$\displaystyle\leq$ $\displaystyle
Cn^{-1/2}(C_{1}n^{1/2}+C_{2}n^{1/2})+\frac{1}{2}n^{-1}(-C^{2}C_{3}n)$
$\displaystyle=$ $\displaystyle
C\left\\{(C_{1}+C_{2})-\frac{C_{3}}{2}C\right\\}$ $\displaystyle<$
$\displaystyle 0$

with probability at least $1-3\epsilon$ when $C>2(C_{1}+C_{2})/C_{3}$, which
proves (S.26).

We next show that the Hessian of
$l\\{{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}$ is negative definite
for all ${\bm{\beta}}$, hence the local maximizer inside the disk with radius
$Cn^{-1/2}$ is in fact the maximizer $\widehat{\bm{\beta}}$, i.e.
$\|\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$. We have

$\displaystyle\begin{bmatrix}\partial^{2}l\\{{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}/\partial{\bm{\beta}}\partial{\bm{\beta}}^{\rm
T}&\partial^{2}l\\{{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}/\partial{\bm{\beta}}\partial{\bm{\gamma}}^{\rm
T}\\\
\partial^{2}l\\{{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}/\partial{\bm{\gamma}}\partial{\bm{\beta}}^{\rm
T}&\partial^{2}l\\{{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}/\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}\end{bmatrix}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\begin{bmatrix}{\bf x}_{i}{\bf x}_{i}^{\rm
T}\hbox{var}^{*}\\{Y|{\bf
x}_{i},{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}&{\bf
x}_{i}\hbox{cov}^{*}\\{Y,{\bf B}(Y)|{\bf
x}_{i},{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}\\\ [{\bf
x}_{i}\hbox{cov}^{*}\\{Y,{\bf B}(Y)|{\bf
x}_{i},{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}]^{\rm
T}&\hbox{cov}^{*}\\{{\bf B}(Y),{\bf B}(Y)|{\bf
x}_{i},{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}\end{bmatrix}$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\hbox{cov}^{*}\left\\{\begin{bmatrix}{\bf
x}_{i}Y\\\ {\bf B}(Y)\end{bmatrix},\begin{bmatrix}{\bf x}_{i}Y\\\ {\bf
B}(Y)\end{bmatrix}\Bigg{|}{\bf
x}_{i},{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\right\\}$

is negative definite at all ${\bm{\beta}}$. Then since the Hessian is the
Schur complement in the above matrix, it is also negative definite for all
${\bm{\beta}}$. Hence we conclude that
$\|\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$.

Now, we analyze the asymptotic properties of $\widehat{\bm{\beta}}$. Since
$\widehat{\bm{\beta}}$ maximizes
$l\\{{\bm{\beta}},\widehat{\bm{\gamma}}({\bm{\beta}})\\}$, we have

$\displaystyle{\bf 0}$ $\displaystyle=$
$\displaystyle\frac{dl\\{\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}}{d{\bm{\beta}}}$
$\displaystyle=$
$\displaystyle\frac{dl\\{{\bm{\beta}}_{0},\widehat{\bm{\gamma}}({\bm{\beta}}_{0})\\}}{d{\bm{\beta}}}+\frac{d^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{d{\bm{\beta}}d{\bm{\beta}}^{\rm
T}}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0}),$

by the Taylor expansion with
${\bm{\beta}}^{*}\equiv\alpha_{1}\widehat{\bm{\beta}}+(1-\alpha_{1}){\bm{\beta}}_{0}$
for some $\alpha_{1}\in(0,1)$. This implies

$\displaystyle\widehat{\bm{\beta}}-{\bm{\beta}}_{0}=\left[-n^{-1}\frac{d^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{d{\bm{\beta}}d{\bm{\beta}}^{\rm
T}}\right]^{-1}n^{-1}\frac{dl\\{{\bm{\beta}}_{0},\widehat{\bm{\gamma}}({\bm{\beta}}_{0})\\}}{d{\bm{\beta}}}.$

Recall that for
${\bm{\gamma}}^{*}=\alpha_{2}\widehat{\bm{\gamma}}({\bm{\beta}}_{0})+(1-\alpha_{2}){\bm{\gamma}}_{0}$
for some $\alpha_{2}\in(0,1)$,

$\displaystyle\frac{dl\\{{\bm{\beta}}_{0},\widehat{\bm{\gamma}}({\bm{\beta}}_{0})\\}}{d{\bm{\beta}}}=\frac{\partial
l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})}{\partial{\bm{\beta}}}+\frac{\partial^{2}l({\bm{\beta}}_{0},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}\partial{\bm{\gamma}}^{\rm
T}}\\{\widehat{\bm{\gamma}}({\bm{\beta}}_{0})-{\bm{\gamma}}_{0}\\}.$

Then by (S.30), (S.32), (S.35) and Proposition 1, we can write the gradient as

$\displaystyle\frac{dl\\{{\bm{\beta}}_{0},\widehat{\bm{\gamma}}({\bm{\beta}}_{0})\\}}{d{\bm{\beta}}}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n}{\bf x}_{i}\\{y_{i}-E(Y|{\bf
x}_{i})\\}$
$\displaystyle-{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}\sum_{i=1}^{n}[{\bf
B}(y_{i})-E\\{{\bf B}(Y)|{\bf x}_{i}\\}]+{\bf r}_{2}^{\prime},$

where $\|{\bf r}_{2}^{\prime}\|_{2}=o_{p}(n^{1/2})$. Hence by (S.40),

$\displaystyle\widehat{\bm{\beta}}-{\bm{\beta}}_{0}$ $\displaystyle=$
$\displaystyle{\bf\Sigma}^{*-1}n^{-1}\sum_{i=1}^{n}{\bf
x}_{i}\\{y_{i}-E(Y|{\bf x}_{i})\\}$
$\displaystyle-{\bf\Sigma}^{*-1}{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}n^{-1}\sum_{i=1}^{n}[{\bf
B}(y_{i})-E\\{{\bf B}(Y)|{\bf x}_{i}\\}]+{\bf r}_{2},$

where $\|{\bf r}_{2}\|_{2}=o_{p}(n^{-1/2})$. Thus by the central limit theorem
and the Slutsky’s theorem, we get

$\displaystyle{{\bf\Sigma}^{*}}^{1/2}\sqrt{n}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})\to
N({\bf 0},{\bf I})$

in distribution when $n\to\infty$. ∎

###  S.6 Lemmas

We introduce another lemmas for asymptotic properties of the profile
estimators.

######  Lemma 6.

Under Conditions (C1)-(C7),
$\|\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\|_{2}=O_{p}\\{(nh)^{-1/2}\\}$
and

$\displaystyle\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}$
$\displaystyle=$
$\displaystyle-{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}{\bf\Sigma}^{*-1}n^{-1}\sum_{i=1}^{n}{\bf
x}_{i}\\{y_{i}-E(Y|{\bf x}_{i})\\}$
$\displaystyle+({\bf\Sigma}_{22}^{-1}+{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}{\bf\Sigma}^{*-1}{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1})n^{-1}\sum_{i=1}^{n}[{\bf
B}(y_{i})-E\\{{\bf B}(Y)|{\bf x}_{i}\\}]+{\bf r}_{3},$

where $\|{\bf r}_{3}\|_{2}=o_{p}\\{(nh)^{-1/2}\\}$.

###### Proof.

Based on (S.28) and the Taylor expansion of
$\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})$, letting ${\bm{\beta}}^{*}$ be
on the line connecting ${\bm{\beta}}_{0}$ and $\widehat{\bm{\beta}}$, we get

$\displaystyle\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}$
$\displaystyle=$
$\displaystyle\widehat{\bm{\gamma}}({\bm{\beta}}_{0})-{\bm{\gamma}}_{0}-\left[\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}}\right]^{-1}\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\beta}}^{\rm
T}}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0}).$ (S.42)

It is easy to verify that
$\|\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\|_{2}=O_{p}\\{(nh)^{-1/2}\\}$
because
$\|\widehat{\bm{\gamma}}({\bm{\beta}}_{0})-{\bm{\gamma}}_{0}\|_{2}=O_{p}\\{(nh)^{-1/2}\\}$
by Proposition 1,
$\|\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$ by Proposition
2 and (S.36). Now recall that $\|{\bf\Sigma}_{22}^{-1}\|_{2}\asymp h^{-1}$ by
(S.23) and Condition (C6). In addition, for any ${\bf u}\in\mathbb{R}^{m}$
such that $\|{\bf u}\|_{2}=1$, we get

$\displaystyle\|{\bf\Sigma}_{12}{\bf u}\|_{2}\leq\|E({\bf X})\|_{2}\sup_{{\bf
x}\in\mathcal{X}}|\hbox{cov}\\{Y,{\bf B}(Y)^{\rm T}{\bf u}|{\bf
x}\\}|=O(h^{1/2})$

by Condition (C1) and Lemma 2, which implies
$\|{\bf\Sigma}_{21}\|_{2}=O(h^{1/2})$. Then by (S.38) and (S.39), we get

$\displaystyle{\bf
R}\equiv-\left[\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}}\right]^{-1}\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\beta}}^{\rm
T}}+{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}$

satisfies $\|{\bf R}\|_{2}=o_{p}(h^{-1/2})$. Then by Proposition 2, we have

$\displaystyle-\left[\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}}\right]^{-1}\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\beta}}^{\rm
T}}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})$ $\displaystyle=$
$\displaystyle-{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}{\bf\Sigma}^{*-1}n^{-1}\sum_{i=1}^{n}{\bf
x}_{i}\\{y_{i}-E(Y|{\bf x}_{i})\\}$
$\displaystyle+{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}{\bf\Sigma}^{*-1}{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}n^{-1}\sum_{i=1}^{n}[{\bf
B}(y_{i})-E\\{{\bf B}(Y)|{\bf x}_{i}\\}]+{\bf r}_{3}^{\prime},$

where

$\displaystyle\|{\bf r}_{3}^{\prime}\|_{2}$ $\displaystyle=$
$\displaystyle\left\|{\bf R}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0}-{\bf
r}_{2})-\left[\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}}\right]^{-1}\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\beta}}^{\rm
T}}{\bf r}_{2}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\|{\bf
R}\|_{2}(\|\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\|_{2}+\|{\bf
r}_{2}\|_{2})+\left\|\left[\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}}\right]^{-1}\frac{\partial^{2}l\\{{\bm{\beta}}^{*},\widehat{\bm{\gamma}}({\bm{\beta}}^{*})\\}}{\partial{\bm{\gamma}}\partial{\bm{\beta}}^{\rm
T}}\right\|_{2}\|{\bf r}_{2}\|_{2}$ $\displaystyle\leq$ $\displaystyle
o_{p}(h^{-1/2})O_{p}(n^{-1/2})+O_{p}(h^{-1/2})o_{p}(n^{-1/2})$
$\displaystyle=$ $\displaystyle o_{p}\\{(nh)^{-1/2}\\}$

by Proposition 2 and (S.36). Combining this with Proposition 1 and (S.42), we
get

$\displaystyle\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}$
$\displaystyle=$
$\displaystyle-{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}{\bf\Sigma}^{*-1}n^{-1}\sum_{i=1}^{n}{\bf
x}_{i}\\{y_{i}-E(Y|{\bf x}_{i})\\}$
$\displaystyle+({\bf\Sigma}_{22}^{-1}+{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}{\bf\Sigma}^{*-1}{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1})n^{-1}\sum_{i=1}^{n}[{\bf
B}(y_{i})-E\\{{\bf B}(Y)|{\bf x}_{i}\\}]+{\bf r}_{3},$

where $\|{\bf r}_{3}\|_{2}=o_{p}\\{(nh)^{-1/2}\\}$. ∎

######  Lemma 7.

Under Conditions (C1)-(C7),

$\displaystyle\begin{bmatrix}\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\\\
\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\end{bmatrix}={\bf\Sigma}^{-1}n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y|{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)|{\bf
x}_{i}\\}\end{bmatrix}+\begin{bmatrix}{\bf r}_{2}\\\ {\bf
r}_{3}\end{bmatrix},$

where $\|{\bf r}_{2}\|_{2}=o_{p}(n^{-1/2})$ and $\|{\bf
r}_{3}\|_{2}=o_{p}\\{(nh)^{-1/2}\\}$.

###### Proof.

Note that Proposition 2 and Lemma 6 lead to

$\displaystyle\begin{bmatrix}\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\\\
\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\end{bmatrix}=\begin{bmatrix}{\bf
C}_{11}&{\bf C}_{12}\\\ {\bf C}_{21}&{\bf
C}_{22}\end{bmatrix}n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y|{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)|{\bf
x}_{i}\\}\end{bmatrix}+\begin{bmatrix}{\bf r}_{2}\\\ {\bf
r}_{3}\end{bmatrix},$

where $\|{\bf r}_{2}\|_{2}=o_{p}(n^{-1/2})$, $\|{\bf
r}_{3}\|_{2}=o_{p}\\{(nh)^{-1/2}\\}$, and

$\displaystyle{\bf C}_{11}$ $\displaystyle\equiv$
$\displaystyle{\bf\Sigma}^{*-1},$ $\displaystyle{\bf C}_{12}$
$\displaystyle\equiv$
$\displaystyle-{\bf\Sigma}^{*-1}{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}={\bf
C}_{21}^{\rm T},$ $\displaystyle{\bf C}_{22}$ $\displaystyle\equiv$
$\displaystyle{\bf\Sigma}_{22}^{-1}+{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}{\bf\Sigma}^{*-1}{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}.$

In addition, it is easy to see that

$\displaystyle\begin{bmatrix}{\bf C}_{11}&{\bf C}_{12}\\\ {\bf C}_{21}&{\bf
C}_{22}\end{bmatrix}={\bf\Sigma}^{-1}.$

∎

###  S.7 Proof of Theorem 1

We can write $\widehat{\bm{\xi}}-{\bm{\xi}}_{0}$ as

$\displaystyle\widehat{\bm{\xi}}-{\bm{\xi}}_{0}$ $\displaystyle=$
$\displaystyle(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})\,n^{-1}\sum_{i=1}^{n}\hbox{var}^{*}\\{Y|{\bf
x}_{i},\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}$
(S.43)
$\displaystyle+{\bm{\beta}}_{0}\,n^{-1}\sum_{i=1}^{n}\left[\hbox{var}^{*}\\{Y|{\bf
x}_{i},\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}-\hbox{var}(Y|{\bf
x}_{i})\right]$
$\displaystyle+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}\hbox{var}(Y|{\bf
x}_{i})-E\\{\hbox{var}(Y|{\bf X})\\}\right].$

Noting that $\|\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$ by
Proposition 2 and
$\|\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\|_{2}=O_{p}\\{(nh)^{-1/2}\\}$
by Lemma 6, it is easy to see that

$\displaystyle n^{-1}\sum_{i=1}^{n}\hbox{var}^{*}\\{Y|{\bf
x}_{i},\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}$
$\displaystyle=$ $\displaystyle n^{-1}\sum_{i=1}^{n}[\hbox{var}^{*}\\{Y|{\bf
x}_{i},\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}-\hbox{var}(Y|{\bf
x}_{i})]$ (S.44) $\displaystyle+n^{-1}\sum_{i=1}^{n}[\hbox{var}(Y|{\bf
x}_{i})-E\\{\hbox{var}(Y|{\bf X})\\}]+E\\{\hbox{var}(Y|{\bf X})\\}$
$\displaystyle=$ $\displaystyle E\\{\hbox{var}(Y|{\bf X})\\}+o_{p}(1)$

by Lemma 5. Next, we have

$\displaystyle n^{-1}\sum_{i=1}^{n}\left[\hbox{var}^{*}\\{Y|{\bf
x}_{i},\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}-\hbox{var}(Y|{\bf
x}_{i})\right]$ $\displaystyle=$ $\displaystyle
n^{-1}\sum_{i=1}^{n}\left[\hbox{var}^{*}\\{Y|{\bf
x}_{i},\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}-\hbox{var}^{*}(Y|{\bf
x}_{i},{\bm{\beta}}_{0},{\bm{\gamma}}_{0})+\hbox{var}^{*}(Y|{\bf
x}_{i},{\bm{\beta}}_{0},{\bm{\gamma}}_{0})-\hbox{var}(Y|{\bf x}_{i})\right]$
$\displaystyle=$ $\displaystyle
n^{-1}\sum_{i=1}^{n}\left[\frac{\partial\hbox{var}^{*}(Y|{\bf
x}_{i},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}^{\rm
T}}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})+\frac{\partial\hbox{var}^{*}(Y|{\bf
x}_{i},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}^{\rm
T}}\\{\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\\}\right]+O(h^{q}),$

where $({\bm{\beta}}^{*\rm T},{\bm{\gamma}}^{*\rm T})^{\rm T}$ is a point on
the line connecting $\\{\widehat{\bm{\beta}}^{\rm
T},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})^{\rm T}\\}^{\rm T}$ and
$({\bm{\beta}}_{0}^{\rm T},{\bm{\gamma}}_{0}^{\rm T})^{\rm T}$. The last
equality holds by Lemma 4. Then by Lemma 5, it is easy to check that

$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}\frac{\partial\hbox{var}^{*}(Y|{\bf
x}_{i},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}}-E[\\{Y-E(Y|{\bf
X})\\}^{3}{\bf X}]\right\|_{2}$ (S.46) $\displaystyle=$
$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}E^{*}[\\{Y-E^{*}(Y|{\bf
x}_{i},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{3}|{\bf
x}_{i},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}]{\bf x}_{i}-E[\\{Y-E(Y|{\bf
X})\\}^{3}{\bf X}]\right\|_{2}$ $\displaystyle=$ $\displaystyle o_{p}(1).$

Furthermore, we have

$\displaystyle\frac{\partial\hbox{var}^{*}(Y|{\bf
x},{\bm{\beta}},{\bm{\gamma}})}{\partial{\bm{\gamma}}}=E^{*}(\\{Y-E^{*}(Y|{\bf
x},{\bm{\beta}},{\bm{\gamma}})\\}^{2}[{\bf B}(Y)-E^{*}\\{{\bf B}(Y)|{\bf
x},{\bm{\beta}},{\bm{\gamma}}\\}]|{\bf x},{\bm{\beta}},{\bm{\gamma}}).$ (S.47)

Note that for ${\bf u}\in\mathbb{R}^{m}$ such that $\|{\bf u}\|_{2}=1$,

$\displaystyle E^{*}[\\{Y-E^{*}(Y|{\bf
x},{\bm{\beta}},{\bm{\gamma}})\\}^{2}{\bf B}(Y)|{\bf
x},{\bm{\beta}},{\bm{\gamma}}]^{\rm T}{\bf u}$ $\displaystyle\leq$
$\displaystyle\|{\bf B}(\cdot)^{\rm T}{\bf u}\|_{1}\sup_{y\in[0,1],{\bf
x}\in\mathcal{X}}\\{y-E^{*}(Y|{\bf
x},{\bm{\beta}},{\bm{\gamma}})\\}^{2}f_{Y\mid{\bf X}}^{*}(y,{\bf
x},{\bm{\beta}},{\bm{\gamma}})$ $\displaystyle=$ $\displaystyle O(h^{1/2})$

by Condition (C1) and Lemma 1. Similar argument will show

$\displaystyle E^{*}[\\{Y-E^{*}(Y|{\bf
x},{\bm{\beta}},{\bm{\gamma}})\\}^{2}E^{*}\\{{\bf B}(Y)|{\bf
x},{\bm{\beta}},{\bm{\gamma}}\\}|{\bf x},{\bm{\beta}},{\bm{\gamma}}]^{\rm
T}{\bf u}=O(h^{1/2}),$

and

$\displaystyle E(\\{Y-E(Y|{\bf x})\\}^{2}[{\bf B}(Y)-E\\{{\bf B}(Y)|{\bf
x}\\}]|{\bf x})^{\rm T}{\bf u}=O(h^{1/2}).$ (S.48)

Then (S.47), Condition (C1), and Lemma 5 lead to

$\displaystyle\sup_{{\bf
x}\in\mathcal{X}}\left|\frac{\partial\hbox{var}^{*}(Y|{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}^{\rm T}}{\bf
u}-E(\\{Y-E(Y|{\bf x})\\}^{2}[{\bf B}(Y)-E\\{{\bf B}(Y)|{\bf x}\\}]|{\bf
x})^{\rm T}{\bf u}\right|$ $\displaystyle=$ $\displaystyle o_{p}(h^{1/2}).$

Thus, we have

$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}\frac{\partial\hbox{var}^{*}(Y|{\bf
x}_{i},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}}-E(\\{Y-E(Y|{\bf
X})\\}^{2}[{\bf B}(Y)-E\\{{\bf B}(Y)|{\bf X}\\}])\right\|_{2}=o_{p}(h^{1/2}).$

Using the fact that
$\|\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$ and
$\|\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\|_{2}=O_{p}\\{(nh)^{-1/2}\\}$,
we combine the above with (S.7) and (S.46) so get

$\displaystyle n^{-1}\sum_{i=1}^{n}\left[\hbox{var}^{*}\\{Y|{\bf
x}_{i},\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}-\hbox{var}(Y|{\bf
x}_{i})\right]$ $\displaystyle=$ $\displaystyle E[\\{Y-E(Y|{\bf X})\\}^{3}{\bf
X}]^{\rm T}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})$
$\displaystyle+E(\\{Y-E(Y|{\bf X})\\}^{2}[{\bf B}(Y)-E\\{{\bf B}(Y)|{\bf
X}\\}])^{\rm
T}\\{\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\\}+o_{p}(n^{-1/2}).$

Then, we can rewrite (S.43) using (S.44) as

$\displaystyle\widehat{\bm{\xi}}-{\bm{\xi}}_{0}$ $\displaystyle=$
$\displaystyle(E\\{\hbox{var}(Y|{\bf X})\\}{\bf
I}+{\bm{\beta}}_{0}E[\\{Y-E(Y|{\bf X})\\}^{3}{\bf X}^{\rm
T}])(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})$ (S.49)
$\displaystyle+{\bm{\beta}}_{0}E(\\{Y-E(Y|{\bf X})\\}^{2}[{\bf B}(Y)-E\\{{\bf
B}(Y)|{\bf X}\\}])^{\rm
T}\\{\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\\}$
$\displaystyle+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\left[\hbox{var}(Y|{\bf
x}_{i})-E\\{\hbox{var}(Y|{\bf X})\\}\right]+{\bf r},$

where $\|{\bf r}\|_{2}=o_{p}(n^{-1/2})$. Hence we get
$\|\widehat{\bm{\xi}}-{\bm{\xi}}_{0}\|_{2}=O_{p}(n^{-1/2})$ by (S.48).

Now we derive the asymptotic distribution of $\widehat{\bm{\xi}}$. Recall from
Theorem 1 that ${\bf A}=[{\bf A}_{1},{\bf A}_{2}]$ and

$\displaystyle{\bf A}_{1}$ $\displaystyle=$ $\displaystyle
E\\{\hbox{var}(Y|{\bf X})\\}{\bf I}+{\bm{\beta}}_{0}E[\\{Y-E(Y|{\bf
X})\\}^{3}{\bf X}^{\rm T}],$ $\displaystyle{\bf A}_{2}$ $\displaystyle=$
$\displaystyle{\bm{\beta}}_{0}E(\\{Y-E(Y|{\bf X})\\}^{2}[{\bf B}(Y)-E\\{{\bf
B}(Y)|{\bf X}\\}])^{\rm T}.$

Then by Lemma 7, (S.49) equals to

$\displaystyle\widehat{\bm{\xi}}-{\bm{\xi}}_{0}$ $\displaystyle=$
$\displaystyle{\bf A}{\bf\Sigma}^{-1}n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y|{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)|{\bf
x}_{i}\\}\end{bmatrix}$
$\displaystyle+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\left[\hbox{var}(Y|{\bf
x}_{i})-E\\{\hbox{var}(Y|{\bf X})\\}\right]$ $\displaystyle+{\bf A}_{1}{\bf
r}_{2}+{\bf A}_{2}{\bf r}_{3}+{\bf r}.$

We obviously have

$\displaystyle\hbox{cov}[{\bf X}\\{Y-E(Y|{\bf X})\\},\hbox{var}(Y|{\bf X})]$
$\displaystyle=$ $\displaystyle{\bf 0}_{p},$ $\displaystyle\hbox{cov}[{\bf
B}(Y)-E\\{{\bf B}(Y)|{\bf X}\\},\hbox{var}(Y|{\bf X})]$ $\displaystyle=$
$\displaystyle{\bf 0}_{m}.$

In addition, recall that $\|{\bf r}_{2}\|_{2}=o_{p}(n^{-1/2})$ and $\|{\bf
r}_{3}\|_{2}=o_{p}\\{(nh)^{-1/2}\\}$ by Lemma 7, and $\|{\bf
A}_{2}\|_{2}=O(h^{1/2})$ by (S.48), then the remainders satisfy

$\displaystyle\|{\bf A}_{1}{\bf r}_{2}+{\bf A}_{2}{\bf r}_{3}+{\bf r}\|_{2}$
$\displaystyle\leq$ $\displaystyle\|{\bf A}_{1}\|_{2}\|{\bf
r}_{2}\|_{2}+\|{\bf A}_{2}\|_{2}\|{\bf r}_{3}\|_{2}+\|{\bf r}\|_{2}$
$\displaystyle=$ $\displaystyle
O(1)o_{p}(n^{-1/2})+O(h^{1/2})o_{p}\\{(nh)^{-1/2}\\}+o_{p}(n^{-1/2})$
$\displaystyle=$ $\displaystyle o_{p}(n^{-1/2}).$

Hence, letting ${\bf\Sigma}_{\bm{\xi}}\equiv{\bf A}{\bf\Sigma}^{-1}{\bf
A}^{\rm T}+{\bm{\beta}}_{0}{\bm{\beta}}_{0}^{\rm
T}\hbox{var}\\{\hbox{var}(Y|{\bf X})\\}$,
${\bf\Sigma}_{\bm{\xi}}^{-1/2}\sqrt{n}(\widehat{\bm{\xi}}-{\bm{\xi}}_{0})$
converges to the normal distribution with mean ${\bf 0}$ and variance ${\bf
I}$. ∎

###  S.8 Proof of Theorem 2

For notational brevity, we denote $\nu\equiv{\bf x}^{\rm T}{\bm{\beta}}$,
$q^{*}(\nu,{\bm{\gamma}})$ be such that

$\displaystyle\int_{0}^{q^{*}(\nu,{\bm{\gamma}})}\exp\\{t\nu+{\bf B}(t)^{\rm
T}{\bm{\gamma}}\\}dt=\tau\int_{0}^{1}\exp\\{t\nu+{\bf B}(t)^{\rm
T}{\bm{\gamma}}\\}dt,$

and

$\displaystyle{q^{*}}^{\prime}(\nu,{\bm{\gamma}})$ $\displaystyle\equiv$
$\displaystyle\frac{\partial q^{*}(\nu,{\bm{\gamma}})}{\partial\nu}$
$\displaystyle=$ $\displaystyle\frac{\tau\int_{0}^{1}t\exp\\{t\nu+{\bf
B}(t)^{\rm T}{\bm{\gamma}}\\}dt-\int_{0}^{y}t\exp\\{t\nu+{\bf B}(t)^{\rm
T}{\bm{\gamma}}\\}dt}{\exp\\{y\nu+{\bf B}(y)^{\rm
T}{\bm{\gamma}}\\}}\bigg{|}_{y=q^{*}(\nu,{\bm{\gamma}})}$ $\displaystyle=$
$\displaystyle\frac{E^{*}([\tau-I\\{Y<q^{*}(\nu,{\bm{\gamma}})\\}]Y|{\bf
x},{\bm{\beta}},{\bm{\gamma}})}{f_{Y\mid{\bf
X}}^{*}\\{q^{*}(\nu,{\bm{\gamma}}),{\bf x},{\bm{\beta}},{\bm{\gamma}}\\}}.$

Let ${\bm{\beta}}^{*}$ and ${\bm{\gamma}}^{*}$ be such that
$\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}=o_{p}(1)$ and
$\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}=o_{p}(1)$ respectively. First
note that

$\displaystyle\sup_{{\bf x}\in\mathcal{X}}|q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-q({\bf x}^{\rm
T}{\bm{\beta}}_{0})|=o_{p}(1).$ (S.50)

This is because

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\int_{0}^{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}f_{Y\mid{\bf X}}^{*}(y,{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})dy-\int_{0}^{q({\bf x}^{\rm
T}{\bm{\beta}}_{0})}f_{Y\mid{\bf X}}(y|{\bf x})dy$ $\displaystyle=$
$\displaystyle\int_{0}^{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}\\{f_{Y\mid{\bf X}}^{*}(y,{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-f_{Y\mid{\bf X}}(y|{\bf
x})\\}dy+\int_{q({\bf x}^{\rm T}{\bm{\beta}}_{0})}^{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}f_{Y\mid{\bf X}}(y|{\bf x})dy,$

which implies, by Condition (C1) and Lemma 5, that

$\displaystyle c_{f}|q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-q({\bf x}^{\rm T}{\bm{\beta}}_{0})|$
$\displaystyle\leq$ $\displaystyle\left|\int_{0}^{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}\\{f_{Y\mid{\bf X}}^{*}(y,{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-f_{Y\mid{\bf X}}(y|{\bf
x})\\}dy\right|$ $\displaystyle=$ $\displaystyle o_{p}(1)$

uniformly in ${\bf x}$, where $c_{f}=\inf_{y\in[0,1],{\bf
x}\in\mathcal{X}}f_{Y\mid{\bf X}}(y|{\bf x})$. Condition (C1) and Lemma 5
further lead to

$\displaystyle\sup_{{\bf x}\in\mathcal{X}}|{q^{*}}^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-q^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}_{0})|=o_{p}(1),$ (S.51)

since uniformly in ${\bf x},y$,

$\displaystyle\left|f_{Y\mid{\bf X}}^{*}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}),{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}-f_{Y\mid{\bf X}}\\{q({\bf x}^{\rm
T}{\bm{\beta}}_{0})|{\bf x}\\}\right|$ (S.52) $\displaystyle\leq$
$\displaystyle\left|f_{Y\mid{\bf X}}^{*}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}),{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}-f_{Y\mid{\bf X}}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})|{\bf x}\\}\right|$
$\displaystyle+\left|f_{Y\mid{\bf X}}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})|{\bf x}\\}-f_{Y\mid{\bf X}}\\{q({\bf
x}^{\rm T}{\bm{\beta}}_{0})|{\bf x}\\}\right|$ $\displaystyle=$ $\displaystyle
o_{p}(1),$

$|E^{*}(Y|{\bf x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-E(Y|{\bf x})|=o_{p}(1)$,
and for an arbitrary $g(\cdot)$ such that $\|g(\cdot)\|_{2}<\infty$,

$\displaystyle\left|E^{*}\left[I\\{Y<q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}g(Y)|{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\right]-E\left[I\\{Y<q({\bm{\beta}}_{0}^{\rm
T}{\bf x})\\}g(Y)|{\bf x}\right]\right|$ (S.53) $\displaystyle\leq$
$\displaystyle\left|\int_{0}^{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}g(y)\\{f_{Y\mid{\bf X}}^{*}(y,{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-f_{Y\mid{\bf X}}(y|{\bf
x})\\}dy\right|+\left|\int_{q({\bf x}^{\rm T}{\bm{\beta}}_{0})}^{q^{*}({\bf
x}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}g(y)f_{Y\mid{\bf X}}(y|{\bf
x})dy\right|$ $\displaystyle\leq$ $\displaystyle
o_{p}(1)\|g(\cdot)\|_{2}+\left|\int_{q({\bf x}^{\rm
T}{\bm{\beta}}_{0})}^{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}\\{g(y)\\}^{2}dy\int_{q({\bf x}^{\rm
T}{\bm{\beta}}_{0})}^{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}\\{f_{Y\mid{\bf X}}(y|{\bf
x})\\}^{2}dy\right|^{1/2}$ $\displaystyle\leq$ $\displaystyle
o_{p}(1)\|g(\cdot)\|_{2}+C_{f}|q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-q({\bf x}^{\rm
T}{\bm{\beta}}_{0})|^{1/2}\|g(\cdot)\|_{2}$ $\displaystyle=$ $\displaystyle
o_{p}(1)\|g(\cdot)\|_{2}.$

Similarly, we also have

$\displaystyle\sup_{{\bf x}\in\mathcal{X}}|{q^{*}}^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}_{0},{\bm{\gamma}}_{0})-q^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}_{0})|=O(h^{q})$ (S.54)

by Condition (C1) and Lemma 4.

In addition, we can show that

$\displaystyle\frac{\partial{q^{*}}^{\prime}(\nu,{\bm{\gamma}})}{\partial{\bm{\beta}}}$
$\displaystyle=$ $\displaystyle\frac{\tau\int_{0}^{1}{\bf
x}t^{2}\exp\\{t\nu+{\bf B}(t)^{\rm
T}{\bm{\gamma}}\\}dt}{\exp[q^{*}(\nu,{\bm{\gamma}})\nu+{\bf
B}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}]}$
$\displaystyle-\frac{\int_{0}^{q^{*}(\nu,{\bm{\gamma}})}{\bf
x}t^{2}\exp\\{t\nu+{\bf B}(t)^{\rm
T}{\bm{\gamma}}\\}dt+q^{*}(\nu,{\bm{\gamma}})\exp[q^{*}(\nu,{\bm{\gamma}})\nu+{\bf
B}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}]{\bf
x}{q^{*}}^{\prime}(\nu,{\bm{\gamma}})}{\exp[q^{*}(\nu,{\bm{\gamma}})\nu+{\bf
B}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}]}$
$\displaystyle-\frac{\tau\int_{0}^{1}t\exp\\{t\nu+{\bf B}(t)^{\rm
T}{\bm{\gamma}}\\}dt-\int_{0}^{q^{*}(\nu,{\bm{\gamma}})}t\exp\\{t\nu+{\bf
B}(t)^{\rm T}{\bm{\gamma}}\\}dt}{\exp[q^{*}(\nu,{\bm{\gamma}})\nu+{\bf
B}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}]}$
$\displaystyle\times\left[{\bf
x}{q^{*}}^{\prime}(\nu,{\bm{\gamma}})\nu+q^{*}(\nu,{\bm{\gamma}}){\bf x}+{\bf
B}^{\prime}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}{\bf
x}{q^{*}}^{\prime}(\nu,{\bm{\gamma}})\right]$ $\displaystyle=$
$\displaystyle{\bf x}\Bigg{\\{}\frac{E^{*}([\tau-I\\{Y\leq
q^{*}(\nu,{\bm{\gamma}})\\}]Y^{2}|{\bf
x},{\bm{\beta}},{\bm{\gamma}})}{f_{Y\mid{\bf
X}}^{*}\\{q^{*}(\nu,{\bm{\gamma}}),{\bf x},{\bm{\beta}},{\bm{\gamma}}\\}}$
$\displaystyle-2{q^{*}}^{\prime}(\nu,{\bm{\gamma}})q^{*}(\nu,{\bm{\gamma}})-\\{{q^{*}}^{\prime}(\nu,{\bm{\gamma}})\\}^{2}[\nu+{\bf
B}^{\prime}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}]\Bigg{\\}},$
$\displaystyle\frac{\partial q^{*}(\nu,{\bm{\gamma}})}{\partial{\bm{\gamma}}}$
$\displaystyle=$ $\displaystyle\frac{\tau\int_{0}^{1}{\bf
B}(t)\exp\\{t\nu+{\bf B}(t)^{\rm T}{\bm{\gamma}}\\}dt-\int_{0}^{y}{\bf
B}(t)\exp\\{t\nu+{\bf B}(t)^{\rm T}{\bm{\gamma}}\\}dt}{\exp\\{y\nu+{\bf
B}(y)^{\rm T}{\bm{\gamma}}\\}}\bigg{|}_{y=q^{*}(\nu,{\bm{\gamma}})}$
$\displaystyle=$ $\displaystyle\frac{E^{*}([\tau-I\\{Y\leq
q^{*}(\nu,{\bm{\gamma}})\\}]{\bf B}(Y)|{\bf
x},{\bm{\beta}},{\bm{\gamma}})}{f_{Y\mid{\bf
X}}^{*}\\{q^{*}(\nu,{\bm{\gamma}}),{\bf x},{\bm{\beta}},{\bm{\gamma}}\\}},$
(S.56)

and

$\displaystyle\frac{\partial{q^{*}}^{\prime}(\nu,{\bm{\gamma}})}{\partial{\bm{\gamma}}}$
$\displaystyle=$ $\displaystyle\frac{\tau\int_{0}^{1}t{\bf
B}(t)\exp\\{t\nu+{\bf B}(t)^{\rm
T}{\bm{\gamma}}\\}dt}{\exp\\{q^{*}(\nu,{\bm{\gamma}})\nu+{\bf B}(y)^{\rm
T}{\bm{\gamma}}\\}}$
$\displaystyle-\frac{\int_{0}^{q^{*}(\nu,{\bm{\gamma}})}t{\bf
B}(t)\exp\\{t\nu+{\bf B}(t)^{\rm
T}{\bm{\gamma}}\\}dt+q^{*}(\nu,{\bm{\gamma}})\exp[q^{*}(\nu,{\bm{\gamma}})\nu+{\bf
B}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}]\frac{\partial
q^{*}(\nu,{\bm{\gamma}})}{\partial{\bm{\gamma}}}}{\exp[q^{*}(\nu,{\bm{\gamma}})\nu+{\bf
B}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}]}$
$\displaystyle-\frac{\tau\int_{0}^{1}t\exp\\{t\nu+{\bf B}(t)^{\rm
T}{\bm{\gamma}}\\}dt-\int_{0}^{q^{*}(\nu,{\bm{\gamma}})}t\exp\\{t\nu+{\bf
B}(t)^{\rm T}{\bm{\gamma}}\\}dt}{\exp[q^{*}(\nu,{\bm{\gamma}})\nu+{\bf
B}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}]}$
$\displaystyle\times\left[\frac{\partial
q^{*}(\nu,{\bm{\gamma}})}{\partial{\bm{\gamma}}}\nu+{\bf
B}^{\prime}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}\frac{\partial
q^{*}(\nu,{\bm{\gamma}})}{\partial{\bm{\gamma}}}+{\bf
B}\\{q^{*}(\nu,{\bm{\gamma}})\\}\right]$ $\displaystyle=$
$\displaystyle\frac{E^{*}([\tau-I\\{Y\leq q^{*}(\nu,{\bm{\gamma}})\\}]Y{\bf
B}(Y)|{\bf x},{\bm{\beta}},{\bm{\gamma}})}{f_{Y\mid{\bf
X}}^{*}\\{q^{*}(\nu,{\bm{\gamma}}),{\bf x},{\bm{\beta}},{\bm{\gamma}}\\}}$
$\displaystyle-{q^{*}}^{\prime}(\nu,{\bm{\gamma}}){\bf
B}\\{q^{*}(\nu,{\bm{\gamma}})\\}-\frac{\partial
q^{*}(\nu,{\bm{\gamma}})}{\partial{\bm{\gamma}}}[q^{*}(\nu,{\bm{\gamma}})+{q^{*}}^{\prime}(\nu,{\bm{\gamma}})\nu+{q^{*}}^{\prime}(\nu,{\bm{\gamma}}){\bf
B}^{\prime}\\{q^{*}(\nu,{\bm{\gamma}})\\}^{\rm T}{\bm{\gamma}}].$

Now using (S.54), we write $\widehat{{\bm{\eta}}_{\tau}}-{\bm{\eta}}_{\tau 0}$
as

$\displaystyle\widehat{{\bm{\eta}}_{\tau}}-{\bm{\eta}}_{\tau 0}$
$\displaystyle=$
$\displaystyle(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})n^{-1}\sum_{i=1}^{n}{q^{*}}^{\prime}\\{{\bf
x}_{i}^{\rm
T}\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\left[{q^{*}}^{\prime}\\{{\bf
x}_{i}^{\rm
T}\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}-{q^{*}}^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}_{0},{\bm{\gamma}}_{0})\right]$ (S.58)
$\displaystyle+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\left\\{{q^{*}}^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}_{0},{\bm{\gamma}}_{0})-q^{\prime}({\bf x}_{i}^{\rm
T}{\bm{\beta}}_{0})\right\\}+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}q^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}_{0})-E\\{q^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}_{0})\\}\right]$ $\displaystyle=$
$\displaystyle\left[n^{-1}\sum_{i=1}^{n}{q^{*}}^{\prime}\\{{\bf x}_{i}^{\rm
T}\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}{\bf
I}+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\frac{\partial{q^{*}}^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}^{\rm
T}}\right](\widehat{\bm{\beta}}-{\bm{\beta}}_{0})$
$\displaystyle+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\frac{\partial{q^{*}}^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}^{\rm
T}}\\{\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\\}$
$\displaystyle+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}q^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}_{0})-E\\{q^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}_{0})\\}\right]+O(h^{q}),$

where $({{\bm{\beta}}^{*}}^{\rm T},{{\bm{\gamma}}^{*}}^{\rm T})^{\rm T}$ is a
point on the line connecting $\\{\widehat{\bm{\beta}}^{\rm
T},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})^{\rm T}\\}^{\rm T}$ and
$({\bm{\beta}}_{0}^{\rm T},{\bm{\gamma}}_{0}^{\rm T})^{\rm T}$. To treat the
first term in (S.58), we first obtain

$\displaystyle\left|n^{-1}\sum_{i=1}^{n}{q^{*}}^{\prime}\\{{\bf x}_{i}^{\rm
T}\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}-E\\{q^{\prime}({\bf
X}^{\rm T}{\bm{\beta}}_{0})\\}\right|$ (S.59) $\displaystyle\leq$
$\displaystyle n^{-1}\sum_{i=1}^{n}\left|{q^{*}}^{\prime}\\{{\bf x}_{i}^{\rm
T}\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}-q^{\prime}({\bf
x}_{i}^{\rm
T}{\bm{\beta}}_{0})\right|+\left|n^{-1}\sum_{i=1}^{n}q^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}_{0})-E\\{q^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}_{0})\\}\right|$ $\displaystyle=$ $\displaystyle o_{p}(1)$

by (S.51). Now we will show

$\displaystyle\Bigg{\|}n^{-1}\sum_{i=1}^{n}\frac{\partial{q^{*}}^{\prime}({\bf
x}_{i}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}}-E\Bigg{[}{\bf
X}\Bigg{\\{}\frac{E([\tau-I\\{Y\leq q({\bf X}^{\rm
T}{\bm{\beta}}_{0})\\}]Y^{2}|{\bf X})}{f_{Y\mid{\bf X}}\\{q({\bf X}^{\rm
T}{\bm{\beta}}_{0})|{\bf X}\\}}$ (S.60) $\displaystyle-2q^{\prime}({\bf
X}^{\rm T}{\bm{\beta}}_{0})q({\bf X}^{\rm
T}{\bm{\beta}}_{0})-\\{q^{\prime}({\bf X}^{\rm T}{\bm{\beta}}_{0})\\}^{2}[{\bf
X}^{\rm T}{\bm{\beta}}_{0}+c^{\prime}\\{q({\bf X}^{\rm
T}{\bm{\beta}}_{0})\\}]\\}]\|_{2}$ $\displaystyle=$ $\displaystyle o_{p}(1).$

We prove (S.60) through proving

$\displaystyle\Bigg{\|}\frac{\partial{q^{*}}^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}}-{\bf
x}\Bigg{\\{}\frac{E([\tau-I\\{Y\leq q({\bf x}^{\rm
T}{\bm{\beta}}_{0})\\}]Y^{2}|{\bf x})}{f_{Y\mid{\bf X}}\\{q({\bf x}^{\rm
T}{\bm{\beta}}_{0})|{\bf x}\\}}$ (S.61) $\displaystyle-2q^{\prime}({\bf
x}^{\rm T}{\bm{\beta}}_{0})q({\bf x}^{\rm
T}{\bm{\beta}}_{0})-\\{q^{\prime}({\bf x}^{\rm T}{\bm{\beta}}_{0})\\}^{2}[{\bf
x}^{\rm T}{\bm{\beta}}_{0}+c^{\prime}\\{q({\bf x}^{\rm
T}{\bm{\beta}}_{0})\\}]\Bigg{\\}}\Bigg{\|}_{2}$ $\displaystyle=$
$\displaystyle o_{p}(1)$

uniformly in ${\bf x}$ under Condition (C1), where
$\partial{q^{*}}^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})/\partial{\bm{\beta}}$ is given in (S.8).
To prove (S.61), first note that

$\displaystyle\left|\frac{E^{*}([\tau-I\\{Y\leq q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}]Y^{2}|{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{f_{Y\mid{\bf X}}^{*}\\{q^{*}({\bf
x}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}),{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}}-\frac{E([\tau-I\\{Y\leq q({\bf
x}^{\rm T}{\bm{\beta}}_{0})\\}]Y^{2}|{\bf x})}{f_{Y\mid{\bf X}}\\{q({\bf
x}^{\rm T}{\bm{\beta}}_{0})|{\bf x}\\}}\right|$ (S.62) $\displaystyle=$
$\displaystyle o_{p}(1),$

because $\left|E^{*}(Y^{2}|{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-E(Y^{2}|{\bf x})\right|=o_{p}(1)$ by
Lemma 5,

$\displaystyle|E^{*}[I\\{Y\leq q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}Y^{2}|{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}]-E[I\\{Y\leq q({\bf x}^{\rm
T}{\bm{\beta}}_{0})\\}Y^{2}|{\bf x}]|=o_{p}(1)$

by (S.53), and

$\displaystyle\left|\frac{1}{f_{Y\mid{\bf X}}^{*}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}),{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}}-\frac{1}{f_{Y\mid{\bf X}}\\{q({\bf
x}^{\rm T}{\bm{\beta}}_{0})|{\bf x}\\}}\right|=o_{p}(1)$

by (S.52). Furthermore, we have

$\displaystyle|q{{}^{*}}^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})-q^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}_{0})q({\bf x}^{\rm T}{\bm{\beta}}_{0})|=o_{p}(1)$ (S.63)

by (S.50) and (S.51). We additionally get

$\displaystyle|{\bf B}^{\prime}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{\rm
T}{\bm{\gamma}}^{*}-c^{\prime}\\{q({\bf x}^{\rm T}{\bm{\beta}}_{0})\\}|$
(S.64) $\displaystyle\leq$ $\displaystyle\|{\bf B}^{\prime}\\{q^{*}({\bf
x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}\|_{2}\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}+|{\bf
B}^{\prime}\\{q^{*}({\bf x}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{\rm
T}{\bm{\gamma}}_{0}-c^{\prime}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}|$
$\displaystyle+|c^{\prime}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}-c^{\prime}\\{q({\bf x}^{\rm
T}{\bm{\beta}}_{0})\\}|$ $\displaystyle=$ $\displaystyle o_{p}(1),$

where, to bound the first term, we used
$\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}=O_{p}\\{(nh)^{-1/2}\\}$ by Lemma
6, for the second term, we used $|{\bf B}^{\prime}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{\rm
T}{\bm{\gamma}}_{0}-c^{\prime}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}|=O(h^{q/2})$ under Conditions (C5),
(C8) and (C9), and for the third term, we noted that $c^{\prime}(\cdot)$ is
continuous under Condition (C1) and used (S.50). The results in (S.64) and
(S.51), together with the fact
$\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$ directly lead to

$\displaystyle|{q^{*}}^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{2}[{\bf x}^{\rm
T}{\bm{\beta}}^{*}+{\bf B}^{\prime}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{\rm
T}{\bm{\gamma}}^{*}]-\\{q^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}_{0})\\}^{2}[{\bf x}^{\rm T}{\bm{\beta}}_{0}+c^{\prime}\\{q({\bf
x}^{\rm T}{\bm{\beta}}_{0})\\}]|$ (S.65) $\displaystyle=$ $\displaystyle
o_{p}(1).$

Combining the results in (S.62), (S.63), and (S.65), and taking into account
the form in (S.8) lead to (S.61), and subsequently (S.60). We now combine
(S.59) and (S.60) to get

$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}{q^{*}}^{\prime}\\{{\bf x}_{i}^{\rm
T}\widehat{\bm{\beta}},\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})\\}{\bf
I}+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\frac{\partial{q^{*}}^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}^{\rm
T}}-{\bf C}_{1}\right\|_{2}=o_{p}(1).$ (S.66)

Next, we handle the second term in (S.58). By (S.56) and ${\bf u}\in
U\equiv\\{{\bf u}\in\mathbb{R}^{m}:\|{\bf u}\|_{2}=1\\}$, we have

$\displaystyle\left|\left\\{\frac{\partial q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}}-\frac{E([\tau-I\\{Y\leq
q({\bf x}^{\rm T}{\bm{\beta}}_{0})\\}]{\bf B}(Y)|{\bf x})}{f_{Y\mid{\bf
X}}\\{q({\bf x}^{\rm T}{\bm{\beta}}_{0})|{\bf x}\\}}\right\\}^{\rm T}{\bf
u}\right|$ $\displaystyle\leq$ $\displaystyle\Bigg{|}\frac{1}{f_{Y\mid{\bf
X}}^{*}\\{q^{*}({\bf x}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}),{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}}\big{\\{}\tau[E^{*}\\{{\bf B}(Y)^{\rm
T}{\bf u}|{\bf x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}-E\\{{\bf B}(Y)^{\rm
T}{\bf u}|{\bf x}\\}]$ $\displaystyle-(E^{*}[I\\{Y\leq q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}{\bf B}(Y)^{\rm T}{\bf u}|{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}]-E[I\\{Y\leq q({\bf x}^{\rm
T}{\bm{\beta}}_{0})\\}{\bf B}(Y)^{\rm T}{\bf u}|{\bf x}])\big{\\}}\Bigg{|}$
$\displaystyle+\Bigg{|}\left[\frac{1}{f_{Y\mid{\bf X}}^{*}\\{q^{*}({\bf
x}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}),{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}}-\frac{1}{f_{Y\mid{\bf X}}\\{q({\bf
x}^{\rm T}{\bm{\beta}}_{0})|{\bf x}\\}}\right]$ $\displaystyle\times
E([\tau-I\\{Y\leq q({\bf x}^{\rm T}{\bm{\beta}}_{0})\\}]{\bf B}(Y)^{\rm T}{\bf
u}|{\bf x})\Bigg{|}$ $\displaystyle=$ $\displaystyle o_{p}(h^{1/2}).$

The last equality holds because $\left|E^{*}\\{{\bf B}(Y)^{\rm T}{\bf u}|{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}-E\\{{\bf B}(Y)^{\rm T}{\bf u}|{\bf
x}\\}\right|=o_{p}(h^{1/2})$ by Lemmas 1 and 5,

$\displaystyle\left|E^{*}[I\\{Y\leq q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}{\bf B}(Y)^{\rm T}{\bf u}|{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}]-E[I\\{Y\leq q({\bf x}^{\rm
T}{\bm{\beta}}_{0})\\}{\bf B}(Y)^{\rm T}{\bf u}|{\bf
x}]\right|=o_{p}(h^{1/2})$

by (S.53) and Lemma 1,

$\displaystyle\left|\frac{1}{f_{Y\mid{\bf X}}^{*}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}),{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}}-\frac{1}{f_{Y\mid{\bf X}}\\{q({\bf
x}^{\rm T}{\bm{\beta}}_{0})|{\bf x}\\}}\right|=o_{p}(1)$

by (S.52), and

$\displaystyle|E([\tau-I\\{Y\leq q({\bf x}^{\rm T}{\bm{\beta}}_{0})\\}]{\bf
B}(Y)^{\rm T}{\bf u}|{\bf x})|\leq E\\{|{\bf B}(Y)^{\rm T}{\bf u}|\mid{\bf
x}\\}=O(h^{1/2})$ (S.67)

by Lemma 2. Hence we get

$\displaystyle\left\|\frac{\partial q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}}-\frac{E([\tau-I\\{Y\leq
q({\bf x}^{\rm T}{\bm{\beta}}_{0})\\}]{\bf B}(Y)|{\bf x})}{f_{Y\mid{\bf
X}}\\{q({\bf x}^{\rm T}{\bm{\beta}}_{0})|{\bf
x}\\}}\right\|_{2}=o_{p}(h^{1/2})$ (S.68)

uniformly in ${\bf x}$ under Condition (C1). Similarly, we have

$\displaystyle|E([\tau-I\\{Y\leq q({\bf x}^{\rm T}{\bm{\beta}}_{0})\\}]Y{\bf
B}(Y)^{\rm T}{\bf u}|{\bf x})|=O(h^{1/2})$ (S.69)

and

$\displaystyle\Bigg{\|}\frac{E^{*}([\tau-I\\{Y\leq q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}]Y{\bf B}(Y)|{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{f_{Y\mid{\bf X}}^{*}\\{q^{*}({\bf
x}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}),{\bf
x},{\bm{\beta}}^{*},{\bm{\gamma}}^{*}\\}}$ (S.70)
$\displaystyle-\frac{E([\tau-I\\{Y\leq q({\bf x}^{\rm
T}{\bm{\beta}}_{0})\\}]Y{\bf B}(Y)|{\bf x})}{f_{Y\mid{\bf X}}\\{q({\bf x}^{\rm
T}{\bm{\beta}}_{0})|{\bf x}\\}}\Bigg{\|}_{2}$ $\displaystyle=$ $\displaystyle
o_{p}(h^{1/2})$

uniformly in ${\bf x}$. In addition, (S.50), (S.51) and Condition (C9) imply

$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}{q^{*}}^{\prime}({\bf x}_{i}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}){\bf B}\\{q^{*}({\bf x}_{i}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}-E\left[q^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}_{0}){\bf B}\\{q({\bf X}^{\rm
T}{\bm{\beta}}_{0})\\}\right]\right\|_{2}$ (S.71) $\displaystyle\leq$
$\displaystyle\sup_{{\bf u}\in
U}\left|n^{-1}\sum_{i=1}^{n}{q^{*}}^{\prime}({\bf x}_{i}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}){\bf B}\\{q^{*}({\bf x}_{i}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{\rm T}{\bf
u}-E\left[{q^{*}}^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}){\bf B}\\{q^{*}({\bf X}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{\rm T}{\bf u}\right]\right|$
$\displaystyle+\sup_{{\bf u}\in U}\left|E\left[{q^{*}}^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}){\bf B}\\{q^{*}({\bf X}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{\rm T}{\bf
u}\right]-E\left[q^{\prime}({\bf X}^{\rm T}{\bm{\beta}}_{0}){\bf B}\\{q({\bf
X}^{\rm T}{\bm{\beta}}_{0})\\}^{\rm T}{\bf u}\right]\right|$ $\displaystyle=$
$\displaystyle o_{p}(h^{1/2}).$

The last equality in (S.71) is because

$\displaystyle\left|E\left[q^{\prime}({\bf X}^{\rm T}{\bm{\beta}}_{0}){\bf
B}\\{q({\bf X}^{\rm T}{\bm{\beta}}_{0})\\}^{\rm T}{\bf u}\right]\right|$
(S.72) $\displaystyle=$ $\displaystyle\left|\int_{\\{\nu:\nu={\bf x}^{\rm
T}{\bm{\beta}}_{0},{\bf x}\in\mathcal{X}\\}}{\bf B}\\{q(\nu)\\}^{\rm T}{\bf
u}\left\\{\int_{\\{{\bf x}:{\bf x}^{\rm T}{\bm{\beta}}_{0}=\nu\\}}f_{{\bf
X}}({\bf x})d{\bf x}\right\\}q^{\prime}(\nu)d\nu\right|$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{1}{\bf B}(t)^{\rm T}{\bf u}\left\\{\int_{\\{{\bf
x}:q({\bf x}^{\rm T}{\bm{\beta}}_{0})=t\\}}f_{{\bf X}}({\bf x})d{\bf
x}\right\\}dt\right|$ $\displaystyle\leq$ $\displaystyle\|{\bf B}(\cdot)^{\rm
T}{\bf u}\|_{1}\sup_{{\bf x}\in\mathcal{X}}f_{{\bf X}}({\bf x})$
$\displaystyle=$ $\displaystyle O(h^{1/2}).$

by Lemma 1, and similarly $|E\left[{q^{*}}^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}){\bf B}\\{q^{*}({\bf X}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{\rm T}{\bf u}\right]|=O(h^{1/2})$,
which leads to the last step in (S.71) under Condition (C1). Furthermore, we
have

$\displaystyle|[q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})+{q^{*}}^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}){\bf x}^{\rm
T}{\bm{\beta}}^{*}+{q^{*}}^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*}){\bf B}^{\prime}\\{q^{*}({\bf x}^{\rm
T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})\\}^{\rm T}{\bm{\gamma}}^{*}]$ (S.73)
$\displaystyle-[q({\bf x}^{\rm T}{\bm{\beta}}_{0})+q^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}_{0}){\bf x}^{\rm T}{\bm{\beta}}_{0}+q^{\prime}({\bf x}^{\rm
T}{\bm{\beta}}_{0})c^{\prime}\\{q({\bf x}^{\rm T}{\bm{\beta}}_{0})\\}]|$
$\displaystyle=$ $\displaystyle o_{p}(1)$

by (S.50), (S.51), (S.64), and
$\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$. Now, combining
(S.67), (S.69) and (S.72), we obtain

$\displaystyle\|{\bf C}_{2}\|_{2}=O(h^{1/2}).$ (S.74)

Combining (S.8), (S.68), (S.70), (S.71) and (S.73), we further get

$\displaystyle\left\|{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\frac{\partial{q^{*}}^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}^{\rm
T}}-{\bf C}_{2}\right\|_{2}=o_{p}(h^{1/2}).$ (S.75)

Inserting (S.66) and (S.75) in (S.58), using Lemma 7, we get

$\displaystyle\widehat{{\bm{\eta}}_{\tau}}-{\bm{\eta}}_{\tau 0}$
$\displaystyle=$ $\displaystyle{\bf
C}_{1}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})+{\bf
C}_{2}\\{\widehat{\bm{\gamma}}(\widehat{\bm{\beta}})-{\bm{\gamma}}_{0}\\}$
$\displaystyle+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}q^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}_{0})-E\\{q^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}_{0})\\}\right]+{\bf r}$ $\displaystyle=$ $\displaystyle{\bf
C}{\bf\Sigma}^{-1}n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y|{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)|{\bf
x}_{i}\\}\end{bmatrix}$
$\displaystyle+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}q^{\prime}({\bf
x}_{i}^{\rm T}{\bm{\beta}}_{0})-E\\{q^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}_{0})\\}\right]+{\bf C}_{1}{\bf r}_{2}+{\bf C}_{2}{\bf
r}_{3}+{\bf r},$

where $\|{\bf r}\|_{2}=o_{p}(n^{-1/2})$ by Proposition 2 and Lemma 6, hence

$\displaystyle\|{\bf C}_{1}{\bf r}_{2}+{\bf C}_{2}{\bf r}_{3}+{\bf
r}\|_{2}=o_{p}(n^{-1/2})$

by Lemma 7 and (S.74). In addition, we obviously have

$\displaystyle\hbox{cov}[{\bf X}\\{Y-E(Y|{\bf X})\\},q^{\prime}({\bf X}^{\rm
T}{\bm{\beta}}_{0})]$ $\displaystyle=$ $\displaystyle{\bf 0}_{p},$
$\displaystyle\hbox{cov}[{\bf B}(Y)-E\\{{\bf B}(Y)|{\bf X}\\},q^{\prime}({\bf
X}^{\rm T}{\bm{\beta}}_{0})]$ $\displaystyle=$ $\displaystyle{\bf 0}_{m}.$

Thus with ${\bf\Sigma}_{{\bm{\eta}}_{\tau}}={\bf C}{\bf\Sigma}^{-1}{\bf
C}^{\rm T}+{\bm{\beta}}_{0}{\bm{\beta}}_{0}^{\rm
T}\hbox{var}\\{q^{\prime}({\bf X}^{\rm T}{\bm{\beta}}_{0})\\}$ defined in
Theorem 2,
${\bf\Sigma}_{{\bm{\eta}}_{\tau}}^{-1/2}\sqrt{n}(\widehat{\bm{\eta}}_{\tau}-{\bm{\eta}}_{\tau
0})$ converges to the normal distribution with mean ${\bf 0}$ and variance
${\bf I}$. ∎

###  S.9 Proof of Proposition 3

In terms of estimating ${\bm{\beta}}$, the score function is ${\bf
S}_{\bm{\beta}}=y{\bf x}-E(Y{\bf x}\mid{\bf x})$ and the nuisance tangent
space is

$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle[{\bf a}(y)-E\\{{\bf
a}(Y)\mid{\bf x}\\}+{\bf b}({\bf x}):\forall{\bf a}(y),{\bf b}({\bf
x})\in{\cal R}^{p},E\\{{\bf b}({\bf X})\\}={\bf 0}].$

Its orthogonal complement is

$\displaystyle\Lambda^{\perp}=({\bf a}(y,{\bf x})-E\\{{\bf a}(Y,{\bf
x})\mid{\bf x}\\}:E\\{{\bf a}(y,{\bf X})\mid y\\}=E[E\\{{\bf a}(Y,{\bf
X})\mid{\bf X}\\}\mid y]).$

Thus, the efficient score is ${\bf S}_{\rm eff}=y{\bf x}-{\bf
a}_{0}(y)-E\\{Y{\bf x}-{\bf a}_{0}(Y)\mid{\bf x}\\}$, where ${\bf a}_{0}(y)$
satisfies

$\displaystyle{\bf a}_{0}(y)-E[E\\{{\bf a}_{0}(Y)\mid{\bf X}\\}\mid y]=E(y{\bf
X}\mid y)-E\\{E(Y{\bf X}\mid{\bf X})\mid y\\}.$ (S.76)

Thus, the efficient variance is $\\{E({\bf S}_{\rm eff}^{\otimes 2})\\}^{-1}$.

To show that the MLE estimator $\widehat{\bm{\beta}}$ is actually efficient,
we only need to show ${\bf\Sigma}^{*}\to E({\bf S}_{\rm eff}^{\otimes 2})$
when $n\to\infty$. Let ${\bf a}_{0}(y)=\Lambda{\bf B}(y)+O(h^{q})$, where
$\Lambda$ is a $p\times m$ coefficient matrix. Then (S.76) implies that

$\displaystyle\Lambda E\\{{\bf B}(Y)^{\otimes 2}\\}-\Lambda E[E\\{{\bf
B}(Y)\mid{\bf X}\\}^{\otimes 2}]$ $\displaystyle=$ $\displaystyle E[E\\{Y{\bf
X}-E(Y{\bf X}\mid{\bf X}){\bf B}(Y)^{\rm T}\mid{\bf X}\\}]+O(h^{q})E\\{{\bf
B}^{\rm T}(Y)\\}$ $\displaystyle=$ $\displaystyle E[{\bf X}\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf X}\\}]+O(h^{q})E\\{{\bf B}^{\rm T}(Y)\\},$

i.e.

$\displaystyle{\bf a}_{0}(y)$ $\displaystyle=$ $\displaystyle\Lambda{\bf
B}(y)={\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}{\bf B}(y)+O(h^{q})E\\{{\bf B}^{\rm
T}(Y)\\}{\bf\Sigma}_{22}^{-1}{\bf B}(y)$ $\displaystyle=$
$\displaystyle{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}{\bf B}(y)+O(h^{q-1/2}),$

where we used $\|E\\{{\bf
B}(Y)\\}\|_{2}=O(h^{1/2}),\|{\bf\Sigma}_{22}^{-1}\|_{2}\asymp h^{-1}$, and
$\|{\bf B}(\cdot)\|_{2}=O(1)$. Therefore

$\displaystyle E({\bf S}_{\rm eff}^{\otimes 2})-{\bf\Sigma}^{*}$
$\displaystyle=$ $\displaystyle E\left\\{(Y{\bf X}-E(Y{\bf X}\mid{\bf
X})-{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}[{\bf B}(y)-E\\{B(Y)\mid{\bf
X}])^{\otimes 2}\right\\}-{\bf\Sigma}^{*}+O(h^{q-1/2})$ $\displaystyle=$
$\displaystyle o(1).$

∎

###  S.10 Proof of Theorem 3

Since

$\displaystyle E(\bm{\phi}_{\rm eff}^{\otimes 2})$ $\displaystyle=$
$\displaystyle E\left([{\bm{\beta}}v({\bm{\beta}}^{\rm T}{\bf
X})-{\bm{\beta}}E\\{v({\bm{\beta}}^{\rm T}{\bf X})\\}]^{\otimes 2}\right)$
$\displaystyle+E\left([{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf
X}Y-E\\{{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf X}Y\mid{\bf X}\\}]^{\otimes
2}\right),$

and

$\displaystyle{\bf\Sigma}_{\bm{\xi}}={\bf A}{\bf\Sigma}^{-1}{\bf A}^{\rm
T}+{\bm{\beta}}^{\otimes 2}\hbox{var}\\{v({\bm{\beta}}^{\rm T}{\bf X})\\},$

so we only need to show

$\displaystyle E\left([{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf
X}Y-E\\{{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf X}Y\mid{\bf X}\\}]^{\otimes
2}\right)-{\bf A}{\bf\Sigma}^{-1}{\bf A}^{\rm T}\to{\bf 0}.$

Now we have ${\bm{\beta}}y^{2}+{\bf a}(y)={\bf\Lambda}{\bf B}(y)+O(h^{q})$,
where ${\bf\Lambda}\in{\cal R}^{p\times m}$. Then (S.5) and (S.6) imply

$\displaystyle-{\bf\Lambda}E[{\bf B}(y)-E\\{{\bf B}(Y)\mid{\bf X}\\}\mid
y]+{\bm{\beta}}E\\{y^{2}-E(Y^{2}\mid{\bf X})\mid y\\}+O(h^{q})$
$\displaystyle=$ $\displaystyle 2{\bm{\beta}}E[yE(Y\mid{\bf
X})-E\\{YE(Y\mid{\bf X})\mid{\bf X}\\}\mid y]+{\bf M}E[{\bf X}\\{y-E(Y\mid{\bf
X})\\}\mid y]$ $\displaystyle=$ $\displaystyle 2{\bm{\beta}}E[yE(Y\mid{\bf
X})-E\\{YE(Y\mid{\bf X})\mid{\bf X}\\}\mid y]$
$\displaystyle+\left(E\\{v({\bm{\beta}}^{\rm T}{\bf X})\\}{\bf
I}-E\left[2{\bm{\beta}}{\bf X}^{\rm T}Yv({\bm{\beta}}^{\rm T}{\bf
X})+\\{{\bf\Lambda}{\bf B}(Y)-{\bm{\beta}}Y^{2}\\}\\{Y-E(Y\mid{\bf X})\\}{\bf
X}^{\rm T}\right]\right)$ $\displaystyle\times{\bf\Sigma}_{11}^{-1}E[{\bf
X}\\{y-E(Y\mid{\bf X})\\}\mid y].$

Multiplying ${\bf B}^{\rm T}(y)$ on both sides and taking expectation lead to

$\displaystyle-{\bf\Lambda}{\bf\Sigma}_{22}+{\bm{\beta}}E[\\{Y^{2}-E(Y^{2}\mid{\bf
X})\\}{\bf B}^{\rm T}(Y)]+O(h^{q})E\\{{\bf B}^{\rm T}(Y)\\}$ $\displaystyle=$
$\displaystyle 2{\bm{\beta}}E[YE(Y\mid{\bf X}){\bf B}^{\rm
T}(Y)-\\{E(Y\mid{\bf X})\\}^{2}{\bf B}^{\rm T}(Y)]$
$\displaystyle+\left(E\\{v({\bm{\beta}}^{\rm T}{\bf X})\\}{\bf
I}-E\left[2{\bm{\beta}}{\bf X}^{\rm T}Yv({\bm{\beta}}^{\rm T}{\bf
X})+\\{{\bf\Lambda}{\bf
B}(Y)-{\bm{\beta}}Y^{2}\\}\\{Y-E(Y\mid{\bm{\beta}}^{\rm T}{\bf X})\\}{\bf
X}^{\rm T}\right]\right)$ $\displaystyle\times{\bf\Sigma}_{11}^{-1}E[{\bf
X}\\{Y-E(Y\mid{\bf X})\\}{\bf B}^{\rm T}(Y)]$ $\displaystyle=$ $\displaystyle
2{\bm{\beta}}E[Y\hbox{cov}\\{Y,{\bf B}(Y)\mid{\bf
X}\\}]+E\\{v({\bm{\beta}}^{\rm T}{\bf
X})\\}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-E\left\\{2{\bm{\beta}}{\bf X}^{\rm
T}Yv({\bm{\beta}}^{\rm T}{\bf
X})\right\\}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}$
$\displaystyle-{\bf\Lambda}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}+{\bm{\beta}}E[Y^{2}\\{Y-E(Y\mid{\bm{\beta}}^{\rm
T}{\bf X})\\}{\bf X}^{\rm T}]{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}$
$\displaystyle=$ $\displaystyle 2{\bm{\beta}}E[Y\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf X}\\}]+{\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf\Lambda}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12},$

hence

$\displaystyle-{\bf\Lambda}{\bf\Sigma}_{22}+{\bf\Lambda}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}$
$\displaystyle=$ $\displaystyle 2{\bm{\beta}}E[Y\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf X}\\}]+{\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bm{\beta}}E[\\{Y^{2}-E(Y^{2}\mid{\bf
X})\\}{\bf B}^{\rm T}(Y)]$ $\displaystyle+O(h^{q})E\\{{\bf B}^{\rm T}(Y)\\}$
$\displaystyle=$ $\displaystyle{\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf A}_{2}+O(h^{q})E\\{{\bf
B}^{\rm T}(Y)\\},$

and

$\displaystyle{\bm{\beta}}y^{2}+{\bf a}(y)$ $\displaystyle=$
$\displaystyle({\bf A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf
A}_{2})({\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf\Sigma}_{22})^{-1}{\bf
B}(y)$ $\displaystyle+O(h^{q})E\\{{\bf B}^{\rm
T}(Y)\\}({\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf\Sigma}_{22})^{-1}{\bf
B}(y)$ $\displaystyle=$ $\displaystyle{\bf U}{\bf B}(y)+O(h^{q-1/2}),$

where, for notational brevity,

$\displaystyle{\bf U}$ $\displaystyle\equiv$ $\displaystyle({\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf
A}_{2})({\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf\Sigma}_{22})^{-1}.$

The above holds by
$\|({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}\|_{2}=O(h^{-1})$.
This is because by Conditions (C6) and (C7), all eigenvalues of ${\bf\Sigma}$
are of order either 1 or $h$, which implies ${\bf\Sigma}^{-1}$ has all
eigenvalues either of order 1 or $h^{-1}$. Since
$({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}$
is a block diagonal element of ${\bf\Sigma}^{-1}$, its eigenvalues are of
order 1 or $h^{-1}$ as well. Then

$\displaystyle{\bf M}$ $\displaystyle=$
$\displaystyle\left(E\\{v({\bm{\beta}}^{\rm T}{\bf X})\\}{\bf
I}-E\\{2{\bm{\beta}}{\bf X}^{\rm T}Yv({\bm{\beta}}^{\rm T}{\bf X})\\}-{\bf
U}E\left[{\bf B}(Y)\\{Y-E(Y\mid{\bm{\beta}}^{\rm T}{\bf X})\\}{\bf X}^{\rm
T}\right]\right.$
$\displaystyle\left.+{\bm{\beta}}E\left[Y^{2}\\{Y-E(Y\mid{\bm{\beta}}^{\rm
T}{\bf X})\\}{\bf X}^{\rm T}\right]\right){\bf\Sigma}_{11}^{-1}+O(h^{q-1/2})$
$\displaystyle=$ $\displaystyle\left(E\\{v({\bm{\beta}}^{\rm T}{\bf X})\\}{\bf
I}-E\\{2{\bm{\beta}}{\bf X}^{\rm T}Yv({\bm{\beta}}^{\rm T}{\bf X})\\}-{\bf
U}{\bf\Sigma}_{21}+{\bm{\beta}}E\left[Y^{2}\\{Y-E(Y\mid{\bf X})\\}{\bf X}^{\rm
T}\right]\right)$ $\displaystyle\times{\bf\Sigma}_{11}^{-1}+O(h^{q-1/2})$
$\displaystyle=$ $\displaystyle({\bf A}_{1}-{\bf
U}{\bf\Sigma}_{21}){\bf\Sigma}_{11}^{-1}+O(h^{q-1/2}).$

Hence

$\displaystyle{\bm{\beta}}y^{2}+{\bf a}(y)-E\\{{\bm{\beta}}Y^{2}+{\bf
a}(Y)\mid{\bf x}\\}={\bf U}[{\bf B}(y)-E\\{{\bf B}(Y)\mid{\bf
x}\\}]+O(h^{q-1/2}),$

and

$\displaystyle E\\{\hbox{var}({\bf M}{\bf X}Y\mid{\bf X})\\}$ $\displaystyle=$
$\displaystyle{\bf M}{\bf\Sigma}_{11}{\bf M}^{\rm T},$ $\displaystyle
E[\hbox{var}\\{{\bm{\beta}}Y^{2}+{\bf a}(Y)\mid{\bf X}\\}]$ $\displaystyle=$
$\displaystyle{\bf U}{\bf\Sigma}_{22}{\bf U}^{\rm T}+O(h^{q-1/2}),$
$\displaystyle E[\hbox{cov}\\{{\bf M}{\bf X}Y,{\bm{\beta}}Y^{2}+{\bf
a}(Y)\mid{\bf X}\\}]$ $\displaystyle=$ $\displaystyle{\bf
M}{\bf\Sigma}_{12}{\bf U}^{\rm T}+O(h^{q-1/2}).$

Thus, noting that ${\bf A}=[{\bf A}_{1},{\bf A}_{2}]$,

$\displaystyle E\left([{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf
X}Y-E\\{{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf X}Y\mid{\bf X}\\}]^{\otimes
2}\right)$ $\displaystyle=$ $\displaystyle{\bf M}{\bf\Sigma}_{11}{\bf M}^{\rm
T}+{\bf U}{\bf\Sigma}_{22}{\bf U}^{\rm T}+{\bf M}{\bf\Sigma}_{12}{\bf U}^{\rm
T}+{\bf U}{\bf\Sigma}_{21}{\bf M}^{\rm T}+O(h^{q-1/2})$ $\displaystyle=$
$\displaystyle({\bf A}_{1}-{\bf U}{\bf\Sigma}_{21}){\bf\Sigma}_{11}^{-1}({\bf
A}_{1}^{\rm T}-{\bf\Sigma}_{12}{\bf U}^{\rm T})+{\bf U}{\bf\Sigma}_{22}{\bf
U}^{\rm T}+({\bf A}_{1}-{\bf
U}{\bf\Sigma}_{21}){\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}{\bf U}^{\rm T}$
$\displaystyle+{\bf U}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}({\bf A}_{1}^{\rm
T}-{\bf\Sigma}_{12}{\bf U}^{\rm T})+O(h^{q-1/2})$ $\displaystyle=$
$\displaystyle{\bf A}_{1}{\bf\Sigma}_{11}^{-1}{\bf A}_{1}^{\rm T}+{\bf
U}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}){\bf
U}^{\rm T}+O(h^{q-1/2})$ $\displaystyle=$ $\displaystyle{\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf A}_{1}^{\rm T}+({\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf
A}_{2})({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}({\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf
A}_{1}^{\rm T}-{\bf A}_{2}^{\rm T})$ $\displaystyle+O(h^{q-1/2})$
$\displaystyle=$ $\displaystyle{\bf
A}\left(\begin{array}[]{cc}{\bf\Sigma}_{11}^{-1}+{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}&-{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}\\\
-({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}&({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}\end{array}\right){\bf
A}^{\rm T}$ $\displaystyle+O(h^{q-1/2})$ $\displaystyle=$ $\displaystyle{\bf
A}{\bf\Sigma}^{-1}{\bf A}^{\rm T}+o(1).$

∎

###  S.11 Proof of Theorem 4

By (S.12)

$\displaystyle E(\bm{\phi}_{\rm eff}^{\otimes 2})$ $\displaystyle=$
$\displaystyle
E\left([{\bm{\beta}}q^{\prime}(\nu)-{\bm{\beta}}E\\{q^{\prime}(\nu)\\}]^{\otimes
2}\right)$ $\displaystyle+E\left([{\bf M}_{1}{\bf X}Y+{\bf a}(Y)-E\\{{\bf
M}_{1}{\bf X}Y+{\bf a}(Y)\mid{\bf X}\\}]^{\otimes 2}\right),$

and

$\displaystyle{\bf\Sigma}_{{\bm{\eta}}_{\tau}}={\bf C}{\bf\Sigma}^{-1}{\bf
C}^{\rm T}+{\bm{\beta}}^{\otimes 2}\hbox{var}\\{q^{\prime}(\nu)\\}$

in Theorem 2, we only need to show

$\displaystyle E\left([{\bf M}_{1}{\bf X}Y+{\bf a}(Y)-E\\{{\bf M}_{1}{\bf
X}Y+{\bf a}(Y)\mid{\bf X}\\}]^{\otimes 2}\right)-{\bf C}{\bf\Sigma}^{-1}{\bf
C}^{\rm T}\to{\bf 0}.$

Now we have ${\bf a}(y)={\bf\Lambda}{\bf B}(y)+O(h^{q})$ where
${\bf\Lambda}\in{\cal R}^{p\times m}$. Then (S.11) and (S.13) imply

$\displaystyle-{\bf\Lambda}E[{\bf B}(y)-E\\{{\bf B}(Y)\mid{\bf X}\\}\mid
y]+O(h^{q})$ $\displaystyle=$ $\displaystyle-{\bm{\beta}}E\\{r(y,\nu)\mid
y\\}+{\bf M}_{1}E[{\bf X}\\{y-E(Y\mid{\bf X})\\}\mid y]$ $\displaystyle=$
$\displaystyle-{\bm{\beta}}E\\{r(y,\nu)\mid y\\}$
$\displaystyle+\left(E\\{q^{\prime}(\nu)\\}{\bf I}+{\bm{\beta}}E\\{{\bf
X}^{\rm T}q^{\prime\prime}(\nu)\\}-E\left[\\{{\bf\Lambda}{\bf
B}(Y)+O(h^{q})\\}{\bf X}^{\rm T}\\{Y-E(Y\mid{\bf X})\\}\right]\right)$
$\displaystyle\times{\bf\Sigma}_{11}^{-1}E[{\bf X}\\{y-E(Y\mid{\bf X})\\}\mid
y].$

Multiplying ${\bf B}^{\rm T}(y)$ on both sides above and taking expectation,
incorporating (S.7), we get

$\displaystyle-{\bf\Lambda}{\bf\Sigma}_{22}+O(h^{q})E\\{{\bf B}^{\rm T}(Y)\\}$
$\displaystyle=$ $\displaystyle-{\bm{\beta}}E\\{r(Y,\nu){\bf B}^{\rm T}(Y)\\}$
$\displaystyle+\left(E\\{q^{\prime}(\nu)\\}{\bf I}+{\bm{\beta}}E\\{{\bf
X}^{\rm T}q^{\prime\prime}(\nu)\\}-E\left[\\{{\bf\Lambda}{\bf
B}(Y)+O(h^{q})\\}{\bf X}^{\rm T}\\{Y-E(Y\mid{\bf X})\\}\right]\right)$
$\displaystyle\times{\bf\Sigma}_{11}^{-1}E[{\bf X}\\{Y-E(Y\mid{\bf X})\\}{\bf
B}^{\rm T}(Y)]$ $\displaystyle=$
$\displaystyle-{\bm{\beta}}E\Bigg{(}\frac{E\\{\epsilon Y{\bf B}^{\rm
T}(Y)|{\bf X}\\}}{f\\{q(\nu),\nu\\}}-q^{\prime}(\nu){\bf B}^{\rm
T}\\{q(\nu)\\}-\frac{E\\{\epsilon{\bf B}^{\rm T}(Y)|{\bf
X}\\}}{f\\{q(\nu),\nu\\}}[q(\nu)+q^{\prime}(\nu)\nu+q^{\prime}(\nu)c^{\prime}\\{q(\nu)\\}]\Bigg{)}$
$\displaystyle+{\bf
C}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf\Lambda}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}+O(h^{q}){\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}$
$\displaystyle=$
$\displaystyle-{\bf\Lambda}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}+{\bf
C}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf
C}_{2}+O(h^{q}){\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12},$

hence

$\displaystyle{\bf\Lambda}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})=-{\bf
C}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}+{\bf C}_{2}+O(h^{q+1/2}),$

since $\|E\\{{\bf B}^{\rm T}(Y)\\}\|_{2}=O(h^{1/2})$ and
$\|{\bf\Sigma}_{12}\|_{2}=O(h^{1/2})$ by Lemma 2. Then

$\displaystyle{\bf a}(y)={\bf U}{\bf B}(y)+O(h^{q-1/2}),$

where for notational brevity,

$\displaystyle{\bf U}\equiv(-{\bf
C}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}+{\bf
C}_{2})({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}.$

The above holds by
$\|({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}\|_{2}=O(h^{-1})$
under Conditions (C6) and (C7). Then by (S.13),

$\displaystyle{\bf M}_{1}$ $\displaystyle=$
$\displaystyle(E\\{q^{\prime}(\nu)\\}{\bf I}+{\bm{\beta}}E\\{{\bf X}^{\rm
T}q^{\prime\prime}(\nu)\\}-{\bf U}E\left[{\bf B}(Y){\bf X}^{\rm
T}\\{Y-E(Y\mid{\bf X})\\}\right]){\bf\Sigma}_{11}^{-1}+O(h^{q-1/2})$
$\displaystyle=$ $\displaystyle({\bf C}_{1}-{\bf
U}{\bf\Sigma}_{21}){\bf\Sigma}_{11}^{-1}+O(h^{q-1/2}).$

Hence

$\displaystyle{\bf a}(y)-E\\{{\bf a}(Y)\mid{\bf x}\\}={\bf U}[{\bf
B}(y)-E\\{{\bf B}(Y)\mid{\bf x}\\}]+O(h^{q-1/2}),$

and

$\displaystyle E\\{\hbox{var}({\bf M}_{1}{\bf X}Y\mid{\bf X})\\}$
$\displaystyle=$ $\displaystyle{\bf M}_{1}{\bf\Sigma}_{11}{\bf M}_{1}^{\rm
T},$ $\displaystyle E[\hbox{cov}\\{{\bf M}{\bf X}Y,{\bf a}(Y)\mid{\bf X}\\}]$
$\displaystyle=$ $\displaystyle{\bf M}_{1}{\bf\Sigma}_{12}{\bf U}^{\rm
T}+O(h^{q-1/2}),$ $\displaystyle E[\hbox{var}\\{{\bf a}(Y)\mid{\bf X}\\}]$
$\displaystyle=$ $\displaystyle{\bf U}{\bf\Sigma}_{22}{\bf U}^{\rm
T}+O(h^{q-1/2}).$

Thus noting that ${\bf C}=[{\bf C}_{1},{\bf C}_{2}]$, we obtain

$\displaystyle E\left([{\bf M}_{1}{\bf X}Y+{\bf a}(Y)-E\\{{\bf M}_{1}{\bf
X}Y+{\bf a}(Y)\mid{\bf X}\\}]^{\otimes 2}\right)$ $\displaystyle=$
$\displaystyle{\bf M}_{1}{\bf\Sigma}_{11}{\bf M}_{1}^{\rm T}+{\bf
M}_{1}{\bf\Sigma}_{12}{\bf U}^{\rm T}+{\bf U}{\bf\Sigma}_{21}{\bf M}_{1}^{\rm
T}+{\bf U}{\bf\Sigma}_{22}{\bf U}^{\rm T}+O(h^{q-1/2})$ $\displaystyle=$
$\displaystyle({\bf C}_{1}-{\bf U}{\bf\Sigma}_{21}){\bf\Sigma}_{11}^{-1}({\bf
C}_{1}^{\rm T}-{\bf\Sigma}_{12}{\bf U}^{\rm T})+({\bf C}_{1}-{\bf
U}{\bf\Sigma}_{21}){\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}{\bf U}^{\rm T}$
$\displaystyle+{\bf U}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}({\bf C}_{1}^{\rm
T}-{\bf\Sigma}_{12}{\bf U}^{\rm T})+{\bf U}{\bf\Sigma}_{22}{\bf U}^{\rm
T}+O(h^{q-1/2})$ $\displaystyle=$ $\displaystyle{\bf
C}_{1}{\bf\Sigma}_{11}^{-1}{\bf C}_{1}^{\rm T}+{\bf
U}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}){\bf
U}^{\rm T}+O(h^{q-1/2})$ $\displaystyle=$ $\displaystyle{\bf
C}_{1}{\bf\Sigma}_{11}^{-1}{\bf C}_{1}^{\rm T}+(-{\bf
C}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}+{\bf
C}_{2})({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}(-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf
C}_{1}^{\rm T}+{\bf C}_{2}^{\rm T})$ $\displaystyle+O(h^{q-1/2})$
$\displaystyle=$ $\displaystyle{\bf
C}\left(\begin{array}[]{cc}{\bf\Sigma}_{11}^{-1}+{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}&-{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}\\\
-({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}&({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}\end{array}\right){\bf
C}^{\rm T}$ $\displaystyle=$ $\displaystyle{\bf C}{\bf\Sigma}^{-1}{\bf C}^{\rm
T}+o(1).$

∎

###  S.12 Lemmas

We now introduce a lemma for the analysis of the discrete response case.

######  Lemma 8.

Let ${\bm{\theta}}^{*}\equiv({\bm{\beta}}^{*\rm T},{\bm{\gamma}}^{*\rm
T})^{\rm T}$ and $\|{\bm{\theta}}^{*}-{\bm{\theta}}_{0}\|_{2}=o(1)$. Under
Conditions (D1)-(D2), uniformly with respect to ${\bf x}$,

  1. (i)

$\|{\bf p}({\bf x},{\bm{\theta}}^{*})-{\bf p}({\bf
x},{\bm{\theta}}_{0})\|_{2}=O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}+\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2})$,

  2. (ii)

$\|{\bf p}_{1}({\bf x},{\bm{\theta}}^{*})-{\bf p}_{1}({\bf
x},{\bm{\theta}}_{0})\|_{2}=O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}+\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2})$,

  3. (iii)

$\|{\bf p}_{2}({\bf x},{\bm{\theta}}^{*})-{\bf p}_{2}({\bf
x},{\bm{\theta}}_{0})\|_{2}=O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}+\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2})$,

  4. (iv)

$\|\hbox{var}\\{{\bf B}(Y)\mid{\bf x},{\bm{\theta}}^{*}\\}-\hbox{var}\\{{\bf
B}(Y)\mid{\bf
x},{\bm{\theta}}_{0}\\}\|_{2}=O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}+\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2})$.

If ${\bm{\theta}}^{*}$ satisfies
$\|{\bm{\theta}}^{*}-{\bm{\theta}}_{0}\|_{2}=o_{p}(1)$, the above results hold
in probability.

###### Proof.

For some $\widetilde{\bm{\theta}}$ on the line connecting ${\bm{\theta}}^{*}$
and ${\bm{\theta}}_{0}$, we have

$\displaystyle{\bf p}({\bf x},{\bm{\theta}}^{*})-{\bf p}({\bf
x},{\bm{\theta}}_{0})$ $\displaystyle=$ $\displaystyle\frac{\partial{\bf
p}({\bf x},\widetilde{\bm{\theta}})}{\partial{\bm{\beta}}^{\rm
T}}({\bm{\beta}}^{*}-{\bm{\beta}}_{0})+\frac{\partial{\bf p}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\gamma}}^{\rm
T}}({\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}).$

First, by Conditions (D1) and (D2),

$\displaystyle\left\|\frac{\partial{\bf p}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\beta}}^{\rm
T}}({\bm{\beta}}^{*}-{\bm{\beta}}_{0})\right\|_{2}$ $\displaystyle=$
$\displaystyle\left\|\left\\{{\bf p}_{1}({\bf
x},\widetilde{\bm{\theta}})-E(Y\mid{\bf x},\widetilde{\bm{\theta}}){\bf
p}({\bf x},\widetilde{\bm{\theta}})\right\\}{\bf x}^{\rm
T}({\bm{\beta}}^{*}-{\bm{\beta}}_{0})\right\|_{2}$ $\displaystyle\leq$
$\displaystyle\left[\sum_{y=1}^{m}\left\\{y-E(Y\mid{\bf
x},\widetilde{\bm{\theta}})\right\\}^{2}\hbox{pr}(Y=y\mid{\bf
x},\widetilde{\bm{\theta}})^{2}\right]^{1/2}O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2})$
$\displaystyle\leq$ $\displaystyle\sum_{y=1}^{m}\left|y-E(Y\mid{\bf
x},\widetilde{\bm{\theta}})\right|\hbox{pr}(Y=y\mid{\bf
x},\widetilde{\bm{\theta}})O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2})$
$\displaystyle\leq$ $\displaystyle 2E(Y\mid{\bf
x},\widetilde{\bm{\theta}})O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2})$
$\displaystyle\leq$ $\displaystyle 2\\{E(Y\mid{\bf
x},{\bm{\theta}}_{0})+O(\|\widetilde{\bm{\theta}}-{\bm{\theta}}_{0}\|_{2})\\}O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2})$
$\displaystyle=$ $\displaystyle O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}).$

The last inequality holds since $E(Y\mid{\bf x},{\bm{\theta}}_{0})$ is bounded
by Condition (D1), and
$\|\widetilde{\bm{\theta}}-{\bm{\theta}}_{0}\|_{2}=o(1)$ by the definition of
$\widetilde{\bm{\theta}}$. In addition, for a vector ${\bf a}$, let
$\hbox{diag}({\bf a})$ be the diagonal matrix with entries equal to the
elements of ${\bf a}$, then

$\displaystyle\left\|\frac{\partial{\bf p}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\gamma}}^{\rm
T}}({\bm{\gamma}}^{*}-{\bm{\gamma}}_{0})\right\|_{2}$ $\displaystyle=$
$\displaystyle\left\|\left[\hbox{diag}\left\\{{\bf p}({\bf
x},\widetilde{\bm{\theta}})\right\\}-{\bf p}({\bf
x},\widetilde{\bm{\theta}}){\bf p}^{\rm T}({\bf
x},\widetilde{\bm{\theta}})\right]({\bm{\gamma}}^{*}-{\bm{\gamma}}_{0})\right\|_{2}$
$\displaystyle\leq$ $\displaystyle\left[\|{\bf p}({\bf
x},\widetilde{\bm{\theta}})\|_{\infty}+\|{\bf p}({\bf
x},\widetilde{\bm{\theta}})\|_{2}^{2}\right]\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}$
$\displaystyle=$ $\displaystyle
O(\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2})$

because $\|{\bf p}({\bf x},\widetilde{\bm{\theta}})\|_{\infty}\leq\|{\bf
p}({\bf x},\widetilde{\bm{\theta}})\|_{2}\leq 1$. Hence, we get

$\displaystyle\|{\bf p}({\bf x},{\bm{\theta}}^{*})-{\bf p}({\bf
x},{\bm{\theta}}_{0})\|_{2}$ $\displaystyle=$ $\displaystyle
O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}+\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}).$

Similarly for some $\widetilde{\bm{\theta}}$ on the line connecting
${\bm{\theta}}^{*}$ and ${\bm{\theta}}_{0}$,

$\displaystyle\left\|\frac{\partial{\bf p}_{1}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\beta}}^{\rm
T}}({\bm{\beta}}^{*}-{\bm{\beta}}_{0})\right\|_{2}$ $\displaystyle=$
$\displaystyle\left\|\left\\{{\bf p}_{2}({\bf
x},\widetilde{\bm{\theta}})-E(Y\mid{\bf x},\widetilde{\bm{\theta}}){\bf
p}_{1}({\bf x},\widetilde{\bm{\theta}})\right\\}{\bf x}^{\rm
T}({\bm{\beta}}^{*}-{\bm{\beta}}_{0})\right\|_{2}$ $\displaystyle\leq$
$\displaystyle\sum_{y=1}^{m}\left|y-E(Y\mid{\bf
x},\widetilde{\bm{\theta}})\right|y\hbox{pr}(Y=y\mid{\bf
x},\widetilde{\bm{\theta}})O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2})$
$\displaystyle\leq$ $\displaystyle\left[E(Y^{2}\mid{\bf
x},{\bm{\theta}}_{0})+\\{E(Y\mid{\bf
x},{\bm{\theta}}_{0})\\}^{2}+O(\|\widetilde{\bm{\theta}}-{\bm{\theta}}_{0}\|_{2})\right]O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2})$
$\displaystyle=$ $\displaystyle O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2})$

by Conditions (D1) and (D2). In addition,

$\displaystyle\left\|\frac{\partial{\bf p}_{1}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\gamma}}^{\rm
T}}({\bm{\gamma}}^{*}-{\bm{\gamma}}_{0})\right\|_{2}$ $\displaystyle=$
$\displaystyle\left\|\left[\hbox{diag}\left\\{{\bf p}_{1}({\bf
x},\widetilde{\bm{\theta}})\right\\}-{\bf p}_{1}({\bf
x},\widetilde{\bm{\theta}}){\bf p}^{\rm T}({\bf
x},\widetilde{\bm{\theta}})\right]({\bm{\gamma}}^{*}-{\bm{\gamma}}_{0})\right\|_{2}$
$\displaystyle\leq$ $\displaystyle\left[\left\|\hbox{diag}\left\\{{\bf
p}_{1}({\bf x},\widetilde{\bm{\theta}})\right\\}\right\|_{2}+\left\|{\bf
p}_{1}({\bf x},\widetilde{\bm{\theta}}){\bf p}^{\rm T}({\bf
x},\widetilde{\bm{\theta}})\right\|_{2}\right]\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}$
$\displaystyle\leq$ $\displaystyle\left[\|{\bf p}_{1}({\bf
x},\widetilde{\theta})\|_{\infty}+\|{\bf p}_{1}({\bf
x},\widetilde{\bm{\theta}})\|_{2}\|{\bf p}({\bf
x},\widetilde{\bm{\theta}})\|_{2}\right]\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}$
$\displaystyle\leq$ $\displaystyle
O(\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}),$

because $\|{\bf p}_{1}({\bf x},\widetilde{\theta})\|_{\infty}\leq\|{\bf
p}_{1}({\bf x},\widetilde{\theta})\|_{2}\leq\|{\bf p}_{1}({\bf
x},\widetilde{\theta})\|_{1}=E(Y\mid{\bf x},\widetilde{\theta})=E(Y\mid{\bf
x})+o(1)$ and $E(Y\mid{\bf x})$ is bounded by Condition (D1). Therefore,

$\displaystyle\|{\bf p}_{1}({\bf x},{\bm{\theta}}^{*})-{\bf p}_{1}({\bf
x},{\bm{\theta}}_{0})\|_{2}$ $\displaystyle=$
$\displaystyle\left\|\frac{\partial{\bf p}_{1}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\beta}}^{\rm
T}}({\bm{\beta}}^{*}-{\bm{\beta}}_{0})+\frac{\partial{\bf p}_{1}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\gamma}}^{\rm
T}}({\bm{\gamma}}^{*}-{\bm{\gamma}}_{0})\right\|_{2}$ $\displaystyle=$
$\displaystyle
O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}+\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}).$

Also, by Conditions (D1) and (D2),

$\displaystyle\left\|\frac{\partial{\bf p}_{2}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\beta}}^{\rm
T}}({\bm{\beta}}^{*}-{\bm{\beta}}_{0})\right\|_{2}$ $\displaystyle=$
$\displaystyle\left\|\left\\{{\bf p}_{3}({\bf
x},\widetilde{\bm{\theta}})-E(Y\mid{\bf x},\widetilde{\bm{\theta}}){\bf
p}_{2}({\bf x},\widetilde{\bm{\theta}})\right\\}{\bf x}^{\rm
T}({\bm{\beta}}^{*}-{\bm{\beta}}_{0})\right\|_{2}$ $\displaystyle\leq$
$\displaystyle\sum_{y=1}^{m}\left|y-E(Y\mid{\bf
x},\widetilde{\bm{\theta}})\right|y^{2}\hbox{pr}(Y=y\mid{\bf
x},\widetilde{\bm{\theta}})O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2})$
$\displaystyle\leq$ $\displaystyle\left[E(Y^{3}\mid{\bf x})+E(Y\mid{\bf
x})E(Y^{2}\mid{\bf
x})+O(\|\widetilde{\bm{\theta}}-{\bm{\theta}}_{0}\|_{2})\right]O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2})$
$\displaystyle=$ $\displaystyle O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}),$

and

$\displaystyle\left\|\frac{\partial{\bf p}_{2}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\gamma}}^{\rm
T}}({\bm{\gamma}}^{*}-{\bm{\gamma}}_{0})\right\|_{2}$ $\displaystyle=$
$\displaystyle\left\|\left[\hbox{diag}\left\\{{\bf p}_{2}({\bf
x},\widetilde{\bm{\theta}})\right\\}-{\bf p}_{2}({\bf
x},\widetilde{\bm{\theta}}){\bf p}^{\rm T}({\bf
x},\widetilde{\bm{\theta}})\right]({\bm{\gamma}}^{*}-{\bm{\gamma}}_{0})\right\|_{2}$
$\displaystyle\leq$ $\displaystyle\left[\left\|\hbox{diag}\left\\{{\bf
p}_{2}({\bf x},\widetilde{\bm{\theta}})\right\\}\right\|_{2}+\left\|{\bf
p}_{2}({\bf x},\widetilde{\bm{\theta}}){\bf p}^{\rm T}({\bf
x},\widetilde{\bm{\theta}})\right\|_{2}\right]\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}$
$\displaystyle\leq$ $\displaystyle\left[\|{\bf p}_{2}({\bf
x},\widetilde{\theta})\|_{\infty}+\|{\bf p}_{2}({\bf
x},\widetilde{\bm{\theta}})\|_{2}\|{\bf p}({\bf
x},\widetilde{\bm{\theta}})\|_{2}\right]\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}$
$\displaystyle\leq$ $\displaystyle
O(\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}).$

The last inequality holds because $\|{\bf p}_{2}({\bf
x},\widetilde{\theta})\|_{\infty}\leq\|{\bf p}_{2}({\bf
x},\widetilde{\theta})\|_{2}\leq\|{\bf p}_{2}({\bf
x},\widetilde{\theta})\|_{1}=E(Y^{2}\mid{\bf
x},\widetilde{\theta})=E(Y^{2}\mid{\bf x})+o(1)$ and $E(Y^{2}\mid{\bf x})$ is
bounded by Condition (D1). Then we get

$\displaystyle\|{\bf p}_{2}({\bf x},{\bm{\theta}}^{*})-{\bf p}_{2}({\bf
x},{\bm{\theta}}_{0})\|_{2}$ $\displaystyle=$
$\displaystyle\left\|\frac{\partial{\bf p}_{2}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\beta}}^{\rm
T}}({\bm{\beta}}^{*}-{\bm{\beta}}_{0})+\frac{\partial{\bf p}_{2}({\bf
x},\widetilde{\bm{\theta}})}{\partial{\bm{\gamma}}^{\rm
T}}({\bm{\gamma}}^{*}-{\bm{\gamma}}_{0})\right\|_{2}$ $\displaystyle=$
$\displaystyle
O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}+\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}).$

Now,

$\displaystyle\left\|\hbox{var}\\{{\bf B}(Y)\mid{\bf
x},{\bm{\theta}}^{*}\\}-\hbox{var}\\{{\bf B}(Y)\mid{\bf
x},{\bm{\theta}}_{0}\\}\right\|_{2}$ $\displaystyle\leq$
$\displaystyle\left\|\hbox{diag}\left\\{{\bf p}({\bf
x},{\bm{\theta}}^{*})-{\bf p}({\bf
x},{\bm{\theta}}_{0})\right\\}\right\|_{2}+\left\|{\bf p}({\bf
x},{\bm{\theta}}^{*}){\bf p}^{\rm T}({\bf x},{\bm{\theta}}^{*})-{\bf p}({\bf
x},{\bm{\theta}}_{0}){\bf p}^{\rm T}({\bf x},{\bm{\theta}}_{0})\right\|_{2}$
$\displaystyle=$ $\displaystyle\|{\bf p}({\bf x},{\bm{\theta}}^{*})-{\bf
p}({\bf x},{\bm{\theta}}_{0})\|_{\infty}$ $\displaystyle+\left\|{\bf p}({\bf
x},{\bm{\theta}}^{*})\\{{\bf p}({\bf x},{\bm{\theta}}^{*})-{\bf p}({\bf
x},{\bm{\theta}}_{0})\\}^{\rm T}+\\{{\bf p}({\bf x},{\bm{\theta}}^{*})-{\bf
p}({\bf x},{\bm{\theta}}_{0})\\}{\bf p}^{\rm T}({\bf
x},{\bm{\theta}}_{0})\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\|{\bf
p}({\bf x},{\bm{\theta}}^{*})-{\bf p}({\bf x},{\bm{\theta}}_{0})\|_{2}$
$\displaystyle+\left|\\{{\bf p}({\bf x},{\bm{\theta}}^{*})-{\bf p}({\bf
x},{\bm{\theta}}_{0})\\}^{\rm T}{\bf p}({\bf
x},{\bm{\theta}}^{*})\right|+\left|{\bf p}^{\rm T}({\bf
x},{\bm{\theta}}_{0})\\{{\bf p}({\bf x},{\bm{\theta}}^{*})-{\bf p}({\bf
x},{\bm{\theta}}_{0})\\}\right|$ $\displaystyle\leq$ $\displaystyle
3\left\|{\bf p}({\bf x},{\bm{\theta}}^{*})-{\bf p}({\bf
x},{\bm{\theta}}_{0})\right\|_{2}$ $\displaystyle=$ $\displaystyle
O(\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}+\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}).$

Since the support of ${\bf x}$ is compact by Condition (D1), the about results
hold uniformly with respect to ${\bf x}$. ∎

###  S.13 Proof of Proposition 4

First, we will show that
$\|\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$ and
$\|\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0}\|_{2}=O_{p}(n^{-1/2}m^{1/2})$. Note
that the Hessian of $l({\bm{\beta}},{\bm{\gamma}})$, i.e.

$\displaystyle\begin{bmatrix}\partial^{2}l({\bm{\beta}},{\bm{\gamma}})/\partial{\bm{\beta}}\partial{\bm{\beta}}^{\rm
T}&\partial^{2}l({\bm{\beta}},{\bm{\gamma}})/\partial{\bm{\beta}}\partial{\bm{\gamma}}^{\rm
T}\\\
\partial^{2}l({\bm{\beta}},{\bm{\gamma}})/\partial{\bm{\gamma}}\partial{\bm{\beta}}^{\rm
T}&\partial^{2}l({\bm{\beta}},{\bm{\gamma}})/\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}\end{bmatrix}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\begin{bmatrix}{\bf x}_{i}{\bf x}_{i}^{\rm
T}\hbox{var}(Y\mid{\bf x}_{i},{\bm{\theta}})&{\bf x}_{i}\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf x}_{i},{\bm{\theta}}\\}\\\ [{\bf x}_{i}\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf x}_{i},{\bm{\theta}}\\}]^{\rm T}&\hbox{var}\\{{\bf B}(Y)\mid{\bf
x}_{i},{\bm{\theta}}\\}\end{bmatrix}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\hbox{cov}\left\\{\begin{bmatrix}{\bf x}_{i}Y\\\
{\bf B}(Y)\end{bmatrix},\begin{bmatrix}{\bf x}_{i}Y\\\ {\bf
B}(Y)\end{bmatrix}\mid{\bf x}_{i},{\bm{\theta}}\right\\}$

is negative definite for any ${\bm{\theta}}$, which implies that a local
maximizer is the global maximizer. Hence similarly to the proof of Proposition
1, it suffices to show for any $\epsilon>0$ there exists constants
$C_{\bm{\beta}},C_{\bm{\gamma}}>0$ such that

$\displaystyle\hbox{pr}\left\\{l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})>\sup_{\|{\bf
v}_{\bm{\beta}}\|_{2}=C_{\bm{\beta}},\|{\bf
v}_{\bm{\gamma}}\|_{2}=C_{\bm{\gamma}}}l({\bm{\beta}}_{0}+n^{-1/2}{\bf
v}_{\bm{\beta}},{\bm{\gamma}}_{0}+n^{-1/2}m^{1/2}{\bf
v}_{\bm{\gamma}})\right\\}\geq 1-5\epsilon$ (S.79)

for a sufficiently large $n$. Now, by the Taylor expansion,

$\displaystyle l({\bm{\beta}}_{0}+n^{-1/2}{\bf
v}_{\bm{\beta}},{\bm{\gamma}}_{0}+n^{-1/2}m^{1/2}{\bf
v}_{\bm{\gamma}})-l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})$ (S.80)
$\displaystyle=$ $\displaystyle n^{-1/2}\frac{\partial
l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})}{\partial{\bm{\beta}}^{\rm T}}{\bf
v}_{\bm{\beta}}+n^{-1/2}m^{1/2}\frac{\partial
l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})}{\partial{\bm{\gamma}}^{\rm T}}{\bf
v}_{\bm{\gamma}}$ $\displaystyle+\frac{1}{2}n^{-1}{\bf v}_{\bm{\beta}}^{\rm
T}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}\partial{\bm{\beta}}^{\rm
T}}{\bf v}_{\bm{\beta}}+\frac{1}{2}n^{-1}m{\bf v}_{\bm{\gamma}}^{\rm
T}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}}{\bf v}_{\bm{\gamma}}+n^{-1}m^{1/2}{\bf v}_{\bm{\beta}}^{\rm
T}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}\partial{\bm{\gamma}}^{\rm
T}}{\bf v}_{\bm{\gamma}}$

where ${\bm{\beta}}^{*}={\bm{\beta}}_{0}+\alpha_{1}n^{-1/2}{\bf
v}_{\bm{\beta}}$ and
${\bm{\gamma}}^{*}={\bm{\gamma}}_{0}+\alpha_{2}n^{-1/2}m^{1/2}{\bf
v}_{\bm{\gamma}}$ for some ${\bf v}_{\bm{\beta}},{\bf v}_{\bm{\gamma}}$ such
that $\|{\bf v}_{\bm{\beta}}\|_{2}=C_{\bm{\beta}},\|{\bf
v}_{\bm{\gamma}}\|_{2}=C_{\bm{\gamma}}$ and some
$\alpha_{1}\in(0,1),\alpha_{2}\in(0,1)$. We first have

$\displaystyle\left\|\frac{\partial
l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})}{\partial{\bm{\beta}}}\right\|_{2}=\left\|\sum_{i=1}^{n}{\bf
x}_{i}\\{y_{i}-E(Y\mid{\bf x}_{i})\\}\right\|_{2}\asymp_{p}n^{1/2}$

by Conditions (D1) and (D5). Therefore, for any $\epsilon>0$ there exists a
constant $0<C_{1}<\infty$ such that

$\displaystyle\hbox{pr}\left\\{\left\|\frac{\partial
l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})}{\partial{\bm{\beta}}}\right\|_{2}\leq
C_{1}n^{1/2}\right\\}\geq 1-\epsilon.$ (S.81)

Also,

$\displaystyle\left\|\frac{\partial
l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})}{\partial{\bm{\gamma}}}\right\|_{2}=\left\|\sum_{i=1}^{n}\\{{\bf
B}(y_{i})-E\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}]\right\|_{2}\asymp_{p}n^{1/2}m^{-1/2},$

because

$\displaystyle\left\|E\left\\{\left(\sum_{i=1}^{n}\left[{\bf
B}(Y_{i})-E\\{{\bf B}(Y)\mid{\bf X}_{i}\\}\right]\right)^{\otimes
2}\right\\}\right\|_{2}$ $\displaystyle=$
$\displaystyle\left\|nE\left(\left[{\bf B}(Y)-E\\{{\bf B}(Y)\mid{\bf
X}\\}\right]^{\otimes 2}\right)\right\|_{2}$ $\displaystyle=$ $\displaystyle
n\left\|E[\hbox{var}\\{{\bf B}(Y)\mid{\bf X}\\}]\right\|_{2}$
$\displaystyle\asymp$ $\displaystyle nm^{-1}$

by Remark 1. Hence, for any $\epsilon>0$ there exists a constant
$0<C_{2}<\infty$ such that

$\displaystyle\hbox{pr}\left\\{\left\|\frac{\partial
l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})}{\partial{\bm{\gamma}}}\right\|_{2}\leq
C_{2}n^{1/2}m^{-1/2}\right\\}\geq 1-\epsilon.$ (S.82)

In addition, noting that
$\|{\bm{\theta}}^{*}-{\bm{\theta}}_{0}\|_{2}=O(n^{-1/2}m^{1/2})=o(1)$ by
Condition (D3), we get

$\displaystyle\left\|-n^{-1}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}\partial{\bm{\beta}}^{\rm
T}}-{\bf\Sigma}_{11}\right\|_{2}$ $\displaystyle\leq$
$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}{\bf x}_{i}{\bf x}_{i}^{\rm
T}\\{\hbox{var}(Y\mid{\bf x}_{i},{\bm{\theta}}^{*})-\hbox{var}(Y\mid{\bf
x}_{i})\\}\right\|_{2}$ (S.83) $\displaystyle+\left\|n^{-1}\sum_{i=1}^{n}{\bf
x}_{i}{\bf x}_{i}^{\rm T}\hbox{var}(Y\mid{\bf x}_{i})-E\\{{\bf X}{\bf X}^{\rm
T}\hbox{var}(Y\mid{\bf X})\\}\right\|_{2}$ $\displaystyle=$ $\displaystyle
o_{p}(1)$

by Conditions (D1) and (D2). Since all eigenvalues of ${\bf\Sigma}_{11}$ are
of constant order by Conditions (D1) and (D5), the above implies there exists
a constant $0<C_{3}<\infty$ such that

$\displaystyle\hbox{pr}\left\\{{\bf v}_{\bm{\beta}}^{\rm
T}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}\partial{\bm{\beta}}^{\rm
T}}{\bf v}_{\bm{\beta}}\leq-C_{3}C_{\bm{\beta}}^{2}n\right\\}\geq 1-\epsilon.$
(S.84)

Further, since $\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}=O(n^{-1/2})$ and
$\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}=O(n^{-1/2}m^{1/2})$,

$\displaystyle\left\|-n^{-1}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}}-{\bf\Sigma}_{22}\right\|_{2}$ $\displaystyle\leq$
$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}[\hbox{var}\\{{\bf B}(Y)\mid{\bf
x}_{i},{\bm{\theta}}^{*}\\}-\hbox{var}\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}]\right\|_{2}$ (S.85)
$\displaystyle+\left\|n^{-1}\sum_{i=1}^{n}\hbox{var}\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}-E[\hbox{var}\\{{\bf B}(Y)\mid{\bf X}\\}]\right\|_{2}$
$\displaystyle=$ $\displaystyle O(n^{-1/2}m^{1/2})+O_{p}(n^{-1/2}m^{-1})$
$\displaystyle=$ $\displaystyle O_{p}(n^{-1/2}m^{1/2})$ $\displaystyle=$
$\displaystyle o_{p}(m^{-1})$

by Lemma 8, Remark 1, and $n^{-1/2}m^{1/2}=o(m^{-1})$ by Condition (D3).
Together with Remark 1, this implies

$\displaystyle\hbox{pr}\left\\{{\bf v}_{\bm{\gamma}}^{\rm
T}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}}{\bf v}_{\bm{\gamma}}\leq-C_{4}C_{\bm{\gamma}}^{2}nm^{-1}\right\\}\geq
1-\epsilon$ (S.86)

for some constant $0<C_{4}<\infty$. Now, we have
$\|{\bf\Sigma}_{12}\|_{2}=O(m^{-1/2})$, because noting that ${\bf\Sigma}_{11}$
and ${\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}$
are positive definite by Condition (D5), for any ${\bf u}\in\mathbb{R}^{m}$
such that $\|{\bf u}\|_{2}=1$ we have

$\displaystyle c\|{\bf\Sigma}_{12}{\bf
u}\|_{2}^{2}\leq\|{\bf\Sigma}_{11}^{-1/2}{\bf\Sigma}_{12}{\bf
u}\|_{2}^{2}={\bf u}^{\rm
T}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}{\bf u}<{\bf u}^{\rm
T}{\bf\Sigma}_{22}{\bf u}\leq\|{\bf\Sigma}_{22}\|_{2}\asymp m^{-1}$ (S.87)

by Remark 1 for some constant $c>0$. Also, since
$\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}=O(n^{-1/2})$ and
$\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}=O(n^{-1/2}m^{1/2})$,

$\displaystyle\left\|-n^{-1}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}\partial{\bm{\gamma}}^{\rm
T}}-{\bf\Sigma}_{12}\right\|_{2}$ $\displaystyle\leq$
$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}{\bf x}_{i}[\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf x}_{i},{\bm{\theta}}^{*}\\}-\hbox{cov}\\{Y,{\bf B}(Y)\mid{\bf
x}_{i}\\}]\right\|_{2}$ (S.88) $\displaystyle+\left\|n^{-1}\sum_{i=1}^{n}{\bf
x}_{i}\hbox{cov}\\{Y,{\bf B}(Y)\mid{\bf x}_{i}\\}-E[{\bf X}\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf X}\\}]\right\|_{2}$ $\displaystyle\leq$
$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}{\bf x}_{i}\\{{\bf p}_{1}^{\rm
T}({\bf x}_{i},{\bm{\theta}}^{*})-{\bf p}_{1}^{\rm T}({\bf
x}_{i},{\bm{\theta}}_{0})\\}\right\|_{2}$
$\displaystyle+\left\|n^{-1}\sum_{i=1}^{n}{\bf x}_{i}\\{E(Y\mid{\bf
x}_{i},{\bm{\theta}}^{*}){\bf p}^{\rm T}({\bf
x}_{i},{\bm{\theta}}^{*})-E(Y\mid{\bf x}_{i},{\bm{\theta}}_{0}){\bf p}^{\rm
T}({\bf x}_{i},{\bm{\theta}}_{0})\\}\right\|_{2}$
$\displaystyle+\left\|n^{-1}\sum_{i=1}^{n}{\bf x}_{i}\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf x}_{i}\\}-E[{\bf X}\hbox{cov}\\{Y,{\bf B}(Y)\mid{\bf
X}\\}]\right\|_{2}$ $\displaystyle=$ $\displaystyle
O(n^{-1/2}m^{1/2})+O_{p}(n^{-1/2}m^{-1/2})$ $\displaystyle=$ $\displaystyle
o_{p}(m^{-1/2})$

by Lemma 8, Condition (D1), and $n^{-1/2}m^{1/2}=o(m^{-1/2})$ by Condition
(D3). Thus we have

$\displaystyle\hbox{pr}\left\\{{\bf v}_{\bm{\beta}}^{\rm
T}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}\partial{\bm{\gamma}}^{\rm
T}}{\bf v}_{\bm{\gamma}}\leq
C_{5}C_{\bm{\beta}}C_{\bm{\gamma}}nm^{-1/2}\right\\}\geq 1-\epsilon$ (S.89)

for some constant $0<C_{5}<\infty$. Combining (S.80), (S.81), (S.82), (S.84),
(S.86), and (S.89), with probability at least $1-5\epsilon$,

$\displaystyle l({\bm{\beta}}_{0}+n^{-1/2}{\bf
v}_{\bm{\beta}},{\bm{\gamma}}_{0}+n^{-1/2}m^{1/2}{\bf
v}_{\bm{\gamma}})-l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})$ $\displaystyle\leq$
$\displaystyle
C_{1}C_{\bm{\beta}}+C_{2}C_{\bm{\gamma}}-\frac{C_{3}}{2}C_{\bm{\beta}}^{2}-\frac{C_{4}}{2}C_{\bm{\gamma}}^{2}+C_{5}C_{\bm{\beta}}C_{\bm{\gamma}}$
$\displaystyle=$ $\displaystyle
C_{\bm{\beta}}\left(C_{1}-\frac{C_{3}}{2}C_{\bm{\beta}}+C_{5}C_{\bm{\gamma}}\right)+C_{\bm{\gamma}}\left(C_{2}-\frac{C_{4}}{2}C_{\bm{\gamma}}\right)$
$\displaystyle<$ $\displaystyle 0$

when $C_{\bm{\beta}}>2(C_{1}+C_{5}C_{\bm{\gamma}})/C_{3}$ and
$C_{\bm{\gamma}}>2C_{2}/C_{4}$, and this proves (S.79). Therefore,
$\|\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$ and
$\|\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0}\|_{2}=O_{p}(n^{-1/2}m^{1/2})$.

Now, we analyze the asymptotic behavior of $(\widehat{\bm{\beta}}^{\rm
T},\widehat{\bm{\gamma}}^{\rm T})^{\rm T}$. Since $(\widehat{\bm{\beta}}^{\rm
T},\widehat{\bm{\gamma}}^{\rm T})^{\rm T}$ is the maximizer of
$l({\bm{\beta}},{\bm{\gamma}})$, letting $({\bm{\beta}}^{*\rm
T},{\bm{\gamma}}^{*\rm T})^{\rm T}$ be on the line connecting
$(\widehat{\bm{\beta}}^{\rm T},\widehat{\bm{\gamma}}^{\rm T})^{\rm T}$ and
$({\bm{\beta}}_{0}^{\rm T},{\bm{\gamma}}_{0}^{\rm T})^{\rm T}$,

$\displaystyle{\bf 0}$ $\displaystyle=$ $\displaystyle n^{-1}\frac{\partial
l({\bm{\beta}}_{0},{\bm{\gamma}}_{0})}{\partial{\bm{\theta}}}-\left\\{-n^{-1}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\theta}}\partial{\bm{\theta}}^{\rm
T}}\right\\}\begin{bmatrix}\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\\\
\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0}\end{bmatrix}$ $\displaystyle=$
$\displaystyle n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y\mid{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}\end{bmatrix}-\left({\bf\Sigma}+\begin{bmatrix}{\bf R}_{11}&{\bf
R}_{12}\\\ {\bf R}_{12}^{\rm T}&{\bf
R}_{22}\end{bmatrix}\right)\begin{bmatrix}\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\\\
\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0}\end{bmatrix},$

where

$\displaystyle{\bf R}_{11}$ $\displaystyle\equiv$
$\displaystyle-n^{-1}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}\partial{\bm{\beta}}^{\rm
T}}-{\bf\Sigma}_{11},$ $\displaystyle{\bf R}_{12}$ $\displaystyle\equiv$
$\displaystyle-n^{-1}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\beta}}\partial{\bm{\gamma}}^{\rm
T}}-{\bf\Sigma}_{12},$ $\displaystyle{\bf R}_{22}$ $\displaystyle\equiv$
$\displaystyle-n^{-1}\frac{\partial^{2}l({\bm{\beta}}^{*},{\bm{\gamma}}^{*})}{\partial{\bm{\gamma}}\partial{\bm{\gamma}}^{\rm
T}}-{\bf\Sigma}_{22}.$

Since ${\bf\Sigma}$ is invertible by Condition (D5), we have

$\displaystyle\begin{bmatrix}\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\\\
\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0}\end{bmatrix}={\bf\Sigma}^{-1}n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y\mid{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}\end{bmatrix}-{\bf\Sigma}^{-1}\begin{bmatrix}{\bf R}_{11}&{\bf
R}_{12}\\\ {\bf R}_{12}^{\rm T}&{\bf
R}_{22}\end{bmatrix}\begin{bmatrix}\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\\\
\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0}\end{bmatrix}.$

Now, note that

$\displaystyle{\bf\Sigma}^{-1}=\begin{bmatrix}{\bf\Sigma}_{\bm{\beta}}&-{\bf\Sigma}_{\bm{\beta}}{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}\\\
-{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}{\bf\Sigma}_{\bm{\beta}}&{\bf\Sigma}_{22}^{-1}+{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}{\bf\Sigma}_{\bm{\beta}}{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}\end{bmatrix}=\begin{bmatrix}O(1)&O(m^{1/2})\\\
O(m^{1/2})&O(m)\end{bmatrix}$ (S.90)

in terms of the 2-norms of the block matrices. This is because
${\bf\Sigma}_{\bm{\beta}}^{-1}={\bf\Sigma}_{11}-{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}{\bf\Sigma}_{21}$
is positive definite by Condition (D5), $\|{\bf\Sigma}_{12}\|_{2}=O(m^{-1/2})$
by (S.87), and $\|{\bf\Sigma}_{22}^{-1}\|_{2}\asymp m$ by Remark 1. In
addition, using the fact that
$\|{\bm{\beta}}^{*}-{\bm{\beta}}_{0}\|_{2}=O_{p}(n^{-1/2})$ and
$\|{\bm{\gamma}}^{*}-{\bm{\gamma}}_{0}\|_{2}=O_{p}(n^{-1/2}m^{1/2})$, similar
arguments to (S.83), (S.85), and (S.88) leads to $\|{\bf
R}_{11}\|_{2}=o_{p}(1)$, $\|{\bf R}_{22}\|_{2}=o_{p}(m^{-1})$ and $\|{\bf
R}_{12}\|_{2}=o_{p}(m^{-1/2})$. Hence, in terms of their 2-norms,

$\displaystyle-{\bf\Sigma}^{-1}\begin{bmatrix}{\bf R}_{11}&{\bf R}_{12}\\\
{\bf R}_{12}^{\rm T}&{\bf
R}_{22}\end{bmatrix}\begin{bmatrix}\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\\\
\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0}\end{bmatrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}O(1)&O(m^{1/2})\\\
O(m^{1/2})&O(m)\end{bmatrix}\begin{bmatrix}o_{p}(1)&o_{p}(m^{-1/2})\\\
o_{p}(m^{-1/2})&o_{p}(m^{-1})\end{bmatrix}\begin{bmatrix}O_{p}(n^{-1/2})\\\
O_{p}(n^{-1/2}m^{1/2})\end{bmatrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}o_{p}(n^{-1/2})\\\
o_{p}(n^{-1/2}m^{1/2})\end{bmatrix}.$

Therefore we get

$\displaystyle\begin{bmatrix}\widehat{\bm{\beta}}-{\bm{\beta}}_{0}\\\
\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0}\end{bmatrix}={\bf\Sigma}^{-1}n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y\mid{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}\end{bmatrix}+\begin{bmatrix}{\bf r}_{1}\\\ {\bf
r}_{2}\end{bmatrix},$

where $\|{\bf r}_{1}\|_{2}=o_{p}(n^{-1/2})$ and $\|{\bf
r}_{2}\|_{2}=o_{p}(n^{-1/2}m^{1/2})$, which proves the second result of
Proposition 4. Also, we can express $\widehat{\bm{\beta}}-{\bm{\beta}}_{0}$ as

$\displaystyle\widehat{\bm{\beta}}-{\bm{\beta}}_{0}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}{\bf I}_{p}&{\bf
0}_{m}\end{bmatrix}{\bf\Sigma}^{-1}n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y\mid{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}\end{bmatrix}+{\bf r}_{1}.$

Then since

$\displaystyle E\left(\begin{bmatrix}{\bf X}\\{Y-E(Y\mid{\bf X})\\}\\\ {\bf
B}(Y)-E\\{{\bf B}(Y)\mid{\bf X}\\}\end{bmatrix}^{\otimes
2}\right)={\bf\Sigma},$

using (S.90) we can conclude
${\bf\Sigma}_{\bm{\beta}}^{-1/2}\sqrt{n}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})\to
N({\bf 0}_{p},{\bf I}_{p})$ in distribution as $n\to\infty$. ∎

###  S.14 Proof of Theorem 5

We can express $\widehat{\bm{\xi}}-{\bm{\xi}}_{0}$ as

$\displaystyle\widehat{\bm{\xi}}-{\bm{\xi}}_{0}$ $\displaystyle=$
$\displaystyle(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})n^{-1}\sum_{i=1}^{n}\hbox{var}(Y\mid{\bf
x}_{i},\widehat{\bm{\theta}})+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\\{\hbox{var}(Y\mid{\bf
x}_{i},\widehat{\bm{\theta}})-\hbox{var}(Y\mid{\bf x}_{i})\\}$ (S.91)
$\displaystyle+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}\hbox{var}(Y\mid{\bf
x}_{i})-E\\{\hbox{var}(Y\mid{\bf X})\\}\right]$ $\displaystyle=$
$\displaystyle\left\\{n^{-1}\sum_{i=1}^{n}\hbox{var}(Y\mid{\bf
x}_{i},\widehat{\bm{\theta}}){\bf
I}+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\frac{\partial\hbox{var}(Y\mid{\bf
x}_{i},{\bm{\theta}}^{*})}{\partial{\bm{\beta}}^{\rm
T}}\right\\}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})$
$\displaystyle+{\bm{\beta}}_{0}n^{-1}\sum_{i=1}^{n}\frac{\partial\hbox{var}(Y\mid{\bf
x}_{i},{\bm{\theta}}^{*})}{\partial{\bm{\gamma}}^{\rm
T}}(\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0})$
$\displaystyle+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}\hbox{var}(Y\mid{\bf
x}_{i})-E\\{\hbox{var}(Y\mid{\bf X})\\}\right],$

where ${\bm{\theta}}^{*}$ is on the line connecting $\widehat{\bm{\theta}}$
and ${\bm{\theta}}_{0}$. Since
$\|\widehat{\bm{\theta}}-{\bm{\theta}}_{0}\|_{2}=O_{p}(n^{-1/2}m^{1/2})$ by
Proposition 4 and $\hbox{var}(Y\mid{\bf x}_{i},{\bm{\theta}})$ is Lipschitz
continuous at ${\bm{\theta}}_{0}$ uniformly with respect to ${\bf x}$ by
Conditions (D1) and (D2), it is easy to see that

$\displaystyle n^{-1}\sum_{i=1}^{n}\hbox{var}(Y\mid{\bf
x}_{i},\widehat{\bm{\theta}})$ $\displaystyle=$ $\displaystyle
n^{-1}\sum_{i=1}^{n}\hbox{var}(Y\mid{\bf x}_{i})+O_{p}(n^{-1/2}m^{1/2})$
(S.92) $\displaystyle=$ $\displaystyle E\\{\hbox{var}(Y\mid{\bf
X})\\}+O_{p}(n^{-1/2}m^{1/2}).$

Similarly, Conditions (D1) and (D2) lead to

$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}\frac{\partial\hbox{var}(Y\mid{\bf
x}_{i},{\bm{\theta}}^{*})}{\partial{\bm{\beta}}}-E\left[{\bf
X}\\{Y-E(Y\mid{\bf X})\\}^{3}\right]\right\|_{2}$ (S.93) $\displaystyle=$
$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}{\bf x}_{i}E[\\{Y-E(Y\mid{\bf
x}_{i},{\bm{\theta}}^{*})\\}^{3}\mid{\bf x}_{i},{\bm{\theta}}^{*}]-E\left[{\bf
X}\\{Y-E(Y\mid{\bf X})\\}^{3}\right]\right\|_{2}$ $\displaystyle=$
$\displaystyle O_{p}(n^{-1/2}m^{1/2}).$

In addition, in terms of the 2-norm, we have

$\displaystyle n^{-1}\sum_{i=1}^{n}\frac{\partial\hbox{var}(Y\mid{\bf
x}_{i},{\bm{\theta}}^{*})}{\partial{\bm{\gamma}}}$ $\displaystyle=$
$\displaystyle n^{-1}\sum_{i=1}^{n}\left[E\\{{\bf B}(Y)Y^{2}\mid{\bf
x}_{i},{\bm{\theta}}^{*}\\}-2E\\{{\bf B}(Y)Y\mid{\bf
x}_{i},{\bm{\theta}}^{*}\\}E(Y\mid{\bf x}_{i},{\bm{\theta}}^{*})\right.$
$\displaystyle\left.+2E\\{{\bf B}(Y)\mid{\bf
x}_{i},{\bm{\theta}}^{*}\\}E(Y\mid{\bf x}_{i},{\bm{\theta}}^{*})^{2}-E\\{{\bf
B}(Y)\mid{\bf x}_{i},{\bm{\theta}}^{*}\\}E(Y^{2}\mid{\bf
x}_{i},{\bm{\theta}}^{*})\right]$ $\displaystyle=$ $\displaystyle
n^{-1}\sum_{i=1}^{n}\left\\{{\bf p}_{2}({\bf x}_{i},{\bm{\theta}}^{*})-2{\bf
p}_{1}({\bf x}_{i},{\bm{\theta}}^{*})E(Y\mid{\bf x}_{i})+2{\bf p}({\bf
x}_{i},{\bm{\theta}}^{*})E(Y\mid{\bf x}_{i})^{2}-{\bf p}({\bf
x}_{i},{\bm{\theta}}^{*})E(Y^{2}\mid{\bf x}_{i})\right\\}$
$\displaystyle+O_{p}(n^{-1/2}m^{1/2})$ $\displaystyle=$ $\displaystyle
n^{-1}\sum_{i=1}^{n}\left\\{{\bf p}_{2}({\bf x}_{i},{\bm{\theta}}_{0})-2{\bf
p}_{1}({\bf x}_{i},{\bm{\theta}}_{0})E(Y\mid{\bf x}_{i})+2{\bf p}({\bf
x}_{i},{\bm{\theta}}_{0})E(Y\mid{\bf x}_{i})^{2}-{\bf p}({\bf
x}_{i},{\bm{\theta}}_{0})E(Y^{2}\mid{\bf x}_{i})\right\\}$
$\displaystyle+O_{p}(n^{-1/2}m^{1/2})$ $\displaystyle=$ $\displaystyle
n^{-1}\sum_{i=1}^{n}E([{\bf B}(Y)-E\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}]\\{Y-E(Y\mid{\bf x}_{i})\\}^{2}\mid{\bf
x}_{i})+O_{p}(n^{-1/2}m^{1/2})$ $\displaystyle=$ $\displaystyle E([{\bf
B}(Y)-E\\{{\bf B}(Y)\mid{\bf X}\\}]\\{Y-E(Y\mid{\bf X})\\}^{2})$
$\displaystyle+n^{-1}\sum_{i=1}^{n}E([{\bf B}(Y)-E\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}]\\{Y-E(Y\mid{\bf x}_{i})\\}^{2}\mid{\bf x}_{i})-E([{\bf
B}(Y)-E\\{{\bf B}(Y)\mid{\bf X}\\}]\\{Y-E(Y\mid{\bf X})\\}^{2})$
$\displaystyle+O_{p}(n^{-1/2}m^{1/2}).$

The second equality holds by Conditions (D1) and (D2), and the third equality
holds by Lemma 8. Now let ${\bf W}_{1}\equiv{\bf B}(Y)-E\\{{\bf B}(Y)\mid{\bf
X}\\}$ and $W_{2}\equiv\\{Y-E(Y\mid{\bf X})\\}^{2}$, then we can show
$\|E({\bf W}_{1}W_{2})\|_{2}=O(m^{-1/2})$ since for any ${\bf
u}\in\mathbb{R}^{m}$ such that $\|{\bf u}\|_{2}=1$,

$\displaystyle\\{E({\bf W}_{1}W_{2})^{\rm T}{\bf u}\\}^{2}$ $\displaystyle=$
$\displaystyle{\bf u}^{\rm T}E\\{\hbox{cov}({\bf W}_{1},W_{2}\mid{\bf
X})\\}E\\{\hbox{cov}(W_{2},{\bf W}_{1}\mid{\bf X})\\}{\bf u}$
$\displaystyle\leq$ $\displaystyle{\bf u}^{\rm T}E\\{\hbox{cov}({\bf
W}_{1},W_{2}\mid{\bf X})\hbox{cov}(W_{2},{\bf W}_{1}\mid{\bf X})\\}{\bf u}$
$\displaystyle\leq$ $\displaystyle{\bf u}^{\rm T}E\\{\hbox{var}({\bf
W}_{1}\mid{\bf X})\hbox{var}(W_{2}\mid{\bf X})\\}{\bf u}$ $\displaystyle\leq$
$\displaystyle C{\bf u}^{\rm T}E\\{\hbox{var}({\bf W}_{1}\mid{\bf X})\\}{\bf
u}$ $\displaystyle=$ $\displaystyle O(m^{-1})$

for some constant $C>0$. The fourth argument holds since
$\hbox{var}(W_{2}\mid{\bf x})$ is uniformly bounded by Condition (D1), and the
last argument holds because $\|E\\{\hbox{var}({\bf W}_{1}\mid{\bf
X})\\}\|_{2}=\|{\bf\Sigma}_{22}\|_{2}\asymp m^{-1}$ by Remark 1. This leads to

$\displaystyle\|{\bf A}_{2}\|_{2}=\|{\bm{\beta}}_{0}E({\bf W}_{1}W_{2})^{\rm
T}\|_{2}=\|{\bm{\beta}}_{0}\|_{2}\|E({\bf W}_{1}W_{2})\|_{2}=O(m^{-1/2}).$
(S.94)

Similarly,

$\displaystyle E\left[\left\\{m^{1/2}E({\bf W}_{1}^{\rm T}{\bf u}W_{2}\mid{\bf
X})\right\\}^{2}\right]$ $\displaystyle=$ $\displaystyle
mE\left[\left\\{\hbox{cov}({\bf W}_{1}^{\rm T}{\bf u},W_{2}\mid{\bf
X})\right\\}^{2}\right]$ $\displaystyle\leq$ $\displaystyle
mE\left\\{\hbox{var}({\bf W}_{1}^{\rm T}{\bf u}\mid{\bf
X})\hbox{var}(W_{2}\mid{\bf X})\right\\}$ $\displaystyle\leq$ $\displaystyle
Cm{\bf u}^{\rm T}E\left\\{\hbox{var}({\bf W}_{1}\mid{\bf X})\right\\}{\bf u}$
$\displaystyle=$ $\displaystyle O(1),$

i.e. the second moment of $m^{1/2}E({\bf W}_{1}^{\rm T}{\bf u}W_{2}\mid{\bf
X})$ is finite, then

$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}E({\bf W}_{1}W_{2}\mid{\bf
x}_{i})-E({\bf W}_{1}W_{2})\right\|_{2}$ $\displaystyle=$ $\displaystyle
m^{-1/2}\sup_{{\bf u}\in{\bf R}^{m}:\|{\bf
u}\|_{2}=1}\left|n^{-1}\sum_{i=1}^{n}m^{1/2}E({\bf W}_{1}^{\rm T}{\bf
u}W_{2}\mid{\bf x}_{i})-m^{1/2}E({\bf W}_{1}^{\rm T}{\bf u}W_{2})\right|$
$\displaystyle=$ $\displaystyle O_{p}(m^{-1/2}n^{-1/2})$ $\displaystyle=$
$\displaystyle o_{p}(m^{-1/2}).$

Thus using $n^{-1/2}m^{1/2}=o(m^{-1/2})$ under Condition (D3), we get

$\displaystyle\left\|n^{-1}\sum_{i=1}^{n}\frac{\partial\hbox{var}(Y\mid{\bf
x}_{i},{\bm{\theta}}^{*})}{\partial{\bm{\gamma}}}-E([{\bf B}(Y)-E\\{{\bf
B}(Y)\mid{\bf X}\\}]\\{Y-E(Y\mid{\bf X})\\}^{2})\right\|_{2}$ (S.95)
$\displaystyle=$ $\displaystyle\left\|n^{-1}\sum_{i=1}^{n}E({\bf
W}_{1}W_{2}\mid{\bf x}_{i})-E({\bf
W}_{1}W_{2})\right\|_{2}+O_{p}(n^{-1/2}m^{1/2})$ $\displaystyle=$
$\displaystyle o_{p}(m^{-1/2}).$

Combining the results of Proposition 4, (S.91), (S.92), (S.93), (S.94), and
(S.95), in terms of the 2-norm we get

$\displaystyle\widehat{\bm{\xi}}-{\bm{\xi}}_{0}$ $\displaystyle=$
$\displaystyle{\bf A}_{1}(\widehat{\bm{\beta}}-{\bm{\beta}}_{0})+{\bf
A}_{2}(\widehat{\bm{\gamma}}-{\bm{\gamma}}_{0})+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}\hbox{var}(Y\mid{\bf
x}_{i})-E\\{\hbox{var}(Y\mid{\bf X})\\}\right]$
$\displaystyle+O_{p}(n^{-1}m^{1/2})+o_{p}(n^{-1/2})$ $\displaystyle=$
$\displaystyle{\bf A}{\bf\Sigma}^{-1}n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y\mid{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}\end{bmatrix}+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}\hbox{var}(Y\mid{\bf
x}_{i})-E\\{\hbox{var}(Y\mid{\bf X})\\}\right]$ $\displaystyle+{\bf A}_{1}{\bf
r}_{1}+{\bf A}_{2}{\bf r}_{2}+o_{p}(n^{-1/2})$ $\displaystyle=$
$\displaystyle{\bf A}{\bf\Sigma}^{-1}n^{-1}\sum_{i=1}^{n}\begin{bmatrix}{\bf
x}_{i}\\{y_{i}-E(Y\mid{\bf x}_{i})\\}\\\ {\bf B}(y_{i})-E\\{{\bf B}(Y)\mid{\bf
x}_{i}\\}\end{bmatrix}+{\bm{\beta}}_{0}\left[n^{-1}\sum_{i=1}^{n}\hbox{var}(Y\mid{\bf
x}_{i})-E\\{\hbox{var}(Y\mid{\bf X})\\}\right]$
$\displaystyle+o_{p}(n^{-1/2})$

since $m=o(n^{1/3})$ by Condition (D3). Since

$\displaystyle E\left(\begin{bmatrix}{\bf X}\\{Y-E(Y\mid{\bf X})\\}\\\ {\bf
B}(Y)-E\\{{\bf B}(Y)\mid{\bf X}\\}\end{bmatrix}^{\otimes 2}\right)$
$\displaystyle=$ $\displaystyle{\bf\Sigma},$ $\displaystyle\hbox{cov}[{\bf
X}\\{Y-E(Y\mid{\bf X})\\},\hbox{var}(Y\mid{\bf X})]$ $\displaystyle=$
$\displaystyle{\bf 0},$ $\displaystyle\hbox{cov}[{\bf B}(Y)-E\\{{\bf
B}(Y)\mid{\bf X}\\},\hbox{var}(Y\mid{\bf X})]$ $\displaystyle=$
$\displaystyle{\bf 0},$

we can conclude
${\bf\Sigma}_{\bm{\xi}}^{-1/2}\sqrt{n}(\widehat{\bm{\xi}}-{\bm{\xi}}_{0})\to
N({\bf 0},{\bf I})$ in distribution as $n\to\infty$. ∎

###  S.15 Proof of Proposition 5

The efficient score for ${\bm{\beta}}$ given in Appendix S.9 is ${\bf S}_{\rm
eff}=y{\bf x}-{\bf a}_{0}(y)-E\\{Y{\bf x}-{\bf a}_{0}(Y)\mid{\bf x}\\}$, where
${\bf a}_{0}(y)$ satisfies

$\displaystyle{\bf a}_{0}(y)-E[E\\{{\bf a}_{0}(Y)\mid{\bf X}\\}\mid y]=E(y{\bf
X}\mid y)-E\\{E(Y{\bf X}\mid{\bf X})\mid y\\}.$ (S.96)

Thus, the efficient variance is $\\{E({\bf S}_{\rm eff}^{\otimes 2})\\}^{-1}$.

To show that the MLE estimator $\widehat{\bm{\beta}}$ in Proposition 4 is
efficient, we need to show ${\bf\Sigma}_{\bm{\beta}}^{-1}=E({\bf S}_{\rm
eff}^{\otimes 2})$. ${\bf a}_{0}(y)$ must be of the general form $\Lambda{\bf
B}(y)$, i.e. ${\bf a}_{0}(y)=\Lambda{\bf B}(y)$, where $\Lambda$ is a $p\times
m$ coefficient matrix. Then (S.96) implies that

$\displaystyle\Lambda E\\{{\bf B}(Y)^{\otimes 2}\\}-\Lambda E[E\\{{\bf
B}(Y)\mid{\bf X}\\}^{\otimes 2}]$ $\displaystyle=$ $\displaystyle E[E\\{Y{\bf
X}-E(Y{\bf X}\mid{\bf X}){\bf B}(Y)^{\rm T}\mid{\bf X}\\}]$ $\displaystyle=$
$\displaystyle E[{\bf X}\hbox{cov}\\{Y,{\bf B}(Y)\mid{\bf X}\\}],$

i.e.,

$\displaystyle{\bf a}_{0}(y)=\Lambda{\bf
B}(y)={\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}{\bf B}(y).$

Therefore,

$\displaystyle E({\bf S}_{\rm eff}^{\otimes 2})=E\left\\{(Y{\bf X}-E(Y{\bf
X}\mid{\bf X})-{\bf\Sigma}_{12}{\bf\Sigma}_{22}^{-1}[{\bf B}(Y)-E\\{{\bf
B}(Y)\mid{\bf X}\\}])^{\otimes 2}\right\\}={\bf\Sigma}_{\bm{\beta}}^{-1}.$

∎

###  S.16 Proof of Theorem 6

Since the efficient score for ${\bm{\xi}}$ given in Section 2.1 leads to

$\displaystyle E(\bm{\phi}_{\rm eff}^{\otimes 2})$ $\displaystyle=$
$\displaystyle E\left([{\bm{\beta}}v({\bm{\beta}}^{\rm T}{\bf
X})-{\bm{\beta}}E\\{v({\bm{\beta}}^{\rm T}{\bf X})\\}]^{\otimes 2}\right)$
$\displaystyle+E\left([{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf
X}Y-E\\{{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf X}Y\mid{\bf X}\\}]^{\otimes
2}\right),$

and the asymptotic variance of $\widehat{\bm{\xi}}$ in Theorem 5 is

$\displaystyle{\bf\Sigma}_{\bm{\xi}}={\bf A}{\bf\Sigma}^{-1}{\bf A}^{\rm
T}+{\bm{\beta}}^{\otimes 2}\hbox{var}\\{v({\bm{\beta}}^{\rm T}{\bf X})\\},$

we only need to show

$\displaystyle E\left([{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf
X}Y-E\\{{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf X}Y\mid{\bf X}\\}]^{\otimes
2}\right)={\bf A}{\bf\Sigma}^{-1}{\bf A}^{\rm T}.$

Now we can always write ${\bm{\beta}}y^{2}+{\bf a}(y)={\bf\Lambda}{\bf B}(y)$,
where ${\bf\Lambda}\in\mathbb{R}^{p\times m}$. Then (S.5) and (S.6) imply

$\displaystyle-{\bf\Lambda}E[{\bf B}(y)-E\\{{\bf B}(Y)\mid{\bf X}\\}\mid
y]+{\bm{\beta}}E\\{y^{2}-E(Y^{2}\mid{\bf X})\mid y\\}$ $\displaystyle=$
$\displaystyle 2{\bm{\beta}}E[yE(Y\mid{\bf X})-E\\{YE(Y\mid{\bf X})\mid{\bf
X}\\}\mid y]+{\bf M}E[{\bf X}\\{y-E(Y\mid{\bf X})\\}\mid y]$ $\displaystyle=$
$\displaystyle 2{\bm{\beta}}E[yE(Y\mid{\bf X})-E\\{YE(Y\mid{\bf X})\mid{\bf
X}\\}\mid y]$ $\displaystyle+\left(E\\{v({\bm{\beta}}^{\rm T}{\bf X})\\}{\bf
I}-E\left[2{\bm{\beta}}{\bf X}^{\rm T}Yv({\bm{\beta}}^{\rm T}{\bf
X})+\\{{\bf\Lambda}{\bf B}(Y)-{\bm{\beta}}Y^{2}\\}\\{Y-E(Y\mid{\bf X})\\}{\bf
X}^{\rm T}\right]\right)$ $\displaystyle\times{\bf\Sigma}_{11}^{-1}E[{\bf
X}\\{y-E(Y\mid{\bf X})\\}\mid y].$

Multiplying ${\bf B}^{\rm T}(y)$ on both sides and taking expectation lead to

$\displaystyle-{\bf\Lambda}{\bf\Sigma}_{22}+{\bm{\beta}}E[\\{Y^{2}-E(Y^{2}\mid{\bf
X})\\}{\bf B}^{\rm T}(Y)]$ $\displaystyle=$ $\displaystyle
2{\bm{\beta}}E[YE(Y\mid{\bf X}){\bf B}^{\rm T}(Y)-\\{E(Y\mid{\bf
X})\\}^{2}{\bf B}^{\rm T}(Y)]$ $\displaystyle+\left(E\\{v({\bm{\beta}}^{\rm
T}{\bf X})\\}{\bf I}-E\left[2{\bm{\beta}}{\bf X}^{\rm T}Yv({\bm{\beta}}^{\rm
T}{\bf X})+\\{{\bf\Lambda}{\bf
B}(Y)-{\bm{\beta}}Y^{2}\\}\\{Y-E(Y\mid{\bm{\beta}}^{\rm T}{\bf X})\\}{\bf
X}^{\rm T}\right]\right)$ $\displaystyle\times{\bf\Sigma}_{11}^{-1}E[{\bf
X}\\{Y-E(Y\mid{\bf X})\\}{\bf B}^{\rm T}(Y)]$ $\displaystyle=$ $\displaystyle
2{\bm{\beta}}E[Y\hbox{cov}\\{Y,{\bf B}(Y)\mid{\bf
X}\\}]+E\\{v({\bm{\beta}}^{\rm T}{\bf
X})\\}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-E\left\\{2{\bm{\beta}}{\bf X}^{\rm
T}Yv({\bm{\beta}}^{\rm T}{\bf
X})\right\\}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}$
$\displaystyle-{\bf\Lambda}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}+{\bm{\beta}}E[Y^{2}\\{Y-E(Y\mid{\bm{\beta}}^{\rm
T}{\bf X})\\}{\bf X}^{\rm T}]{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}$
$\displaystyle=$ $\displaystyle 2{\bm{\beta}}E[Y\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf X}\\}]+{\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf\Lambda}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12},$

where ${\bf A}_{1},{\bf A}_{2}$ are defined in Theorem 5, hence

$\displaystyle-{\bf\Lambda}{\bf\Sigma}_{22}+{\bf\Lambda}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}$
$\displaystyle=$ $\displaystyle 2{\bm{\beta}}E[Y\hbox{cov}\\{Y,{\bf
B}(Y)\mid{\bf X}\\}]+{\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bm{\beta}}E[\\{Y^{2}-E(Y^{2}\mid{\bf
X})\\}{\bf B}^{\rm T}(Y)]$ $\displaystyle=$ $\displaystyle{\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf A}_{2},$

and

$\displaystyle{\bm{\beta}}y^{2}+{\bf a}(y)=({\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf
A}_{2})({\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf\Sigma}_{22})^{-1}{\bf
B}(y)={\bf U}{\bf B}(y),$

where, for notational brevity,

$\displaystyle{\bf U}$ $\displaystyle\equiv$ $\displaystyle({\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf
A}_{2})({\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf\Sigma}_{22})^{-1}.$

Then

$\displaystyle{\bf M}$ $\displaystyle=$
$\displaystyle\left(E\\{v({\bm{\beta}}^{\rm T}{\bf X})\\}{\bf
I}-E\\{2{\bm{\beta}}{\bf X}^{\rm T}Yv({\bm{\beta}}^{\rm T}{\bf X})\\}-{\bf
U}E\left[{\bf B}(Y)\\{Y-E(Y\mid{\bm{\beta}}^{\rm T}{\bf X})\\}{\bf X}^{\rm
T}\right]\right.$
$\displaystyle\left.+{\bm{\beta}}E\left[Y^{2}\\{Y-E(Y\mid{\bm{\beta}}^{\rm
T}{\bf X})\\}{\bf X}^{\rm T}\right]\right){\bf\Sigma}_{11}^{-1}$
$\displaystyle=$ $\displaystyle\left(E\\{v({\bm{\beta}}^{\rm T}{\bf X})\\}{\bf
I}-E\\{2{\bm{\beta}}{\bf X}^{\rm T}Yv({\bm{\beta}}^{\rm T}{\bf X})\\}-{\bf
U}{\bf\Sigma}_{21}+{\bm{\beta}}E\left[Y^{2}\\{Y-E(Y\mid{\bf X})\\}{\bf X}^{\rm
T}\right]\right){\bf\Sigma}_{11}^{-1}$ $\displaystyle=$ $\displaystyle({\bf
A}_{1}-{\bf U}{\bf\Sigma}_{21}){\bf\Sigma}_{11}^{-1}.$

Hence

$\displaystyle{\bm{\beta}}y^{2}+{\bf a}(y)-E\\{{\bm{\beta}}Y^{2}+{\bf
a}(Y)\mid{\bf x}\\}={\bf U}[{\bf B}(y)-E\\{{\bf B}(Y)\mid{\bf x}\\}],$

and

$\displaystyle E\\{\hbox{var}({\bf M}{\bf X}Y\mid{\bf X})\\}$ $\displaystyle=$
$\displaystyle{\bf M}{\bf\Sigma}_{11}{\bf M}^{\rm T},$ $\displaystyle
E[\hbox{var}\\{{\bm{\beta}}Y^{2}+{\bf a}(Y)\mid{\bf X}\\}]$ $\displaystyle=$
$\displaystyle{\bf U}{\bf\Sigma}_{22}{\bf U}^{\rm T},$ $\displaystyle
E[\hbox{cov}\\{{\bf M}{\bf X}Y,{\bm{\beta}}Y^{2}+{\bf a}(Y)\mid{\bf X}\\}]$
$\displaystyle=$ $\displaystyle{\bf M}{\bf\Sigma}_{12}{\bf U}^{\rm T}.$

Thus, noting that ${\bf A}=[{\bf A}_{1},{\bf A}_{2}]$,

$\displaystyle E\left([{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf
X}Y-E\\{{\bm{\beta}}Y^{2}+{\bf a}(Y)+{\bf M}{\bf X}Y\mid{\bf X}\\}]^{\otimes
2}\right)$ $\displaystyle=$ $\displaystyle{\bf M}{\bf\Sigma}_{11}{\bf M}^{\rm
T}+{\bf U}{\bf\Sigma}_{22}{\bf U}^{\rm T}+{\bf M}{\bf\Sigma}_{12}{\bf U}^{\rm
T}+{\bf U}{\bf\Sigma}_{21}{\bf M}^{\rm T}$ $\displaystyle=$
$\displaystyle({\bf A}_{1}-{\bf U}{\bf\Sigma}_{21}){\bf\Sigma}_{11}^{-1}({\bf
A}_{1}^{\rm T}-{\bf\Sigma}_{12}{\bf U}^{\rm T})+{\bf U}{\bf\Sigma}_{22}{\bf
U}^{\rm T}+({\bf A}_{1}-{\bf
U}{\bf\Sigma}_{21}){\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}{\bf U}^{\rm T}$
$\displaystyle+{\bf U}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}({\bf A}_{1}^{\rm
T}-{\bf\Sigma}_{12}{\bf U}^{\rm T})$ $\displaystyle=$ $\displaystyle{\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf A}_{1}^{\rm T}+{\bf
U}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}){\bf
U}^{\rm T}$ $\displaystyle=$ $\displaystyle{\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf A}_{1}^{\rm T}+({\bf
A}_{1}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}-{\bf
A}_{2})({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}({\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf
A}_{1}^{\rm T}-{\bf A}_{2}^{\rm T})$ $\displaystyle=$ $\displaystyle{\bf
A}\left(\begin{array}[]{cc}{\bf\Sigma}_{11}^{-1}+{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}&-{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12}({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}\\\
-({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}&({\bf\Sigma}_{22}-{\bf\Sigma}_{21}{\bf\Sigma}_{11}^{-1}{\bf\Sigma}_{12})^{-1}\end{array}\right){\bf
A}^{\rm T}$ $\displaystyle=$ $\displaystyle{\bf A}{\bf\Sigma}^{-1}{\bf A}^{\rm
T}.$

∎

###  S.17 Additional Tables for Simulation Experiments

Table S.1: ${\bm{\eta}}_{\tau}$ estimation results under the truncated normal distribution. $\tau$ |  | Method |  $|$bias$|$ | $\sigma_{\text{sim}}$ | $\widehat{\sigma}_{\text{est}}$ | C.I.
---|---|---|---|---|---|---
0.05 | $\eta_{1}$ | aMLE | 0.022 | 0.028 | 0.029 | 0.954
| pMLE | 0.026 | 0.029 | - | -
| MLE | 0.035 | 0.035 | 0.035 | 0.893
$\eta_{2}$ | aMLE | 0.029 | 0.036 | 0.034 | 0.937
| pMLE | 0.036 | 0.035 | - | -
| MLE | 0.053 | 0.041 | 0.036 | 0.681
$\eta_{3}$ | aMLE | 0.033 | 0.041 | 0.042 | 0.957
| pMLE | 0.047 | 0.040 | - | -
| MLE | 0.075 | 0.044 | 0.035 | 0.463
0.25 | $\eta_{1}$ | aMLE | 0.023 | 0.029 | 0.029 | 0.950
| pMLE | 0.023 | 0.029 | - | -
| MLE | 0.028 | 0.035 | 0.035 | 0.954
$\eta_{2}$ | aMLE | 0.028 | 0.036 | 0.034 | 0.941
| pMLE | 0.028 | 0.035 | - | -
| MLE | 0.035 | 0.041 | 0.036 | 0.898
$\eta_{3}$ | aMLE | 0.032 | 0.040 | 0.040 | 0.953
| pMLE | 0.032 | 0.039 | - | -
| MLE | 0.040 | 0.044 | 0.035 | 0.839
0.50 | $\eta_{1}$ | aMLE | 0.023 | 0.029 | 0.029 | 0.945
| pMLE | 0.023 | 0.030 | - | -
| MLE | 0.029 | 0.035 | 0.035 | 0.940
$\eta_{2}$ | aMLE | 0.028 | 0.036 | 0.034 | 0.945
| pMLE | 0.029 | 0.035 | - | -
| MLE | 0.039 | 0.041 | 0.036 | 0.841
$\eta_{3}$ | aMLE | 0.032 | 0.040 | 0.040 | 0.957
| pMLE | 0.033 | 0.039 | - | -
| MLE | 0.049 | 0.044 | 0.035 | 0.742
0.75 | $\eta_{1}$ | aMLE | 0.023 | 0.029 | 0.029 | 0.941
| pMLE | 0.023 | 0.029 | - | -
| MLE | 0.028 | 0.035 | 0.035 | 0.953
$\eta_{2}$ | aMLE | 0.028 | 0.036 | 0.034 | 0.940
| pMLE | 0.028 | 0.034 | - | -
| MLE | 0.035 | 0.041 | 0.036 | 0.898
$\eta_{3}$ | aMLE | 0.033 | 0.040 | 0.040 | 0.959
| pMLE | 0.032 | 0.039 | - | -
| MLE | 0.039 | 0.044 | 0.035 | 0.846
0.95 | $\eta_{1}$ | aMLE | 0.023 | 0.028 | 0.029 | 0.940
| pMLE | 0.025 | 0.029 | - | -
| MLE | 0.036 | 0.035 | 0.035 | 0.893
$\eta_{2}$ | aMLE | 0.028 | 0.035 | 0.034 | 0.945
| pMLE | 0.032 | 0.034 | - | -
| MLE | 0.053 | 0.041 | 0.036 | 0.678
$\eta_{3}$ | aMLE | 0.033 | 0.041 | 0.042 | 0.964
| pMLE | 0.041 | 0.038 | - | -
| MLE | 0.075 | 0.044 | 0.035 | 0.458
Table S.2: ${\bm{\eta}}_{\tau}$ estimation results under the normal distribution. $\tau$ |  | Method |  $|$bias$|$ | $\sigma_{\text{sim}}$ | $\widehat{\sigma}_{\text{est}}$ | C.I.
---|---|---|---|---|---|---
0.05 | $\eta_{1}$ | aMLE | 0.026 | 0.033 | 0.032 | 0.947
| pMLE | 0.030 | 0.039 | - | -
| MLE | 0.026 | 0.032 | 0.032 | 0.954
$\eta_{2}$ | aMLE | 0.026 | 0.032 | 0.032 | 0.948
| pMLE | 0.031 | 0.039 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.957
$\eta_{3}$ | aMLE | 0.026 | 0.033 | 0.033 | 0.934
| pMLE | 0.032 | 0.039 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.941
0.25 | $\eta_{1}$ | aMLE | 0.026 | 0.032 | 0.032 | 0.950
| pMLE | 0.030 | 0.038 | - | -
| MLE | 0.026 | 0.032 | 0.032 | 0.954
$\eta_{2}$ | aMLE | 0.025 | 0.032 | 0.032 | 0.950
| pMLE | 0.030 | 0.039 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.957
$\eta_{3}$ | aMLE | 0.025 | 0.032 | 0.032 | 0.940
| pMLE | 0.031 | 0.038 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.941
0.50 | $\eta_{1}$ | aMLE | 0.026 | 0.032 | 0.032 | 0.953
| pMLE | 0.029 | 0.038 | - | -
| MLE | 0.026 | 0.032 | 0.032 | 0.954
$\eta_{2}$ | aMLE | 0.025 | 0.032 | 0.032 | 0.955
| pMLE | 0.030 | 0.039 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.957
$\eta_{3}$ | aMLE | 0.025 | 0.032 | 0.032 | 0.935
| pMLE | 0.030 | 0.038 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.941
0.75 | $\eta_{1}$ | aMLE | 0.026 | 0.032 | 0.032 | 0.953
| pMLE | 0.029 | 0.038 | - | -
| MLE | 0.026 | 0.032 | 0.032 | 0.954
$\eta_{2}$ | aMLE | 0.025 | 0.032 | 0.032 | 0.951
| pMLE | 0.030 | 0.038 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.957
$\eta_{3}$ | aMLE | 0.026 | 0.032 | 0.032 | 0.938
| pMLE | 0.030 | 0.038 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.941
0.95 | $\eta_{1}$ | aMLE | 0.026 | 0.032 | 0.032 | 0.950
| pMLE | 0.029 | 0.038 | - | -
| MLE | 0.026 | 0.032 | 0.032 | 0.954
$\eta_{2}$ | aMLE | 0.026 | 0.032 | 0.032 | 0.951
| pMLE | 0.031 | 0.039 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.957
$\eta_{3}$ | aMLE | 0.027 | 0.034 | 0.033 | 0.937
| pMLE | 0.033 | 0.039 | - | -
| MLE | 0.025 | 0.032 | 0.032 | 0.941
Table S.3: ${\bm{\eta}}_{\tau}$ estimation results under the gamma distribution. $\tau$ |  | Method |  $|$bias$|$ | $\sigma_{\rm sim}$ | $\widehat{\sigma}_{\rm est}$ | C.I.
---|---|---|---|---|---|---
0.05 | $\eta_{1}$ | aMLE | 0.031 | 0.040 | 0.040 | 0.949
| pMLE | 0.561 | 0.690 | - | -
| MLE | 0.021 | 0.026 | 0.025 | 0.950
$\eta_{2}$ | aMLE | 0.038 | 0.049 | 0.049 | 0.939
| pMLE | 0.840 | 0.588 | - | -
| MLE | 0.022 | 0.028 | 0.028 | 0.950
0.25 | $\eta_{1}$ | aMLE | 0.051 | 0.066 | 0.064 | 0.942
| pMLE | 1.070 | 0.953 | - | -
| MLE | 0.035 | 0.044 | 0.043 | 0.949
$\eta_{2}$ | aMLE | 0.058 | 0.074 | 0.073 | 0.943
| pMLE | 1.666 | 0.753 | - | -
| MLE | 0.037 | 0.046 | 0.046 | 0.953
0.50 | $\eta_{1}$ | aMLE | 0.070 | 0.089 | 0.088 | 0.944
| pMLE | 1.259 | 1.080 | - | -
| MLE | 0.048 | 0.060 | 0.059 | 0.948
$\eta_{2}$ | aMLE | 0.077 | 0.096 | 0.096 | 0.950
| pMLE | 1.945 | 0.844 | - | -
| MLE | 0.051 | 0.064 | 0.064 | 0.950
0.75 | $\eta_{1}$ | aMLE | 0.093 | 0.117 | 0.116 | 0.944
| pMLE | 1.246 | 1.410 | - | -
| MLE | 0.064 | 0.081 | 0.080 | 0.948
$\eta_{2}$ | aMLE | 0.103 | 0.127 | 0.126 | 0.944
| pMLE | 1.831 | 0.907 | - | -
| MLE | 0.069 | 0.086 | 0.086 | 0.951
0.95 | $\eta_{1}$ | aMLE | 0.146 | 0.186 | 0.185 | 0.948
| pMLE | 0.828 | 1.529 | - | -
| MLE | 0.094 | 0.118 | 0.116 | 0.949
$\eta_{2}$ | aMLE | 0.178 | 0.227 | 0.232 | 0.957
| pMLE | 0.991 | 1.025 | - | -
| MLE | 0.102 | 0.127 | 0.126 | 0.948