1}^{n}w_{i}{\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }g^{\prime \prime } ({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0}\varphi (F_{j}(\xi _{j})). \end{aligned}$$

Note that $E[w_{i}{\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }[g^{\prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0})] ^{2}E\{f_{j}(\xi _{j}))\varphi ^{\prime }(F_{j}(\xi _{j}))|{\mathbf {x}}_{i}\}] =E\big [w{\mathbf {x}}{\mathbf {x}}^{\tau }\{g^{\prime }({\mathbf {x}}^{\tau }{\varvec{\beta }}_{0}\}^{2} \mu _{j}({\mathbf {x}})\big ]<\infty ,$ by assumptions $(I_1),$ $(I_3)$ and $(I_6).$ Also, $E[w_{i}{\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }g^{\prime \prime }({\mathbf {x}}_{i}^{\tau } {\varvec{\beta }}_{0})E\{\varphi (F_{j}(\xi _{j}))|{\mathbf {x}}_{i}\}]=E[w_{i} {\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }g^{\prime \prime }({\mathbf {x}}_{i}^{\tau }{\varvec{\beta }}_{0})\nu _{j}({\mathbf {x}}_{i})],$ which is also finite by assumptions $(I_1),$ $(I_3)$ and $(I_6).$ Now, a direct application of the SLLN, gives $\nabla _{{\varvec{\beta }}}{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})\rightarrow {\mathbf {W}}_{j}\;a.s.$ If $\varepsilon$ is independent of ${\mathbf {x}},$ $\xi _{j}$ is independent of ${\mathbf {x}}.$ Hence,

$$\begin{aligned} \mu _{j}({\mathbf {x}})=E[f_{j}(\xi _{j})\varphi ^{\prime }(F_{j}(\xi _{j}))|{\mathbf {x}}] =E[f_{j}(\xi _{j})\varphi ^{\prime }(F_{j}(\xi _{j}))]=\int f_{j}(\xi _{j})\varphi ^{\prime }(F_{j}(\xi _{j})) dF_{j}(\xi _{j}). \end{aligned}$$

From integration by parts, $\displaystyle \int f_{j}(\xi _{j})\varphi ^{\prime }(F_{j}(\xi _{j})) dF_{j}(\xi _{j}){=}-\int _{-\infty }^{\infty }f^{\prime }_{j}(\xi _{j}) \varphi (F_{j}(\xi _{j}))d\xi _{j}$ since $f_{j}(\xi _{j})\varphi (F_{j}(\xi _{j}))\rightarrow 0$ as $\xi _{j}\rightarrow \pm \infty .$ Putting $x^{j}=F_{j}(\xi _{j}),$ we have

$$\begin{aligned} \int _{-\infty }^{\infty }f_{j}^{\prime }(\xi _{j})\varphi (F_{j} (\xi _{j}))d\xi _{j}=-\int _{0}^{1}\varphi (x^{j})\varphi _{f_{j}} (x^{j})dx^{j}=-\gamma _{j\varphi }^{-1}. \end{aligned}$$

(2)

Also, $\nu _{j}({\mathbf {x}})=E[\varphi (F_{j}(\xi _{j}))|{\mathbf {x}}],$ from which putting $x^{j}=F_{j}(\xi _{j}),$ $dx^{j}=dF_{j}(\xi _{j})$ and

$$\begin{aligned} \nu _{j}({\mathbf {x}})= & {} E[\varphi (F_{j}(\xi _{j}))|{\mathbf {x}}]=E[\varphi (F_{j} (\xi _{j}))]=\int _{-\infty }^{\infty }\varphi (F_{j}(\xi _{j}))dF_{j} (\xi _{j})\nonumber \\= & {} \int _{0}^{1}\varphi (x^{j})dx^{j}=0, \end{aligned}$$

(3)

by assumption $(I_{1}).$ From Eqs. (2) and (3),

$$\begin{aligned} {\mathbf {W}}_{j}= & {} -E\big [w{\mathbf {x}}{\mathbf {x}}^{\tau }\{g^{\prime }({\mathbf {x}}^{\tau }{\varvec{\beta }}_{0}\}^{2} \mu _{j}({\mathbf {x}})\big ]+E\big [w{\mathbf {x}}{\mathbf {x}}^{\tau }\{g^{\prime \prime }({\mathbf {x}}^{\tau } {\varvec{\beta }}_{0})\}\nu _{j}({\mathbf {x}})\big ]\\= & {} -E\big [w{\mathbf {x}}{\mathbf {x}}^{\tau }\{g^{\prime } ({\mathbf {x}}^{\tau }{\varvec{\beta }}_{0}\}^{2}\mu _{j}({\mathbf {x}})\big ]\\= & {} \gamma _{j\varphi }^{-1}E\big [w{\mathbf {x}}{\mathbf {x}}^{\tau }\{g^{\prime }({\mathbf {x}}^{\tau } {\varvec{\beta }}_{0}\}^{2}\big ]\\= & {} \gamma _{j\varphi }^{-1}\varSigma . \end{aligned}$$

Finally,

$$\begin{aligned}&\nabla _{{\varvec{\beta }}}^{2}{\widetilde{T}}_{jn}(\varvec{\xi }_n)\\&\quad =-\frac{3}{n} \sum _{i=1}^{n} w_{i}({\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }){\mathbf {x}}_{i}g^{\prime \prime } (\varvec{\xi }_{n}^{T}{\mathbf {x}}_{i})g^{\prime }(\varvec{\xi }_{n}^{T}{\mathbf {x}}_{i}) {\tilde{f}}_{i}(\eta _{i}(\varvec{\xi }_{n}))\varphi ^{\prime }({\tilde{F}}_{i} ({\tilde{\eta }}_{i}(\varvec{\xi }_{n})))\\&\qquad +\frac{1}{n}\sum _{i=1}^{n}w_{i}({\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }){\mathbf {x}}_{i} \left( g^{\prime }({\mathbf {x}}_{i}^{\tau }\varvec{\xi }_{n})\right) ^{3}{\tilde{f}}_{i}^{2} ({\tilde{\eta }}_{i}(\varvec{\xi })_{n})\varphi ^{\prime \prime }({\tilde{F}}_{i} ({\tilde{\eta }}_{i}(\varvec{\xi }_{n})))\\&\qquad +\frac{1}{n}\sum _{i=1}^{n}w_{i}({\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }){\mathbf {x}}_{i} \left( g^{\prime }({\mathbf {x}}_{i}^{\tau }\varvec{\xi }_{n})\right) ^{3} {\tilde{f}}_{i}^{\prime }({\tilde{\eta }}_{i}(\varvec{\xi })_{n})\varphi ^{\prime } ({\tilde{F}}_{i}({\tilde{\eta }}_{i}(\varvec{\xi }_{n})))\\&\qquad +\frac{1}{n}\sum _{i=1}^{n}w_{i}({\mathbf {x}}_{i}{\mathbf {x}}_{i}^{\tau }) {\mathbf {x}}_{i}g^{\prime \prime \prime }({\mathbf {x}}_{i}^{\tau }\varvec{\xi }_{n})\varphi ^{\prime } ({\tilde{F}}_{i}({\tilde{\eta }}_{i}(\varvec{\xi }_{n}))). \end{aligned}$$

Assumptions $(I_1)-(I_3)$ and $(I_6)$ together with a direct application of the SLLN, imply that each term to the right hand side of the above equality is bounded in probability, which in turn implies that $\nabla _{{\varvec{\beta }}}^{2}{\widetilde{T}}_{n}(\varvec{\xi }_n)$ is bounded in probability. $\square$

Proof of Theorem 2

From the consistency of $\widehat{{\varvec{\beta }}}_{n}^{j}$ and the boundedness of $\nabla _{{\varvec{\beta }}}^{2}{\widetilde{T}}_{jn}(\varvec{\xi }_n),$ we have

$$\begin{aligned} (\widehat{{\varvec{\beta }}}_{n}^{j}-{\varvec{\beta }}_{0})\nabla _{{\varvec{\beta }}}^{2} {\widetilde{T}}_{jn}(\varvec{\xi }_{n})\rightarrow 0\;a.s. \end{aligned}$$

Then, Eq. (1) is reduced to $0={\widetilde{T}}_{jn}({\varvec{\beta }}_{0})+\nabla _{{\varvec{\beta }}} {\widetilde{T}}_{jn}({\varvec{\beta }}_{0})\cdot (\widehat{{\varvec{\beta }}}_{n}^{j} -{\varvec{\beta }}_{0})+o_{p}(1/\sqrt{n}).$ Thus, taking into account Lemma 3a we have $\sqrt{n}(\widehat{{\varvec{\beta }}}_{n}^{j}-{\varvec{\beta }}_{0})=-{\mathbf {W}}_{j}^{-} \sqrt{n}{\widetilde{T}}_{jn}({\varvec{\beta }}_{0})+o_{p}(1).$ Since $\sqrt{n}{\widetilde{T}}_{jn}({\varvec{\beta }}_0)\ \xrightarrow {{\mathscr {D}}} \ N(\mathbf {0},{\mathbf {V}}_{j})$ as $n\rightarrow \infty ,$ we have $\sqrt{n}(\widehat{{\varvec{\beta }}}_{n}^{j}-{\varvec{\beta }}_{0})\ \xrightarrow {{\mathscr {D}}} \ N\big (\mathbf {0},{\mathbf {W}}_{j}^{-}{\mathbf {V}}_{j}{\mathbf {W}}_{j}^{-}\big ).$ $\square$