$V_{E}\coloneqq\frac{1}{n}\sum_{i\in[n]}(\mathbbm{1}_{M_{i}>0}+w_{i})^{2}\sigma_{D_{i}}^{2}(\pi(X_{i}))$
and standard normal distribution function $\Phi$.

###### Proof.

We apply a Lindeberg–Feller central limit theorem as $E_{i}$, given
$(D^{(n)},PS^{(n)})$, are independently, but not identically, distributed with
mean zero across $i\in[n]$ (the $M_{i},w_{i}$ are constants given
$(D^{(n)},PS^{(n)})$). By Assumption 2, the $\mu^{d}$, and hence
$\varepsilon$, are well-defined. By definition of $V_{E}$,
$\operatorname*{\mathbb{V}}\left[\sum_{i\in[n]}E_{i}/\sqrt{nV_{E}}\mid
D^{(n)},PS^{(n)}\right]=1$. Thus, we only need to verify the Lindeberg–Feller
condition:

$\displaystyle\sum_{i\in[n]}\mathbb{E}\left[\left.(E_{i}/\sqrt{nV_{E}})^{2}\mathbbm{1}_{|E_{i}/\sqrt{nV_{E}}|\geq\eta}\,\right|\,D^{(n)},PS^{(n)}\right]\overset{P}{\longrightarrow}0\quad\text{for
all constants }\eta>0.$ (270)

Since $\mathbbm{1}_{|E_{i}/\sqrt{nV_{E}}|\geq\eta}$ is bounded by
$E_{i}^{2}/(\eta^{2}nV_{E})$ on $\left\\{E_{i}^{2}/(\eta^{2}nV_{E})\geq
1\right\\}$, the expectation in (270) satisfies

$\displaystyle\frac{1}{nV_{E}}\mathbb{E}\left[\left.E_{i}^{2}\mathbbm{1}_{|E_{i}|\geq\eta\sqrt{nV_{E}}}\,\right|\,D^{(n)},PS^{(n)}\right]$
$\displaystyle\leq\frac{1}{nV_{E}}\mathbb{E}\left[\left.E_{i}^{4}/(\eta^{2}nV_{E})\,\right|\,D^{(n)},PS^{(n)}\right]$
$\displaystyle=\frac{\mathbb{E}\left[\left.E_{i}^{4}\,\right|\,D^{(n)},PS^{(n)}\right]}{(\eta
nV_{E})^{2}},$ (271)

where
$\mathbb{E}\left[\left.E_{i}^{4}\,\right|\,D^{(n)},PS^{(n)}\right]=(\mathbbm{1}_{M_{i}>0}+w_{i})^{4}\mathbb{E}\left[\left.\varepsilon_{i}^{4}\,\right|\,D_{i},\pi(X_{i})\right]$,
with
$\mathbb{E}\left[\left.\varepsilon_{i}^{4}\,\right|\,D_{i},\pi(X_{i})\right]\leq\sup_{d\in\mathbb{B},p\in[0,1]}\mathbb{E}\left[\varepsilon_{i}^{4}\mid
D_{i}=d,\pi(X_{i})=p\right]<\infty$ by Assumption 4. By Lemma 4,
$V_{E}\overset{P}{\longrightarrow}V_{\sigma,\pi}\in(0,\infty)$, so (271) is
bounded by
$(1+o_{P}\left(1\right))n^{-1}\left(\frac{1}{n}\sum_{i\in[n]}(\mathbbm{1}_{M_{i}>0}+w_{i})^{4}\right)$
up to a constant factor. Fix a constant $C>0$. By Markov’s inequality,

$\mathbb{P}\left(\frac{1}{n}\sum_{i\in[n]}(\mathbbm{1}_{M_{i}>0}+w_{i})^{4}>C\right)\leq
C^{-1}\max_{i\in[n]}\mathbb{E}(\mathbbm{1}_{M_{i}>0}+w_{i})^{4}.$

Below, we show that
$n^{-\varrho}\max_{i\in[n]}\mathbb{E}(\mathbbm{1}_{M_{i}>0}+w_{i})^{4}=O\left(1\right)$
for any $\varrho>0$, so that

$\frac{1}{n}\sum_{i\in[n]}(\mathbbm{1}_{M_{i}>0}+w_{i})^{4}=O_{P}\left(n^{\varrho}\right).$

Then (271) is bounded by
$(1+o_{P}\left(1\right))O_{P}\left(n^{\varrho-1}\right)$, which is
$o_{P}\left(1\right)$ for $\varrho<1$, so (270) is met.

As $w_{i}\geq 0$ and $(1+x)^{4}\leq(2x)^{4}$ for $x\geq 1$,

$(\mathbbm{1}_{M_{i}>0}+w_{i})^{4}\leq
2^{4}+2^{4}w_{i}^{4}\mathbbm{1}_{w_{i}>1}.$

We have $w_{i}\leq M_{i}\max_{j\in\mathcal{J}(i)}M_{j}^{-1}$ and hence

$\mathbb{E}w_{i}^{4}\mathbbm{1}_{w_{i}>1}\leq\mathbb{E}\left[\mathbbm{1}_{w_{i}>1}M_{i}^{4}\max_{j\in\mathcal{J}(i)}M_{j}^{-4}\right].$

This yields the result by Lemma 3.

When the caliper is
$\delta_{n}=\overline{\underline{\Delta}}_{n}\vee\frac{\log
N_{0}}{N_{0}+1}\vee\frac{\log N_{1}}{N_{1}+1}$, it is
$\sigma\left\\{D^{(n)},PS^{(n)}\right\\}$-measurable, hence the $M_{i},w_{i}$
are constants given $(D^{(n)},PS^{(n)})$. Lemmas 3 and 4 complete the proof. ∎

######  Lemma 3 (Lindeberg–Feller Bound).

Suppose that the caliper $\delta_{n}$ satisfies (3) and that Assumption 2
holds. Then for any finite fixed constant integer $r\geq 2$ and any finite
fixed constant $\varrho>0$,

$\displaystyle\max_{i\in[n]}\mathbb{E}{\mathbbm{1}_{w_{i}>1}\max_{j\in\mathcal{J}(i)}\left(\frac{M_{i}}{M_{j}}\right)^{r}}=o\left(n^{\varrho}\right).$
(272)

###### Proof.

If $w_{i}>1$ for some $i\in[n]$, then $M_{i}\geq 1$ and hence also $M_{j}\geq
1$ for all $j\in\mathcal{J}(i)$, as $j\in\mathcal{J}(i)$ if and only if
$i\in\mathcal{J}(j)$. This in turn implies $N_{0},N_{1}\geq 1$. Thus,
$\mathbbm{1}_{w_{i}>1}\leq\mathbbm{1}_{M_{j}\geq 1}\mathbbm{1}_{N_{0}\geq
1}\mathbbm{1}_{N_{1}\geq 1}$ for all $j\in\mathcal{J}(i)$ for all $i\in[n]$.
By definition,
$M_{i}=N_{1-d}\mathbbm{F}_{N_{1-d}}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]$
if $D_{i}=d$. Then, in the notation of (17) and (56), the left side of (272)
is bounded by

$\displaystyle\sum_{d\in\\{0,1\\}}\max_{i:D_{i}=d}\mathbb{E}\left[\left(\frac{N_{1-d}\mathbbm{1}_{N_{d}\geq
1}}{N_{d}}\right)^{r}\max_{j\in\mathcal{J}(i)}\left(\frac{F_{1-d}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}{F_{d}[\pi(X_{j})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}\frac{1+R_{1-d,i}}{1+R_{d,j}}\mathbbm{1}_{N_{1-d}\geq
1}\mathbbm{1}_{M_{j}\geq 1}\right)^{r}\right]$ (289)
$\displaystyle\lesssim\sum_{d\in\\{0,1\\}}\max_{i:D_{i}=d}\mathbb{E}\left[\left(\frac{N_{1-d}\mathbbm{1}_{N_{d}\geq
1}}{N_{d}}\right)^{r}\max_{j\in\mathcal{J}(i)}\left(\frac{1+R_{1-d,i}}{1+R_{d,j}}\mathbbm{1}_{N_{1-d}\geq
1}\mathbbm{1}_{M_{j}\geq 1}\right)^{r}\right],$ (290)

where the second line follows from Assumption 2: because $f_{0},f_{1}$ have
the same support, are bounded away from zero and infinity, we have, for $c\neq
0$,
$0<\frac{2c\inf_{p\in[\underline{p},\bar{p}]}f_{1-d}(p)}{2c\sup_{p\in[\underline{p},\bar{p}]}f_{d}(p)}\leq
F_{1-d}[a\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}c]/F_{d}[b\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}c]\leq\frac{2c\sup_{p\in[\underline{p},\bar{p}]}f_{1-d}(p)}{2c\inf_{p\in[\underline{p},\bar{p}]}f_{d}(p)}<\infty$.
We address the case $d=0$ in (290); $d=1$ follows by symmetry. For the
expectation in (290), two applications of the Cauchy–Schwarz inequality give

$\displaystyle\mathbb{E}\left[\left(\frac{N_{1}\mathbbm{1}_{N_{0}\geq
1}}{N_{0}}\right)^{r}\max_{j\in\mathcal{J}(i)}\left(\frac{1+R_{1i}}{1+R_{0j}}\mathbbm{1}_{N_{1}\geq
1}\mathbbm{1}_{M_{j}\geq 1}\right)^{r}\right]$
$\displaystyle\leq\sqrt{\mathbb{E}(N_{1}\mathbbm{1}_{N_{0}\geq
1}/N_{0})^{2r}\sqrt{\mathbb{E}(1+R_{1i})^{4r}\mathbbm{1}_{N_{1}\geq
1}\mathbb{E}\left(\max_{j\in\mathcal{J}(i)}(\mathbbm{1}_{M_{j}\geq
1}/(1+R_{0j}))^{r}\right)^{4}}}.$ (291)

Here $\mathbb{E}(N_{1}\mathbbm{1}_{N_{0}\geq 1}/N_{0})^{2r}\leq
n^{2r}\mathbb{E}(\mathbbm{1}_{N_{0}\geq 1}/N_{0})^{2r}$ with

$\displaystyle\mathbb{E}(\mathbbm{1}_{N_{0}\geq 1}/N_{0})^{2r}$
$\displaystyle=\sum_{n_{0}=1}^{n}{n\choose
n_{0}}(1-p_{1})^{n_{0}}p_{1}^{n-n_{0}}\left(\frac{1}{n_{0}}\right)^{2r}$
$\displaystyle=\sum_{n_{0}=0}^{n-1}\frac{n}{n_{0}+1}{n-1\choose
n_{0}}(1-p_{1})^{n_{0}+1}p_{1}^{n-(n_{0}+1)}\left(\frac{1}{n_{0}+1}\right)^{2r}$
$\displaystyle=n\frac{1-p_{1}}{p_{1}}\sum_{n_{0}=0}^{n-1}{n-1\choose
n_{0}}(1-p_{1})^{n_{0}}p_{1}^{n-n_{0}}\left(\frac{1}{n_{0}+1}\right)^{2r+1}.$

The last line is $O\left(n^{-2r}\right)$ by Cribari-Neto et al. (2000), thus
$\mathbb{E}(N_{1}\mathbbm{1}_{N_{0}\geq 1}/N_{0})^{2r}=O\left(1\right)$.

Next, we bound the second factor of (291). We have
$\mathbb{E}(1+R_{1i})^{4r}\mathbbm{1}_{N_{1}\geq 1}\leq
2^{4r}+c\mathbb{E}|R_{1i}|^{4r}\mathbbm{1}_{N_{1}\geq 1}$ for some constant
$c>0$, where $\max_{i:D_{i}=0}|R_{1i}|\leq W_{1}$ for $W_{1}$ in (126).
Because $W_{1}\geq 0$,

$\mathbb{E}\left[\left.W_{1}^{4r}\,\right|\,D^{(n)}\right]=\int_{0}^{\infty}\mathbb{P}\left(W_{1}^{4r}>w\mid
D^{(n)}\right)\mathrm{d}w=\int_{0}^{\infty}\mathbb{P}\left(W_{1}>w^{\frac{1}{4r}}\mid
D^{(n)}\right)\mathrm{d}w.$

By (269) of Lemma 1,

$\displaystyle\mathbb{E}\left[\left.W_{1}^{4r}\,\right|\,D^{(n)}\right]\leq$
$\displaystyle\frac{c_{0}}{N_{1}\gamma_{n}}\int_{0}^{\infty}\frac{1}{w^{\frac{2}{4r}}}e^{-w^{\frac{2}{4r}}N_{1}\gamma_{n}/1024}\mathrm{d}w+c_{1}\int_{0}^{\infty}e^{-w^{\frac{1}{4r}}N_{1}\gamma_{n}/256}\mathrm{d}w$
(292)
$\displaystyle+cN_{1}\int_{0}^{\infty}{w^{\frac{2}{4r}}}e^{-\frac{N_{1}w^{\frac{2}{4r}}}{2}}\mathrm{d}w.$
(293)

The integral in the first term of (292) is
$\frac{c_{0}}{N_{1}\gamma_{n}}\frac{1}{\lambda_{n,N_{1}}}\int_{0}^{\infty}t^{-1}\lambda_{n,N_{1}}e^{-\lambda_{n,N_{1}}t}t^{2r-1}\mathrm{d}t$,
where $\lambda_{n,N_{1}}\coloneqq\frac{N_{1}\gamma_{n}}{1024}$ is strictly
positive for $N_{1}\geq 1$. This integral is the $(2r-2)$th moment of an
Exponential$(\lambda_{n,N_{1}})$ variable, which is well-defined for $r\geq 2$
and for finite integer $r$ is bounded by
$\left(\frac{1}{\lambda_{n,N_{1}}}\right)^{2r-2}$ up to a constant factor.
Hence, the first term of (292) is bounded by
$c_{0}\left(\frac{1}{N_{1}\gamma_{n}}\right)^{2r}\simeq
c_{0}\left(\frac{n}{N_{1}\log n}\right)^{2r}$. Similar arguments show that the
second integrals of (292) and (293) are bounded by
$c_{1}\left(\frac{1}{N_{1}\gamma_{n}}\right)^{4r}$ and
$c\left(\frac{1}{N_{1}}\right)^{2r}$, respectively. Conclude that

$\displaystyle\mathbb{E}(1+R_{1i})^{4r}\mathbbm{1}_{N_{1}\geq 1}\leq
O\left(1\right)+c_{0}n^{2r}\mathbb{E}\left(\frac{\mathbbm{1}_{N_{1}\geq
1}}{N_{1}}\right)^{2r}+c_{1}n^{4r}\mathbb{E}\left(\frac{\mathbbm{1}_{N_{1}\geq
1}}{N_{1}}\right)^{4r}.$

By arguments bounding $\mathbb{E}(N_{1}\mathbbm{1}_{N_{0}\geq 1}/N_{0})^{2r}$
of (291), the right side is $O\left(1\right)$.

Finally, the last factor of (291) satisfies

$\displaystyle\max_{j\in\mathcal{J}(i)}(\mathbbm{1}_{M_{j}\geq
1}/(1+R_{0j}))^{r}=\max_{j\in\mathcal{J}(i)}\left(\frac{N_{0}F_{0}[\pi(X_{j})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}{M_{j}}\mathbbm{1}_{M_{j}\geq
1}\right)^{r}$
$\displaystyle\leq(2n\left\lVert{f_{0}}\right\rVert_{\infty}\delta_{n})^{r}$
(302) $\displaystyle\lesssim(\log n)^{r}.$

As $(\log n)^{r}=o\left(n^{\varrho}\right)$ for any $\varrho>0$, (272)
follows.

When the caliper is
$\delta_{n}=\overline{\underline{\Delta}}_{n}\vee\frac{\log
N_{0}}{N_{0}+1}\vee\frac{\log N_{1}}{N_{1}+1}$, (292) holds with
$\gamma_{n}\coloneqq 2\inf_{p\in[\underline{p},\bar{p}]}f_{1}(p)\frac{\log
N_{1}}{N_{1}+1}$ in view of the proof of Lemma 1, showing
$\mathbb{E}(1+R_{1i})^{4r}\mathbbm{1}_{N_{1}\geq 1}=O\left(1\right)$. Consider
(302), wherein $\delta_{n}^{r}\leq
2^{r}\left(\overline{\underline{\Delta}}_{n}^{r}+(2\log
n)^{r}\left(\frac{1}{N_{0}+1}\right)^{r}+\left(\frac{1}{N_{1}+1}\right)^{r}\right)$.
Here, $\mathbb{E}\left(\frac{1}{N_{d}+1}\right)^{r}=O\left(n^{-r}\right)$ by
Cribari-Neto et al. (2000). Arguments in the proof of Proposition 1 and Lemma
5 imply $n^{r}\mathbb{E}\overline{\underline{\Delta}}_{n}^{r}=O\left((\log
n)^{r}\right)$. Conclude that $n^{r}\mathbb{E}\delta_{n}^{r}=O\left((\log
n)^{r}\right)$ and hence (272) follows from (302). ∎

######  Lemma 4 (Semiparametric Efficiency).

If Assumptions 2 and 4 hold, and the caliper $\delta_{n}$ satisfies (3), then
$V_{E}\overset{P}{\longrightarrow}V_{\sigma,\pi}$ as $n\to\infty$.

###### Proof.

By Proposition 2, $\min_{i\in[n]}M_{i}>0$ with probability tending to one, so
with probability tending to one,

$\displaystyle V_{E}$
$\displaystyle=\frac{1}{n}\sum_{i\in[n]}(1+w_{i})^{2}\sigma_{D_{i}}^{2}(\pi(X_{i}))=\frac{1}{n}\sum_{i\in[n]}(1+2w_{i}+w_{i}^{2})\sigma_{D_{i}}^{2}(\pi(X_{i}))$
$\displaystyle=\mathbb{E}\sigma_{D}^{2}(\pi(X))+o_{P}\left(1\right)+\frac{2}{n}\sum_{i\in[n]}w_{i}\sigma_{D_{i}}^{2}(\pi(X_{i}))+\frac{1}{n}\sum_{i\in[n]}w_{i}^{2}\sigma_{D_{i}}^{2}(\pi(X_{i}))$
(303)

by the law of large of large numbers. In the notation of (17), we have, by
definition,

$\displaystyle\sum_{i\in[n]}w_{i}^{r}\sigma_{D_{i}}^{2}(\pi(X_{i}))=$
$\displaystyle\,\left(\frac{N_{1}}{N_{0}}\right)^{r}\sum_{i:D_{i}=0}\left(\frac{1}{N_{1}}\sum_{j\in\mathcal{J}(i)}\frac{1}{\mathbbm{F}_{N_{0}}[\pi(X_{j})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}\right)^{r}\sigma_{D_{i}}^{2}(\pi(X_{i}))$
(312)
$\displaystyle+\left(\frac{N_{0}}{N_{1}}\right)^{r}\sum_{i:D_{i}=1}\left(\frac{1}{N_{0}}\sum_{j\in\mathcal{J}(i)}\frac{1}{\mathbbm{F}_{N_{1}}[\pi(X_{j})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}\right)^{r}\sigma_{D_{i}}^{2}(\pi(X_{i})).$
(321)

Write
$\mathbbm{F}_{N_{0}}[\pi(X_{j})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]=(1+R_{0j})F_{0}[\pi(X_{j})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]$
and
$F_{0}[\pi(X_{j})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]=2\delta_{n}f_{0}(\pi(X_{j}))(1+\tilde{R}_{0j})$
for $R_{0j},\tilde{R}_{0j}$ of (56), (65). By Lemma 1,

$\displaystyle\max_{j:D_{j}=1}|R_{0j}|=o_{P}\left(1\right)\quad\text{ and
}\quad\max_{j:D_{j}=1}|\tilde{R}_{0j}|=o_{P}\left(1\right).$

Then we can write

$\displaystyle\frac{1}{N_{1}}\sum_{j\in\mathcal{J}(i)}\frac{1}{\mathbbm{F}_{N_{0}}[\pi(X_{j})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}$
$\displaystyle=\frac{1}{N_{1}}\sum_{j\in\mathcal{J}(i)}\frac{1}{F_{0}[\pi(X_{j})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}\frac{1}{1+R_{0j}}$
$\displaystyle=\frac{1}{N_{1}}\sum_{j\in\mathcal{J}(i)}\frac{1}{2\delta_{n}f_{0}(\pi(X_{j}))}\frac{1}{1+\tilde{R}_{0j}}\frac{1}{1+R_{0j}}$
$\displaystyle=\frac{1+o_{P}\left(1\right)}{N_{1}}\sum_{j\in\mathcal{J}(i)}\frac{1}{2\delta_{n}f_{0}(\pi(X_{j}))}$

where the $o_{P}\left(1\right)$ terms are uniform in $i\in[n]$. Write
$f_{0}(\pi(X_{j}))=f_{0}(\pi(X_{i}))(1+\check{R}_{0ji})$ for $\check{R}_{0ji}$
of (66). By Lemma 1,
$\max_{i:D_{i}=0}\max_{j\in\mathcal{J}(i)}|\check{R}_{0ji}|=o_{P}\left(1\right)$.
Then, by the continuous mapping theorem,

$\displaystyle\frac{1}{N_{1}}\sum_{j\in\mathcal{J}(i)}\frac{1}{2\delta_{n}f_{0}(\pi(X_{j}))}$
$\displaystyle=\frac{1}{N_{1}}\sum_{j\in\mathcal{J}(i)}\frac{1}{2\delta_{n}f_{0}(\pi(X_{i}))}\frac{1}{1+\check{R}_{0ji}}$
$\displaystyle=(1+o_{P}\left(1\right))\frac{f_{1}(\pi(X_{i}))}{f_{0}(\pi(X_{i}))}\frac{\mathbbm{F}_{N_{1}}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}{2\delta_{n}f_{1}(\pi(X_{i}))},$

because $\inf_{p\in[\underline{p},\bar{p}]}f_{1}(p)>0$ by Assumption 2. Write

$\frac{\mathbbm{F}_{N_{1}}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}{2\delta_{n}f_{1}(\pi(X_{i}))}=\frac{\mathbbm{F}_{N_{1}}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}{F_{1}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}\frac{F_{1}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}{2\delta_{n}f_{1}(\pi(X_{i}))},$

where
$\max_{i:D_{i}=0}\bigg{|}\frac{\mathbbm{F}_{N_{1}}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}{F_{1}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}-1\bigg{|}=o_{P}\left(1\right)$
and
$\max_{i:D_{i}=0}\bigg{|}\frac{F_{1}[\pi(X_{i})\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}\delta_{n}]}{2\delta_{n}f_{1}(\pi(X_{i}))}-1\bigg{|}=o_{P}\left(1\right)$
by Lemma 1. By symmetry, similar arguments apply to the second term of (321).
Then (321), divided by $n$, is equal to

$\displaystyle\frac{1+o_{P}\left(1\right)}{n}\left[\left(\frac{N_{1}}{N_{0}}\right)^{r}\sum_{i:D_{i}=0}\left(\frac{f_{1}(\pi(X_{i}))}{f_{0}(\pi(X_{i}))}\right)^{r}\sigma_{D_{i}}^{2}(\pi(X_{i}))\right.$
$\displaystyle\left.+\left(\frac{N_{0}}{N_{1}}\right)^{r}\sum_{i:D_{i}=1}\left(\frac{f_{0}(\pi(X_{i}))}{f_{1}(\pi(X_{i}))}\right)^{r}\sigma_{D_{i}}^{2}(\pi(X_{i}))\right]$
$\displaystyle\overset{P}{\longrightarrow}\mathbb{E}\left[(1-D)\left(\frac{p_{1}}{1-p_{1}}\frac{f_{1}(\pi(X))}{f_{0}(\pi(X))}\right)^{r}\sigma_{D}^{2}(\pi(X))\right]$
(322)
$\displaystyle+\mathbb{E}\left[D\left(\frac{1-p_{1}}{p_{1}}\frac{f_{0}(\pi(X))}{f_{1}(\pi(X))}\right)^{r}\sigma_{D}^{2}(\pi(X))\right],$
(323)

by the weak law of large numbers as
$(N_{d}/N_{1-d})^{r}\overset{a.s.}{\longrightarrow}\left(\frac{p_{d}}{1-p_{d}}\right)^{r}$
and the $f_{d}/f_{1-d}$ are uniformly bounded by Assumption 2 and the
$\sigma_{d}^{2}$ are bounded by Assumption 4. For $r=1$, (322) is

$\displaystyle\mathbb{E}\left[\left.\frac{p_{1}}{1-p_{1}}\frac{f_{1}(\pi(X))}{f_{0}(\pi(X))}\sigma_{0}^{2}(\pi(X))\,\right|\,D=0\right](1-p_{1})$
$\displaystyle=\int_{\underline{p}}^{\bar{p}}p_{1}\frac{f_{1}(p)}{f_{0}(p)}\sigma_{0}^{2}(p)f_{0}(p)\mathrm{d}p$
$\displaystyle=\mathbb{E}\pi(X)\sigma_{0}^{2}(\pi(X)),$

where we used that $f_{1}(p)=\frac{p}{p_{1}}f_{\pi(X)}(p)$, with $f_{\pi(X)}$
being the density of $\pi(X)$. Noting that
$f_{0}(p)=\frac{1-p}{1-p_{1}}f_{\pi(X)}(p)$, (323) is
$\mathbb{E}(1-\pi(X))\sigma_{1}^{2}(\pi(X))$. For $r=2$, the same arguments
yield $\mathbb{E}\frac{\pi(X)^{2}}{1-\pi(X)}\sigma_{0}^{2}(\pi(X))$ for (322),
and $\mathbb{E}\frac{(1-\pi(X))^{2}}{\pi(X)}\sigma_{1}^{2}(\pi(X))$ for (323).
Note that
$\mathbb{E}\sigma_{D}^{2}(\pi(X))=\mathbb{E}(1-\pi(X))\sigma_{0}^{2}(\pi(X))+\mathbb{E}\pi(X)\sigma_{1}^{2}(\pi(X))$.
Collect the terms to get the assertion, which also holds for
$\delta_{n}=\overline{\underline{\Delta}}_{n}\vee\frac{\log
N_{0}}{N_{0}+1}\vee\frac{\log N_{1}}{N_{1}+1}$ in view of Lemma 1. ∎

## Supplement

###### Proof of Proposition 3.

The semiparametric lower bound is the variance of the efficient influence
function. When the observed sample is $((Y_{i},D_{i},X_{i}))_{i\in[n]}$, the
efficient influence function of ATE is

$\displaystyle\chi_{\mathcal{X}}(Y,D,X)\coloneqq\frac{D(Y-\mu_{\mathcal{X}}^{1}(X))}{\pi(X)}-\frac{(1-D)(Y-\mu_{\mathcal{X}}^{0}(X))}{1-\pi(X)}+\mu_{\mathcal{X}}^{1}(X)-\mu_{\mathcal{X}}^{0}(X)-\tau,$

see Hahn (1998). Then $V=V_{\mathrm{eff}}$ follows under (10) in Proposition
3. One can verify that
$V=\operatorname*{\mathbb{V}}\left[\chi(Y,D,\pi(X))\right]$, where

$\displaystyle\chi(Y,D,\pi(X))\coloneqq$
$\displaystyle\,\frac{D(Y-\mu^{1}(\pi(X)))}{\pi(X)}-\frac{(1-D)(Y-\mu^{0}(\pi(X)))}{1-\pi(X)}$
$\displaystyle+\mu^{1}(\pi(X))-\mu^{0}(\pi(X))-\tau.$

Assumption 1 and $\mathbb{E}\left[\left.D\,\right|\,X\right]=\pi(X)$ imply
that

$\mathbb{E}\chi_{\mathcal{X}}(Y,D,X)(\chi(Y,D,\pi(X))-\chi_{\mathcal{X}}(Y,D,X))=0,$

thus

$\displaystyle V=\operatorname*{\mathbb{V}}\left[\chi(Y,D,\pi(X))\right]$
$\displaystyle=\operatorname*{\mathbb{V}}\left[\chi(Y,D,\pi(X))-\chi_{\mathcal{X}}(Y,D,X)\right]+\operatorname*{\mathbb{V}}\left[\chi_{\mathcal{X}}(Y,D,X)\right]$
$\displaystyle\geq\operatorname*{\mathbb{V}}\left[\chi_{\mathcal{X}}(Y,D,X)\right]=V_{\mathrm{eff}}.$

For ATT, $V_{\mathrm{t},\mathrm{eff}}\leq V_{\mathrm{t}}$ follows similarly as
the efficient influence function of ATT under unknown propensity score (Hahn,
1998) is

$\displaystyle\chi_{\mathrm{t},\mathcal{X}}(Y,D,X)\coloneqq$
$\displaystyle\,\frac{D}{p_{1}}(Y-\mu_{\mathcal{X}}^{1}(X))-\frac{1-D}{p_{1}}\frac{\pi(X)}{1-\pi(X)}(Y-\mu_{\mathcal{X}}^{0}(X))$
$\displaystyle+\frac{D}{p_{1}}(\mu_{\mathcal{X}}^{1}(X)-\mu_{\mathcal{X}}^{0}(X)-\tau_{\mathrm{t}}).$

Under Assumption 1, one verifies that
$V_{\mathrm{t}}=\operatorname*{\mathbb{V}}\left[\chi_{\mathrm{t}}(Y,D,\pi(X))\right]$,
where

$\displaystyle\chi_{\mathrm{t}}(Y,D,\pi(X))\coloneqq$
$\displaystyle\,\frac{D}{p_{1}}(Y-\mu^{1}(\pi(X)))-\frac{1-D}{p_{1}}\frac{\pi(X)}{1-\pi(X)}(Y-\mu^{0}(\pi(X)))$
$\displaystyle+\frac{D}{p_{1}}(\mu^{1}(\pi(X))-\mu^{0}(\pi(X))-\tau_{\mathrm{t}}),$

and that

$\mathbb{E}\chi_{\mathrm{t},\mathcal{X}}(Y,D,X)(\chi_{\mathrm{t}}(Y,D,\pi(X))-\chi_{\mathrm{t},\mathcal{X}}(Y,D,X))=0,$

thus proving the assertion.

∎

######  Lemma 5 (Moment Bounds of Ordered Uniform Spacings).

Let the order statistics of
$(U_{1},U_{2},\ldots,U_{n})\overset{\text{i.i.d.}}{\sim}\text{Uniform}(0,1)$
be $U_{(1)}\leq U_{(2)}\leq\ldots\leq U_{(n)}$. Let $\tilde{U}_{1}\coloneqq
U_{(1)}$, $\tilde{U}_{i}\coloneqq U_{(i)}-U_{(i-1)}$ for $i=2,\ldots,n$ and
$\tilde{U}_{n+1}\coloneqq 1-U_{(n)}$ be the spacings generated by
$(U_{i})_{i\in[n]}$. Let
$\tilde{U}_{(1)}\leq\tilde{U}_{(2)}\leq\ldots\leq\tilde{U}_{(n+1)}$ be the
ordered spacings. Then for any finite fixed integer $1\leq a<\frac{n+1}{2}$
and $n\geq 2$,

$\displaystyle\mathbb{E}\tilde{U}_{(r)}^{a}$
$\displaystyle=\begin{cases}O\left(\left(\frac{1}{n}\right)^{2a}\right)\quad&\text{
for }r=1\\\ O\left(\left(\frac{\log r}{n}\right)^{a}\right)\quad&\text{ for
all }r=2,3,\ldots,n+1.\end{cases}$

In particular, $\mathbb{E}\tilde{U}_{(r)}^{a}=o\left(1\right)$ for all
$r\in[n+1]$ and finite fixed integer $1\leq a<\frac{n+1}{2}$.

###### Proof.

By Shorack and Wellner (2009, Chapter 21), $\tilde{U}_{(r)}$ is distributed as
$\frac{Z_{r:n+1}}{\sum_{i\in[n+1]}Z_{i}}$, for
$(Z_{1},Z_{2},\ldots,Z_{n+1})\overset{\text{i.i.d.}}{\sim}\text{Exponential}(1)$
with order statistics $Z_{1:n+1}\leq Z_{2:n+1}\leq\ldots\leq Z_{n+1:n+1}$. The
Cauchy–Schwarz inequality gives

$\mathbb{E}\tilde{U}_{(r)}^{a}\leq\sqrt{\mathbb{E}Z_{r:n+1}^{2a}\mathbb{E}\left(\sum_{i\in[n+1]}Z_{i}\right)^{-2a}}.$

Here, $\mathbb{E}Z_{1:n+1}^{2a}=O\left(n^{-2a}\right)$ and
$\mathbb{E}Z_{r:n+1}^{2a}=O\left(\left(\log r\right)^{2a}\right)$ for $r\geq
2$ and for finite fixed integer $a\geq 1$ by Lemma 6. The sum of $n+1$ i.i.d.
$\text{Exponential}(1)$ variates follows a $\text{Gamma}(n+1,1)$ distribution.
Thus, $(\sum_{i\in[n+1]}Z_{i})^{-1}$ follows an $\text{Inverse-Gamma}(n+1,1)$
distribution, whose $2a$-th moment is equal to $\frac{(n-2a)!}{n!}\lesssim
n^{-2a}$ for $2a<n+1$, where the last inequality follows from
$\sqrt{2\pi}n^{n+1/2}e^{-n}\leq n!\leq en^{n+1/2}e^{-n}$ (Robbins, 1955),
because $n-2a+1/2\geq 0$ for a positive integer $a$. ∎

######  Lemma 6 (Moments of Exponential$(1)$ Order Statistics).

Let $(Z_{1},Z_{2},\ldots,Z_{n})\overset{\text{i.i.d.}}{\sim}$ Exponential(1)
with order statistics $Z_{1:n}\leq Z_{2:n+1}\leq\ldots\leq Z_{n:n}$. Then for
finite fixed integer $k\geq 1$, we have for all $n\geq 2$,

$\displaystyle\mathbb{E}Z_{r:n}^{k}=k!\sum_{t_{1}=1}^{r}\frac{1}{n+1-t_{1}}\sum_{t_{2}=1}^{t_{1}}\frac{1}{n+1-t_{2}}\cdots\sum_{t_{k-1}=1}^{t_{k-2}}\frac{1}{n+1-t_{k-1}}\sum_{t_{k}=1}^{t_{k-1}}\frac{1}{n+1-t_{k}}$
(324)

for all $r=1,2,\ldots,n$. The right side in (324) is $O\left((\log
r)^{k}\right)$ for $r\geq 2$, and
$\mathbb{E}Z_{1:n}^{k}=O\left(n^{-k}\right)$.

###### Proof.

The right side of (324) and $\mathbb{E}Z_{1:n}^{k}=O\left(n^{-k}\right)$ are
obtained by solving the recursion in Balakrishnan and Gupta (1998, Theorems 1
and 2). The innermost sum in (324) satisfies
$\sum_{t_{k}=1}^{t_{k-1}}\frac{1}{n+1-t_{k}}\leq\sum_{j=1}^{r}\frac{1}{n+1-j}$
as $t_{k}\leq r$. Because every fraction in (324) is positive, we can upper
bound the right side of (324) by

$k!\left(\sum_{j=1}^{r}\frac{1}{n+1-j}\right)\left(\sum_{t_{1}=1}^{r}\frac{1}{n+1-t_{1}}\sum_{t_{2}=1}^{t_{1}}\frac{1}{n+1-t_{2}}\cdots\sum_{t_{k-1}=1}^{t_{k-2}}\frac{1}{n+1-t_{k-1}}\right).$

Apply the same bound for the remaining $k-1$ sums noting that $1\leq t_{1}\leq
t_{1}\leq\ldots\leq t_{k-1}\leq r$, to obtain the bound
$k!\left(\sum_{j=1}^{r}\frac{1}{n+1-j}\right)^{k}$ on (324). The proof is
complete as $\sum_{j=1}^{r}\frac{1}{n+1-j}$ is $O\left(\log r\right)$ for
$r\in[n]$ as $n\to\infty$. ∎

######  Lemma 7 (Conditional Martingale Central Limit Theorem).

Let $(\Omega_{n},\mathcal{F}_{n},\mathbb{P}_{n})$ be a sequence of probability
spaces. Let $\xi_{n1},\xi_{n2},\ldots,\xi_{nn}:\Omega_{n}\to\real$ be
martingale differences with respect to sub-$\sigma$-algebras
$\mathcal{F}_{n1}\subset\mathcal{F}_{n2}\subset\ldots\subset\mathcal{F}_{nn}\subset\mathcal{F}_{n}$.
Let $\mathcal{F}_{n0}\subset\mathcal{F}_{n1}$ be a sub-$\sigma$-algebra. For
$k=1,2,\ldots,n$, let
$\sigma_{nk}^{2}\coloneqq{\mathbb{E}}_{n}\left[\left.\xi_{nk}^{2}\,\right|\,\mathcal{F}_{n,k-1}\right]$.
If there exists a finite constant $\sigma>0$ such that

$\displaystyle\mathbb{P}_{n}\left(\left.\bigg{|}\sum_{k=1}^{n}\sigma_{nk}^{2}-\sigma^{2}\bigg{|}>\epsilon\,\right|\,\mathcal{F}_{n0}\right)\overset{\mathbb{P}_{n}}{\longrightarrow}0\quad\text{for
all constants $\epsilon>0$ and}$ (325)
$\displaystyle\sum_{k=1}^{n}{\mathbb{E}}_{n}\left[\left.\xi_{nk}^{2}\mathbbm{1}_{|\xi_{nk}|\geq\eta}\,\right|\,\mathcal{F}_{n0}\right]\overset{\mathbb{P}_{n}}{\longrightarrow}0\quad\text{for
all constants $\eta>0$,}$ (326)

then $\mathbb{P}_{n}\left(\left.\sigma^{-1}\sum_{k=1}^{n}\xi_{nk}\leq
x\,\right|\,\mathcal{F}_{n0}\right)\overset{\mathbb{P}_{n}}{\longrightarrow}\Phi(x)$
as $n\to\infty$ for all $x\in\real$, where $\Phi$ is the standard normal
distribution function.

###### Proof.

Follows from Billingsley (1995, Theorem 35.12) by conditioning on
$\mathcal{F}_{n0}$ throughout. ∎

###### Proof of Proposition 7.

_Existence of $A_{n},\hat{A}_{n}$_. We show that $A_{n},\hat{A}_{n}$ are well-
defined with probability tending to one. For $A_{n}$, this happens if and only
if

$\mathbb{P}\left(\underline{p}_{\hat{\theta}}+a_{n}<\bar{p}_{\hat{\theta}}-a_{n}\right)=\mathbb{P}\left(2a_{n}<\bar{p}_{\hat{\theta}}-\underline{p}_{\hat{\theta}}\right)\to
1.$

If $\underline{p}_{\hat{\theta}}=\underline{p}+o_{P}\left(1\right)$ and
$\bar{p}_{\hat{\theta}}=\bar{p}+o_{P}\left(1\right)$, the probability is
$\mathbb{P}\left(2a_{n}<\bar{p}-\underline{p}+o_{P}\left(1\right)\right)=\mathbb{P}\left(2<a_{n}^{-1}(\bar{p}-\underline{p})+o_{P}\left(a_{n}^{-1}\right)\right)$,
which goes to one as $\bar{p}>\underline{p}$ and $a_{n}\downarrow 0$. To show
that these conditions hold, define
$T_{n}(x)\coloneqq\hat{\theta}^{\intercal}x$ and
$T(x)\coloneqq\theta_{0}^{\intercal}x$, so by definition
$\bar{p}=\sup_{x\in\mathcal{X}}g(T(x))=g(\sup_{x\in\mathcal{X}}T(x))$ and
$\bar{p}_{\hat{\theta}}=\sup_{x\in\mathcal{X}}g(T_{n}(x))=g(\sup_{x\in\mathcal{X}}T_{n}(x))$
because $g$ is increasing by Assumption 11. Since $g$ is continuous by
Assumption 11, it suffices by the continuous mapping theorem to show
$\sup_{x\in\mathcal{X}}T_{n}(x)\overset{P}{\longrightarrow}\sup_{x\in\mathcal{X}}T(x)$.
Because $\mathcal{X}$ is bounded and
$\hat{\theta}\overset{P}{\longrightarrow}\theta_{0}$ by Assumption 9,
$\sup_{x\in\mathcal{X}}|T_{n}(x)-T(x)|\lesssim\lVert{\hat{\theta}-\theta_{0}}\rVert=o_{P}\left(1\right)$,
so that $\bar{p}_{\hat{\theta}}=\bar{p}+o_{P}\left(1\right)$. Similar
arguments yield
$\underline{p}_{\hat{\theta}}=\underline{p}+o_{P}\left(1\right)$.

For $\hat{A}_{n}$, the desired result follows from that for $A_{n}$ above, and
that
$\min_{i\in[n]}g(\hat{\theta}^{\intercal}X_{i})=\underline{p}_{\hat{\theta}}+o_{P}\left(1\right)$
and
$\max_{i\in[n]}g(\hat{\theta}^{\intercal}X_{i})=\bar{p}_{\hat{\theta}}+o_{P}\left(1\right)$.
Because $F_{\hat{\theta}}^{-1}$, the inverse of
$F_{\hat{\theta}}(p)=p_{1}F_{1,\hat{\theta}}(p)+(1-p_{1})F_{0,\hat{\theta}}(p)$
which is the distribution of $(\pi(X,\hat{\theta})\mid\hat{\theta})$ under
Assumption 9, is strictly increasing by Assumption 6,
$(\min_{i\in[n]}\pi(X_{i},\hat{\theta})\mid\hat{\theta})$ is distributed as
$F_{\hat{\theta}}^{-1}(U_{(1)})$, where $U_{(1)}$ is the sample minimum of
$((U_{1},U_{2},\ldots,U_{n})\mid\hat{\theta})\overset{\text{i.i.d.}}{\sim}\text{Uniform}(0,1)$.
Then
$(\min_{i\in[n]}\pi(X_{i},\hat{\theta})-\underline{p}_{\hat{\theta}}\mid\hat{\theta})$
is distributed as
$F_{\hat{\theta}}^{-1}(U_{(1)})-F_{\hat{\theta}}^{-1}(F_{\hat{\theta}}(\underline{p}_{\hat{\theta}}))=F_{\hat{\theta}}^{-1}(U_{(1)})-F_{\hat{\theta}}^{-1}(0)$
given $\hat{\theta}.$ By Assumption 6, $F_{\hat{\theta}}^{-1}$ is Lipschitz
with constant
$\left\lVert{(F_{\hat{\theta}}^{-1})^{\prime}}\right\rVert_{\infty}$ with
$(F_{\hat{\theta}}^{-1})^{\prime}(u)=\frac{1}{f_{\hat{\theta}}(F_{\hat{\theta}}^{-1}(u))}$
finite for
$\inf_{p\in\underline{p}_{\hat{\theta}},\bar{p}_{\hat{\theta}}}f_{\hat{\theta}}(p)>0$
by Assumption 6 for $\hat{\theta}\in\textup{Nb}(\theta_{0},\epsilon)$. Thus,
$F_{\hat{\theta}}^{-1}(U_{(1)})-F_{\hat{\theta}}^{-1}(0)\lesssim U_{(1)}$.
Here, $\mathbb{E}U_{(1)}$ goes to zero by the proof of Proposition 1. Hence,
by Assumption 11,
$\min_{i\in[n]}g(\hat{\theta}^{\intercal}X_{i})=\underline{p}_{\hat{\theta}}+o_{P}\left(1\right)$
and
$\max_{i\in[n]}g(\hat{\theta}^{\intercal}X_{i})=\bar{p}_{\hat{\theta}}+o_{P}\left(1\right)$
similarly. In the following, we prove the consistency of the variance
component estimators for $V_{\hat{\pi}}$; similar arguments give the result
for $V_{\mathrm{t},\hat{\pi}}$. We show below that
$\hat{N}/n\overset{P}{\longrightarrow}1$ (since (331) is
$o_{P}\left(1\right)$). Therefore, in the following, we prove the consistency
of the estimators
$\hat{V}_{\tau},\hat{V}_{\tau_{\mathrm{t}}},\hat{V}_{\sigma,\pi},\hat{V}_{\mathrm{t},\sigma,\pi},\hat{q}_{d}$
and $\hat{q}_{\mathrm{t},d}$ normalised by $n$ rather than $\hat{N}$.

_Consistency of $\hat{V}_{\tau}$._ By Theorem 3 and the continuous mapping
theorem, $(\hat{\tau}_{\hat{\pi}})^{2}\overset{P}{\longrightarrow}\tau^{2}$.
For short, put $\pi_{i}\coloneqq g(\theta_{0}^{\intercal}X_{i})$,
$\hat{\pi}_{i}\coloneqq g(\hat{\theta}^{\intercal}X_{i})$ and
$\hat{\mathbbm{1}}_{i}\coloneqq\mathbbm{1}_{g(\hat{\theta}^{\intercal}X_{i})\in\hat{A}_{n}}$.
The first term in (16) is

$\displaystyle\frac{1}{n}\sum_{i\in[n]}[\mu^{1}(\hat{\theta},\hat{\pi}_{i})-\mu^{0}(\hat{\theta},\hat{\pi}_{i})]^{2}\hat{\mathbbm{1}}_{i}$
(327)
$\displaystyle+\frac{1}{n}\sum_{i\in[n]}[\hat{\mu}^{1}(\hat{\theta},\hat{\pi}_{i})-\hat{\mu}^{0}(\hat{\theta},\hat{\pi}_{i})-(\mu^{1}(\hat{\theta},\hat{\pi}_{i})-\mu^{0}(\hat{\theta},\hat{\pi}_{i}))]^{2}\hat{\mathbbm{1}}_{i}$
(328)
$\displaystyle+\frac{2}{n}\sum_{i\in[n]}[\hat{\mu}^{1}(\hat{\theta},\hat{\pi}_{i})-\hat{\mu}^{0}(\hat{\theta},\hat{\pi}_{i})-(\mu^{1}(\hat{\theta},\hat{\pi}_{i})-\mu^{0}(\hat{\theta},\hat{\pi}_{i}))][\mu^{1}(\hat{\theta},\hat{\pi}_{i})-\mu^{0}(\hat{\theta},\hat{\pi}_{i})]\hat{\mathbbm{1}}_{i}.$
(329)

Here, (327) converges to
$\mathbb{E}(\mu^{1}(\theta_{0},\pi_{i})-\mu^{0}(\theta_{0},\pi_{i}))^{2}$. To
see this, first note that under Assumption 9,

$\displaystyle\bigg{|}\frac{1}{n}\sum_{i\in[n]}[\mu^{1}(\hat{\theta},\hat{\pi}_{i})-\mu^{0}(\hat{\theta},\hat{\pi}_{i})]^{2}\hat{\mathbbm{1}}_{i}-\frac{1}{n}\sum_{i\in[n]}[\mu^{1}(\theta_{0},\pi_{i})-\mu^{0}(\theta_{0},\pi_{i})]^{2}\hat{\mathbbm{1}}_{i}\bigg{|}\overset{P}{\longrightarrow}0,$

which follows from a mean-value expansion of
$[\mu^{1}(\hat{\theta},\hat{\pi}_{i})-\mu^{0}(\hat{\theta},\hat{\pi}_{i})]^{2}$
in $\hat{\theta}$, similarly to the treatment of (34) in the proof of Theorem
3. Second, as the $\mu^{d}(\theta,\cdot)$,
$\theta\in\textup{Nb}(\theta_{0},\epsilon)$, are bounded by Assumption 5,

$\displaystyle\bigg{|}\frac{1}{n}\sum_{i\in[n]}[\mu^{1}(\theta_{0},\pi_{i})-\mu^{0}(\theta_{0},\pi_{i})]^{2}\hat{\mathbbm{1}}_{i}-\frac{1}{n}\sum_{i\in[n]}[\mu^{1}(\theta_{0},\pi_{i})-\mu^{0}(\theta_{0},\pi_{i})]^{2}\bigg{|}$
(330)

is of the order

$\displaystyle\frac{1}{n}\sum_{i\in[n]}(1-\hat{\mathbbm{1}}_{i})=$
$\displaystyle\,\frac{1}{n}\sum_{i\in[n]}\left(\mathbbm{1}_{g(\hat{\theta}^{\intercal}X_{i})\notin\hat{A}_{n}}-\mathbb{E}\left[\left.\mathbbm{1}_{g(\hat{\theta}^{\intercal}X_{i})\notin\hat{A}_{n}}\,\right|\,\hat{\theta}\right]\right)$
$\displaystyle+\frac{1}{n}\sum_{i\in[n]}\mathbb{E}\left[\left.\mathbbm{1}_{g(\hat{\theta}^{\intercal}X_{i})\notin\hat{A}_{n}}\,\right|\,\hat{\theta}\right].$
(331)

with probability tending to one. Here the first term has mean zero and
variance bounded by $1/n$, so it converges to zero in the first mean, and then
so in probability. By Assumption 9, the second term of (331) is

$\displaystyle\mathbb{P}\left(\left.g(\hat{\theta}^{\intercal}X_{i})\notin\hat{A}_{n}\,\right|\,\hat{\theta}\right)=$
$\displaystyle\mathbb{P}\left(\left.g(\hat{\theta}^{\intercal}X_{i})<\min_{i\in[n]}g(\hat{\theta}^{\intercal}X_{i})+a_{n}\,\right|\,\hat{\theta}\right)$
(332)
$\displaystyle+\mathbb{P}\left(\left.\max_{i\in[n]}g(\hat{\theta}^{\intercal}X_{i}))-a_{n}<g(\hat{\theta}^{\intercal}X_{i})\,\right|\,\hat{\theta}\right)$
(333)

for some $i\in[n]$. Let
$\underline{G}_{\hat{\theta}}\coloneqq\min_{i\in[n]}g(\hat{\theta}^{\intercal}X_{i})$.
To bound (332), we have, by Shanmugam and Arnold (1988),

$\displaystyle\mathbb{P}\left(\left.g(\hat{\theta}^{\intercal}X_{i})\leq\underline{G}_{\hat{\theta}}+a_{n}\,\right|\,\hat{\theta},\underline{G}_{\hat{\theta}}\right)=\frac{1}{n}+\frac{n-1}{n}\frac{F_{\hat{\theta}}(\underline{G}_{\hat{\theta}}+a_{n})-F_{\hat{\theta}}(\underline{G}_{\hat{\theta}})}{1-F_{\hat{\theta}}(\underline{G}_{\hat{\theta}})}$
(334)

under Assumptions 9 and 11, where $F_{\hat{\theta}}\coloneqq
p_{1}F_{\hat{\theta},1}+(1-p_{1})F_{\hat{\theta},0}$ is the distribution
function of $g(\hat{\theta}^{\intercal}X)$ given $\hat{\theta}$. By Assumption
6, $F_{\hat{\theta}}$ is continuous, and by arguments on $\hat{A}_{n}$ above
$\underline{G}_{\hat{\theta}}=\underline{p}_{\hat{\theta}}+o_{P}\left(1\right)$.
Then $a_{n}\downarrow 0$ implies that (334) is $o_{P}\left(1\right)$, which in
turn implies that (332), being bounded by one, is also $o_{P}\left(1\right)$.
Term (333) is $o_{P}\left(1\right)$ by similar arguments, noting that
$\left\\{\max_{i\in[n]}g(\hat{\theta}^{\intercal}X_{i}))-a_{n}<g(\hat{\theta}^{\intercal}X_{i})\right\\}=\left\\{\min_{i\in[n]}-g(\hat{\theta}^{\intercal}X_{i}))+a_{n}>-g(\hat{\theta}^{\intercal}X_{i})\right\\}$.
Thus, (331) is $o_{P}\left(1\right)$. Conclude that (327) converges in
probability to $V_{\tau}+\tau^{2}$. Write (328) as

$\displaystyle\frac{1}{n}\sum_{i\in[n]}[\hat{\mu}^{1}(\hat{\theta},\hat{\pi}_{i})-\mu^{1}(\hat{\theta},\hat{\pi}_{i})]^{2}\hat{\mathbbm{1}}_{i}+\frac{2}{n}\sum_{i\in[n]}(\hat{\mu}^{1}(\hat{\theta},\hat{\pi}_{i})-\mu^{1}(\hat{\theta},\hat{\pi}_{i}))(\hat{\mu}^{0}(\hat{\theta},\hat{\pi}_{i})-\mu^{0}(\hat{\theta},\hat{\pi}_{i}))\hat{\mathbbm{1}}_{i}$
$\displaystyle+\frac{1}{n}\sum_{i\in[n]}[\hat{\mu}^{0}(\hat{\theta},\hat{\pi}_{i})-\mu^{0}(\hat{\theta},\hat{\pi}_{i})]^{2}\hat{\mathbbm{1}}_{i}.$

As Assumption 10(i) bounds both $\hat{\mu}^{d}$ and $\mu^{d}$, $\sup_{p\in
A_{n}}|\hat{\mu}^{d}(\hat{\theta},p)-\mu^{d}(\hat{\theta},p)|\overset{P}{\longrightarrow}0$
implies that all three terms in the last display are $o_{P}\left(1\right)$
(note that $\hat{A}_{n}\subset A_{n}$). This convergence can be established
along the same lines as that of other estimators, which are detailed below.
Similar arguments show that (329) is $o_{P}\left(1\right)$. Conclude that
$\hat{V}_{\tau}\overset{P}{\longrightarrow}V_{\tau}$.

_Consistency of $\hat{V}_{\sigma,\pi}$._ First,
$0<g(\hat{\theta}^{\intercal}x)<1$ for all $x$ in compact $\mathcal{X}$ and
for all $\hat{\theta}\in\textup{Nb}(\theta_{0},\epsilon)$. Then under
Assumption 11, a mean-value expansion and
$\hat{\theta}\overset{P}{\longrightarrow}\theta_{0}$, implied by Assumption 9,
gives that
$\sup_{x\in\mathcal{X}}|1/g(\hat{\theta}^{\intercal}x)-1/g(\theta_{0}^{\intercal}x)|=o_{P}\left(1\right)$
and similarly for $1/(1-g(\hat{\theta}^{\intercal}x))$. Second, the Lipschitz
condition in Assumption 8(i) implies

$\displaystyle\bigg{|}\frac{1}{n}\sum_{i\in[n]}\sigma_{d}^{2}(\hat{\theta},\hat{\pi}_{i})\hat{\mathbbm{1}}_{i}-\frac{1}{n}\sum_{i\in[n]}\sigma_{d}^{2}(\theta_{0},\pi_{i})\hat{\mathbbm{1}}_{i}\bigg{|}\overset{P}{\longrightarrow}0,$

and, as in (330) above, we also have

$\displaystyle\bigg{|}\frac{1}{n}\sum_{i\in[n]}\sigma_{d}^{2}(\theta_{0},\pi_{i})\hat{\mathbbm{1}}_{i}-\frac{1}{n}\sum_{i\in[n]}\sigma_{d}^{2}(\theta_{0},\pi_{i})\bigg{|}\overset{P}{\longrightarrow}0.$

Thus, if $\sup_{p\in
A_{n}}|\hat{\sigma}_{d}^{2}(\hat{\theta},p)-\sigma_{d}^{2}(\hat{\theta},p)|\overset{P}{\longrightarrow}0$,
then the law of large numbers gives that
$\hat{V}_{\sigma,\pi}\overset{P}{\longrightarrow}V_{\sigma,\pi}$. This holds
if both $\sup_{p\in
A_{n}}|\hat{\mu}^{d}(\hat{\theta},p)-\mu^{d}(\hat{\theta},p)|\overset{P}{\longrightarrow}0$
and $\sup_{p\in
A_{n}}|\hat{\mu}_{2}^{d}(\hat{\theta},p)-\mu_{2}^{d}(\hat{\theta},p)|\overset{P}{\longrightarrow}0$
since the former implies $\sup_{p\in
A_{n}}|(\hat{\mu}^{d}(\hat{\theta},p))^{2}-(\mu^{d}(\hat{\theta},p))^{2}|\overset{P}{\longrightarrow}0$
under Assumption 10(i). See below for a proof of such convergence (e.g.
establishing (372)).

_Consistency of $\hat{q}_{d}$._ By definition of $q_{d}$, the conditions

$\displaystyle\lVert{\frac{1}{n}\sum_{i\in[n]}\Lambda^{d}(\hat{\theta},X_{i})-\frac{1}{n}\sum_{i\in[n]}\Lambda^{d}(\hat{\theta},X_{i})\hat{\mathbbm{1}}_{i}}\rVert\overset{P}{\longrightarrow}0$
(335) $\displaystyle\sup_{p\in
A_{n}}\bigg{|}\widehat{\left(\frac{\partial\mu^{d}}{\partial
p}\right)}(\hat{\theta},p)-\frac{\partial\mu^{d}}{\partial
p}(\hat{\theta},p)\bigg{|}\overset{P}{\longrightarrow}0$ (336)
$\displaystyle\sup_{p\in
A_{n}}\bigg{|}\widehat{\left(\frac{\partial\mu^{d}}{\partial\theta_{k}}\right)}(\hat{\theta},p)-\frac{\partial\mu^{d}}{\partial\theta_{k}}(\hat{\theta},p)\bigg{|}\overset{P}{\longrightarrow}0\quad\text{for
all }k=1,2,\ldots,K,$ (337)

together with Assumptions 10 and 11, implying the boundedness of $\mathcal{X}$
and $g^{\prime}$, ensure $\hat{q}_{d}\overset{P}{\longrightarrow}q_{d}$.
Assumption 7, 11(i), and Assumption 10 imply that $\Lambda^{d}$ is bounded
uniformly. Therefore, (335) is satisfied, because the arguments treating
$\hat{\mathbbm{1}}_{i}$ in the case of $\hat{V}_{\tau}$ above (e.g. (330))
apply.

In the following, we show that (336) and (337) hold, wherein we also give a
detailed proof of the uniform consistency of the nonparametric estimators
assumed above. For simplicity of exposition, we only give the proof for the
(derivatives of)
$\mu(\theta,p)\coloneqq\mathbb{E}\left[\left.Y\,\right|\,g(\theta^{\intercal}X)=p\right]=\mathbb{E}\left[\left.Y\,\right|\,\theta^{\intercal}X=g^{-1}(p)\right]$
and
$\mu_{2}(\theta,p)\coloneqq\mathbb{E}\left[\left.Y^{2}\,\right|\,g(\theta^{\intercal}X)=p\right]$.
The proof when we also condition on $D=d$ follows along the same lines. To
this end, let $h(\theta,p)\coloneqq
p_{1}h_{1}(\theta,p)+(1-p_{1})h_{0}(\theta,p)$ be the density of
$\pi(X,\theta)$, $q_{\mu}(\theta,p)\coloneqq\mu(\theta,p)h(\theta,p)$,
$q_{\mu_{2}}(\theta,p)\coloneqq\mu_{2}(\theta,p)h(\theta,p)$ and their
corresponding estimators be obtained by setting
$\mathbbm{1}_{D_{j}=d}\coloneqq 1$ for all $j\in[n]$ in the formulae for
$\hat{h}_{d}$, $\hat{q}_{\mu,d}$, $\hat{q}_{\mu_{2},d}$, respectively. For
short, we also let $h^{\prime}(\theta,p)\coloneqq(\partial/\partial
p)h(\theta,p)$, $\hat{h}^{\prime}(\theta,p)\coloneqq(\partial/\partial
p)\hat{h}(\theta,p)$ and likewise for
$q_{\mu}^{\prime},\hat{q}_{\mu}^{\prime}$. Furthermore, we let
$\widehat{\left(\frac{\partial h}{\partial\theta_{k}}\right)}$ and
$\widehat{\left(\frac{\partial q_{\mu}}{\partial\theta_{k}}\right)}$ be
obtained by setting $\mathbbm{1}_{D_{j}=d}\coloneqq 1$ in the formula for
$\widehat{\left(\frac{\partial h_{d}}{\partial\theta_{k}}\right)}$ and
$\widehat{\left(\frac{\partial q_{\mu,d}}{\partial\theta_{k}}\right)}$,
respectively. For short, we put
$\dot{h}_{k}(\theta,p)\coloneqq(\partial/\partial\theta_{k})h(\theta,p)$,
$\hat{\dot{h}}_{k}(\theta,p)\coloneqq\widehat{\left(\frac{\partial
h}{\partial\theta_{k}}\right)}(\theta,p)$ and likewise for
$\dot{q}_{\mu,k}(\theta,p),\hat{\dot{q}}_{\mu,k}(\theta,p)$. For a function
$r:\Theta\times[0,1]\to\real$, we let
$\lVert{r}\rVert_{A_{n}}\coloneqq\sup_{p\in A_{n}}|r(\hat{\theta},p)|$.

_Condition ( 336)._ First we show that the numerator of

$\displaystyle\widehat{\left(\frac{\partial\mu}{\partial
p}\right)}(\hat{\theta},p)\coloneqq\frac{\hat{q}_{\mu}^{\prime}(\hat{\theta},p)\hat{h}(\hat{\theta},p)-\hat{q}_{\mu}(\hat{\theta},p)\hat{h}^{\prime}(\hat{\theta},p)}{(\hat{h}(\hat{\theta},p))^{2}}$
(338)

converges to that of

$\displaystyle\frac{\partial\mu}{\partial
p}(\hat{\theta},p)=\frac{q_{\mu}^{\prime}(\hat{\theta},p)h(\hat{\theta},p)-q_{\mu}(\hat{\theta},p)h^{\prime}(\hat{\theta},p)}{h(\hat{\theta},p)^{2}}.$
(339)

Consider

$\displaystyle\lVert{\hat{q}_{\mu}^{\prime}\hat{h}-\hat{q}_{\mu}\hat{h}^{\prime}-(q_{\mu}^{\prime}h-q_{\mu}h^{\prime})}\rVert_{A_{n}}\leq$
$\displaystyle\,\lVert{\hat{q}_{\mu}^{\prime}\hat{h}-q_{\mu}^{\prime}h}\rVert_{A_{n}}+\lVert{\hat{q}_{\mu}\hat{h}^{\prime}-q_{\mu}h^{\prime}}\rVert_{A_{n}}$
$\displaystyle\lVert{\hat{q}_{\mu}^{\prime}\hat{h}-q_{\mu}^{\prime}h}\rVert_{A_{n}}=$
$\displaystyle\,\lVert{(\hat{q}_{\mu}^{\prime}-q_{\mu}^{\prime}+q_{\mu}^{\prime})(\hat{h}-h+h)-q_{\mu}^{\prime}h}\rVert_{A_{n}}$
$\displaystyle\leq$
$\displaystyle\,\lVert{\hat{q}^{\prime}_{\mu}-q_{\mu}^{\prime}}\rVert_{A_{n}}\lVert{\hat{h}-h}\rVert_{A_{n}}+\lVert{\hat{q}_{\mu}^{\prime}-q_{\mu}^{\prime}}\rVert_{A_{n}}\lVert{h}\rVert_{A_{n}}$
$\displaystyle+\lVert{q^{\prime}_{\mu}}\rVert_{A_{n}}\lVert{\hat{h}-h}\rVert_{A_{n}}$
$\displaystyle\lVert{\hat{q}_{\mu}\hat{h}^{\prime}-q_{\mu}h^{\prime}}\rVert_{A_{n}}\leq$
$\displaystyle\,\lVert{\hat{h}^{\prime}-h^{\prime}}\rVert_{A_{n}}\lVert{\hat{q}_{\mu}-q_{\mu}}\rVert_{A_{n}}+\lVert{\hat{h}^{\prime}-h^{\prime}}\rVert_{A_{n}}\lVert{q_{\mu}}\rVert_{A_{n}}$
$\displaystyle+\lVert{h^{\prime}}\rVert_{A_{n}}\lVert{\hat{q}_{\mu}-q_{\mu}}\rVert_{A_{n}}.$

By Assumption 6(iv), $\lVert{h}\rVert_{A_{n}}$ is bounded with probability
tending to one as $\hat{\theta}\overset{P}{\longrightarrow}\theta_{0}$ and
combining it with Assumptions 7 and 11(ii), the same holds for
$\lVert{h^{\prime}}\rVert_{A_{n}}$, $\lVert{q_{\mu}}\rVert_{A_{n}}=\lVert{\mu
h}\rVert_{A_{n}}$ and
$\lVert{q_{\mu}^{\prime}}\rVert_{A_{n}}=\lVert{\mu^{\prime}h+\mu
h^{\prime}}\rVert_{A_{n}}$. It follows that if all

$\displaystyle\lVert{\hat{q}^{\prime}_{\mu}-q_{\mu}^{\prime}}\rVert_{A_{n}}$
$\displaystyle\overset{P}{\longrightarrow}0,$ (340)
$\displaystyle\lVert{\hat{h}^{\prime}-h^{\prime}}\rVert_{A_{n}}$
$\displaystyle\overset{P}{\longrightarrow}0,$ (341)
$\displaystyle\lVert{\hat{q}_{\mu}-q_{\mu}}\rVert_{A_{n}}$
$\displaystyle\overset{P}{\longrightarrow}0,$ (342)
$\displaystyle\lVert{\hat{h}-h}\rVert_{A_{n}}$
$\displaystyle\overset{P}{\longrightarrow}0,$ (343)

then the numerator of (338) converges to that of (339) uniformly in $p\in
A_{n}$. The denominator of (338) satisfies

$\displaystyle\lVert{(\hat{h})^{2}-h^{2}}\rVert_{A_{n}}$
$\displaystyle=\lVert{(\hat{h}-h)(\hat{h}+h+h-h)}\rVert_{A_{n}}\leq\lVert{\hat{h}-h}\rVert_{A_{n}}\left\\{\lVert{\hat{h}-h}\rVert_{A_{n}}+2\lVert{h}\rVert_{A_{n}}\right\\}.$

By Assumption 6(iv), $\lVert{h}\rVert_{A_{n}}<\infty$ with probability tending
to one as $\hat{\theta}\overset{P}{\longrightarrow}\theta_{0}$, hence (343)
implies that the denominator of (338) converges to that of (339) uniformly in
$p\in A_{n}$. Thus, both the numerator and the denominator of (338) converges
to those of (339). Because
$\inf_{p\in[\underline{p}_{\hat{\theta}},\bar{p}_{\hat{\theta}}]}h(\hat{\theta},p)>0$
for all $\hat{\theta}\in\textup{Nb}(\theta_{0},\epsilon)$, it follows that
(338) converges to (339) uniformly in $p\in A_{n}$. In the following, we show
that (340)–(343) hold.

_Condition ( 336), part (340)._ The proof consists in showing

$\displaystyle\mathbb{E}\mathbb{E}\left[\left.\sup_{p\in
A_{n}}\bigg{|}\hat{q}_{\mu}^{\prime}(\hat{\theta},p)-\mathbb{E}\left[\left.\hat{q}_{\mu}^{\prime}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]\bigg{|}\,\right|\,\hat{\theta}\right]$
$\displaystyle\to 0\quad\text{ and }$ (344) $\displaystyle\sup_{p\in
A_{n}}\bigg{|}\mathbb{E}\left[\left.\hat{q}_{\mu}^{\prime}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]-q_{\mu}^{\prime}(\hat{\theta},p)\bigg{|}$
$\displaystyle\overset{P}{\longrightarrow}0.$ (345)

We show (344) following Bierens (1994). For the imaginary unit $i$, let
$\psi(t)\coloneqq\int e^{itx}K(x)\mathrm{d}x$, $t\in\real$, be the
characteristic function of $K$, which is $\psi(t)=e^{-t^{2}/2}$ for the
Gaussian kernel $K$, so that $K(x)=(2\pi)^{-1}\int e^{-itx}\psi(t)\mathrm{d}t$
by the inversion formula for characteristic functions. Then
$K^{\prime}(x)=(2\pi)^{-1}\int(-it)e^{-itx}\psi(t)\mathrm{d}t$, hence

$\displaystyle\hat{q}_{\mu}^{\prime}(\hat{\theta},p)$
$\displaystyle=-\frac{1}{n\gamma_{n}^{2}}\sum_{j\in[n]}Y_{j}(2\pi)^{-1}\int(-it)e^{-it(g(\hat{\theta}^{\intercal}X_{j})-p)/\gamma_{n}}\psi(t)\mathrm{d}t$
$\displaystyle=(2\pi)^{-1}\int\left(\frac{1}{n}\sum_{j\in[n]}Y_{j}e^{-itg(\hat{\theta}^{\intercal}X_{j})}\right)e^{itp}it\psi(\gamma_{n}t)\mathrm{d}t,$

where we used a change of variables and Fubini’s theorem (the integral is
bounded as $|Y_{j}|<\bar{y}$ almost surely by Assumption 10(i), $|it|\leq 1$,
$\sup_{t\in\real}|\psi(t)|<\infty$ and the exponential is bounded too). As
$|e^{itp}|\leq 1$ and $|it|\leq|t|$, we have that

$\displaystyle\mathbb{E}\left[\left.\sup_{p\in
A_{n}}\bigg{|}\hat{q}_{\mu}^{\prime}(\hat{\theta},p)-\mathbb{E}\left[\left.\hat{q}_{\mu}^{\prime}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]\bigg{|}\,\right|\,\hat{\theta}\right]$
$\displaystyle\leq(2\pi)^{-1}\int\mathbb{E}\left[\left.\bigg{|}\left(\frac{1}{n}\sum_{j\in[n]}Y_{j}e^{-itg(\hat{\theta}^{\intercal}X_{j})}\right)-\mathbb{E}\left[\left.Y_{j}e^{-itg(\hat{\theta}^{\intercal}X_{j})}\,\right|\,\hat{\theta}\right]\bigg{|}\,\right|\,\hat{\theta}\right]|t||\psi(\gamma_{n}t)|\mathrm{d}t.$

As $e^{-ia}=\cos(a)-i\sin(a)$ for $a\in\real$, and
$\mathbb{E}|W|\leq\sqrt{\mathbb{E}W^{2}}$ for any square-integrable random
variable $W$, the expectation on the right of the last display is bounded by

$\displaystyle\operatorname*{\mathbb{V}}\left[\left.\frac{1}{n}\sum_{j\in[n]}Y_{j}\cos(g(\hat{\theta}^{\intercal}X_{j}))\,\right|\,\hat{\theta}\right]^{1/2}+\operatorname*{\mathbb{V}}\left[\left.\frac{1}{n}\sum_{j\in[n]}Y_{j}\sin(g(\hat{\theta}^{\intercal}X_{j}))\,\right|\,\hat{\theta}\right]^{1/2}\leq
2\sqrt{\frac{\mathbb{E}Y_{j}^{2}}{n}},$

where we used that by Assumption 9(ii) the elements in the sum are i.i.d.
given $\hat{\theta}$, so the covariances are zero, and that
$\operatorname*{\mathbb{V}}\left[\left.Y_{j}\cos(g(\hat{\theta}^{\intercal}X_{j}))\,\right|\,\hat{\theta}\right]\leq\mathbb{E}\left[\left.Y_{j}^{2}\,\right|\,\hat{\theta}\right]=\mathbb{E}Y_{j}^{2}$.
Thus,

$\displaystyle\mathbb{E}\left[\left.\sup_{p\in
A_{n}}\bigg{|}\hat{q}_{\mu}^{\prime}(\hat{\theta},p)-\mathbb{E}\left[\left.\hat{q}_{\mu}^{\prime}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]\bigg{|}\,\right|\,\hat{\theta}\right]$
$\displaystyle\leq\sqrt{\frac{\mathbb{E}Y_{j}^{2}}{\pi^{2}n}}\int|t||\psi(\gamma_{n}t)|\mathrm{d}t$
$\displaystyle\leq\sqrt{\frac{\mathbb{E}Y_{j}^{2}}{\pi^{2}n\gamma_{n}^{4}}}\int|t||\psi(t)|\mathrm{d}t.$

As $\mathbb{E}Y_{j}^{2}<\infty$ by Assumption 10(i) and
$\int|t||\psi(t)|\mathrm{d}t=\int|t|e^{-t^{2}/2}<\infty$ for the Gaussian $K$,
the right side is of the order
$1/(\gamma_{n}^{2}\sqrt{n})=(\kappa_{0})^{-2}n^{2\beta-1/2}=o\left(1\right)$
for $\beta<1/4$.

Next, we show (345). As the summands are identically distributed given
$\hat{\theta}$ by Assumption 9(ii), the tower property of expectations gives

$\displaystyle\mathbb{E}\left[\left.\hat{q}_{\mu}^{\prime}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]$
$\displaystyle=-\frac{1}{\gamma_{n}^{2}}\mathbb{E}\left[\left.\mathbb{E}\left[\left.Y\,\right|\,g(\hat{\theta}^{\intercal}X),\hat{\theta}\right]K^{\prime}((g(\hat{\theta}X)-p)/\gamma_{n})\,\right|\,\hat{\theta}\right]$
$\displaystyle=-\frac{1}{\gamma_{n}^{2}}\mathbb{E}\left[\left.\mu(\hat{\theta},g(\hat{\theta}^{\intercal}X))K^{\prime}((g(\hat{\theta}X)-p)/\gamma_{n})\,\right|\,\hat{\theta}\right]$
$\displaystyle=-\frac{1}{\gamma_{n}^{2}}\int_{\underline{p}_{\hat{\theta}}}^{\bar{p}_{\hat{\theta}}}\mu(\hat{\theta},\tilde{p})K^{\prime}((\tilde{p}-p)/\gamma_{n})h(\hat{\theta},\tilde{p})\mathrm{d}\tilde{p}$
$\displaystyle=-\frac{1}{\gamma_{n}^{2}}\int_{\underline{p}_{\hat{\theta}}}^{\bar{p}_{\hat{\theta}}}q_{\mu}(\hat{\theta},\tilde{p})K^{\prime}((\tilde{p}-p)/\gamma_{n})\mathrm{d}\tilde{p}$
$\displaystyle=-\frac{1}{\gamma_{n}}\int_{(\underline{p}_{\hat{\theta}}-p)/\gamma_{n}}^{(\bar{p}_{\hat{\theta}}-p)/\gamma_{n}}q_{\mu}(\hat{\theta},\gamma_{n}v+p)K^{\prime}(v)\mathrm{d}v$

by definition of $h(\hat{\theta},\cdot)$ as the density of
$(g(\hat{\theta}^{\intercal}X)\mid\hat{\theta})$ under Assumptions 9(ii) and
11(i), and $q_{\mu}=\mu h$. Integration by parts gives

$\displaystyle\mathbb{E}\left[\left.\hat{q}_{\mu}^{\prime}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]=$
$\displaystyle\,-\frac{1}{\gamma_{n}}\left\\{q(\hat{\theta},\bar{p}_{\hat{\theta}})K((\bar{p}_{\hat{\theta}}-p)/\gamma_{n})-q(\hat{\theta},\underline{p}_{\hat{\theta}})K((\underline{p}_{\hat{\theta}}-p)/\gamma_{n})\right\\}$
(346)
$\displaystyle+\int_{(\underline{p}_{\hat{\theta}}-p)/\gamma_{n}}^{(\bar{p}_{\hat{\theta}}-p)/\gamma_{n}}q_{\mu}^{\prime}(\hat{\theta},\gamma_{n}v+p)K(v)\mathrm{d}v.$
(347)

As $q_{\mu}(\hat{\theta},\cdot)$ is bounded for
$\hat{\theta}\in\textup{Nb}(\theta_{0},\epsilon)$, with probability tending to
one (346) is of the order

$\gamma_{n}^{-1}\left[\sup_{p\in
A_{n}}K((\bar{p}_{\hat{\theta}}-p)/\gamma_{n})+\sup_{p\in
A_{n}}K((\underline{p}_{\hat{\theta}}-p)/\gamma_{n})\right]=2\gamma_{n}^{-1}K(a_{n}/\gamma_{n}).$

As $\gamma_{n}^{-1}K(a_{n}/\gamma_{n})\to 0$, (346) is $o_{P}\left(1\right)$.
Now we show that (347) converges uniformly to
$q_{\mu}^{\prime}(\hat{\theta},p)$ adapting the proof of Schuster and Yakowitz
(1979, Lemma 1). The kernel $K$ being a density integrating to one implies

$\displaystyle q_{\mu}^{\prime}(\hat{\theta},p)$
$\displaystyle=q_{\mu}^{\prime}(\hat{\theta},p)\gamma_{n}^{-1}\left\\{\int_{-\infty}^{p-\bar{p}_{\hat{\theta}}}K(u/\gamma_{n})\mathrm{d}u+\int_{p-\bar{p}_{\hat{\theta}}}^{p-\underline{p}_{\hat{\theta}}}K(u/\gamma_{n})\mathrm{d}u+\int_{p-\underline{p}_{\hat{\theta}}}^{\infty}K(u/\gamma_{n})\mathrm{d}u\right\\}.$

Combine this with a change of variables in (347) (with
$u\coloneqq-\gamma_{n}v$ noting that $K$ is symmetric about zero), to get

$\displaystyle\sup_{p\in
A_{n}}\bigg{|}\int_{(\underline{p}_{\hat{\theta}}-p)/\gamma_{n}}^{(\bar{p}_{\hat{\theta}}-p)/\gamma_{n}}q_{\mu}^{\prime}(\hat{\theta},\gamma_{n}v+p)K(v)\mathrm{d}v-q_{\mu}^{\prime}(\hat{\theta},p)\bigg{|}$
$\displaystyle\leq\sup_{p\in
A_{n}}\bigg{|}\int_{-\infty}^{p-\bar{p}_{\hat{\theta}}}q_{\mu}^{\prime}(\hat{\theta},p)\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}+\sup_{p\in
A_{n}}\bigg{|}\int_{p-\underline{p}_{\hat{\theta}}}^{\infty}q_{\mu}^{\prime}(\hat{\theta},p)\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}$
(348) $\displaystyle+\sup_{p\in
A_{n}}\bigg{|}\int_{p-\bar{p}_{\hat{\theta}}}^{p-\underline{p}_{\hat{\theta}}}[q_{\mu}^{\prime}(\hat{\theta},p-u)-q_{\mu}^{\prime}(\hat{\theta},p)]\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}.$
(349)

The two terms in (348) are $o_{P}\left(1\right)$. The first one is bounded by

$\sup_{p\in A_{n}}|q_{\mu}^{\prime}(\hat{\theta},p)|\sup_{p\in
A_{n}}\int_{-\infty}^{(p-\bar{p}_{\hat{\theta}})/\gamma_{n}}K(v)\mathrm{d}v=\sup_{p\in
A_{n}}|q_{\mu}^{\prime}(\hat{\theta},p)|\sup_{p\in
A_{n}}\int_{-\infty}^{-a_{n}/\gamma_{n}}K(v)\mathrm{d}v,$

which vanishes as, on one hand, $\sup_{p\in
A_{n}}|q_{\mu}^{\prime}(\hat{\theta},p)|<\infty$ with probability tending to
one as $\hat{\theta}\overset{P}{\longrightarrow}\theta_{0}$ by Assumptions
6(iv) and 7, and, on the other hand, $a_{n}/\gamma_{n}\to\infty$ implies that
the integral of the Gaussian $K$ goes to zero. The second term in (348) is
bounded by

$\sup_{p\in A_{n}}|q_{\mu}^{\prime}(\hat{\theta},p)|\sup_{p\in
A_{n}}\int_{(p-\underline{p}_{\hat{\theta}})/\gamma_{n}}^{\infty}K(v)\mathrm{d}v=\sup_{p\in
A_{n}}|q_{\mu}^{\prime}(\hat{\theta},p)|\sup_{p\in
A_{n}}\int_{a_{n}/\gamma_{n}}^{\infty}K(v)\mathrm{d}v,$

so it is also $o_{P}\left(1\right)$ by the same argument. To show that (349)
also vanishes, let $\rho_{n}>0$ be a sequence satisfying
$\rho_{n}/\gamma_{n}\to\infty$ and $\rho_{n}<a_{n}$, i.e.
$\gamma_{n}\ll\rho_{n}<a_{n}$ (as $\gamma_{n}=\kappa_{0}n^{-\beta}$ and
$a_{n}=\kappa_{1}n^{-\alpha}$, $\beta>\alpha$, we can take
$\rho_{n}=\kappa_{2}n^{-(\beta+\alpha)/2}$, $0<\kappa_{2}<\kappa_{1}$). Then
(349) is bounded by

$\displaystyle\sup_{p\in
A_{n}}\bigg{|}\int_{[p-\bar{p}_{\hat{\theta}},p-\underline{p}_{\hat{\theta}}]\cap\left\\{|u|\leq\rho_{n}\right\\}}[q_{\mu}^{\prime}(\hat{\theta},p-u)-q_{\mu}^{\prime}(\hat{\theta},p)]\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}$
(350) $\displaystyle+\sup_{p\in
A_{n}}\bigg{|}\int_{[p-\bar{p}_{\hat{\theta}},p-\underline{p}_{\hat{\theta}}]\cap\left\\{|u|>\rho_{n}\right\\}}[q_{\mu}^{\prime}(\hat{\theta},p-u)-q_{\mu}^{\prime}(\hat{\theta},p)]u^{-1}u\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}.$
(351)

Here, (350) is bounded by

$\displaystyle\sup_{p\in
A_{n}}\sup_{|u|\leq\rho_{n}}|q_{\mu}^{\prime}(\hat{\theta},p-u)-q_{\mu}^{\prime}(\hat{\theta},p)|\sup_{p\in
A_{n}}\int_{-\infty}^{\infty}\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u$
$\displaystyle\leq\sup_{p\in
A_{n}}\sup_{|u|\leq\rho_{n}}|q_{\mu}^{\prime}(\hat{\theta},p-u)-q_{\mu}^{\prime}(\hat{\theta},p)|.$

By Assumptions 6(iv) and 7, $q_{\mu}^{\prime}(\hat{\theta},\cdot)$ is
continuous on the compact set
$[\underline{p}_{\hat{\theta}},\bar{p}_{\hat{\theta}}]\supset A_{n}$ and is
therefore uniformly continuous. Thus, $|u|\leq\rho_{n}$ for a small enough
$\rho_{n}<a_{n}$ implies that
$p-u,p\in[\underline{p}_{\hat{\theta}},\bar{p}_{\hat{\theta}}]$ for all $p\in
A_{n}$. Hence, $\sup_{p\in
A_{n}}\sup_{|u|\leq\rho_{n}}|q_{\mu}^{\prime}(\hat{\theta},p-u)-q_{\mu}^{\prime}(\hat{\theta},p)|=o_{P}\left(1\right)$.
In the integral of (351),
$u\in[p-\bar{p}_{\hat{\theta}},p-\underline{p}_{\hat{\theta}}]$, so
$p-u\in[\underline{p}_{\hat{\theta}},\bar{p}_{\hat{\theta}}]$, and thus (351)
is bounded by
$2\sup_{p\in[\underline{p}_{\hat{\theta}},\bar{p}_{\hat{\theta}}]}|q_{\mu}^{\prime}(\hat{\theta},p)|$
times

$\displaystyle\sup_{p\in
A_{n}}\bigg{|}\int_{[p-\bar{p}_{\hat{\theta}},p-\underline{p}_{\hat{\theta}}]\cap\left\\{|u|>\rho_{n}\right\\}}\frac{K(u/\gamma_{n})}{\gamma_{n}}\mathrm{d}u\bigg{|}$
$\displaystyle\leq\sup_{p\in
A_{n}}\int_{\left\\{|u|>\rho_{n}\right\\}}\frac{K(u/\gamma_{n})}{\gamma_{n}}\mathrm{d}u$
$\displaystyle\leq\int_{\left\\{|v|>\rho_{n}/\gamma_{n}\right\\}}K(v)\mathrm{d}v,$

where we used that $K>0$. As $\gamma_{n}\ll\rho_{n}$, and $K(v)\downarrow 0$
as $|v|\to\infty$, the right integral tends to zero. As Assumptions 6(iv) and
7 control
$\sup_{p\in[\underline{p}_{\hat{\theta}},\bar{p}_{\hat{\theta}}]}|q_{\mu}^{\prime}(\hat{\theta},p)|$,
(351) is $o_{P}\left(1\right)$. Thus, (345) holds. Conclude that (340) holds.

_Condition ( 336), part (341)._ Follows directly along the lines of (340),
setting $Y_{j}\coloneqq 1$ for all $j\in[n]$, in the formulae of (340).

_Condition ( 336), part (342)._ Analogously to $\hat{q}_{\mu}^{\prime}$ above,
we can write

$\displaystyle\hat{q}_{\mu}(\hat{\theta},p)$
$\displaystyle=\frac{1}{n\gamma_{n}}\sum_{j\in[n]}Y_{j}(2\pi)^{-1}\int
e^{-it(g(\hat{\theta}^{\intercal}X_{j})-p)/\gamma_{n}}\psi(t)\mathrm{d}t$
$\displaystyle=(2\pi)^{-1}\int\left(\frac{1}{n}\sum_{j\in[n]}Y_{j}e^{-itg(\hat{\theta}^{\intercal}X_{j})}\right)e^{itp}\psi(\gamma_{n}t)\mathrm{d}t,$

and then

$\displaystyle\mathbb{E}\left[\left.\sup_{p\in
A_{n}}\bigg{|}\hat{q}_{\mu}(\hat{\theta},p)-\mathbb{E}\left[\left.\hat{q}_{\mu}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]\bigg{|}\,\right|\,\hat{\theta}\right]\leq\sqrt{\frac{\mathbb{E}Y_{j}^{2}}{\pi^{2}n}}\int|\psi(\gamma_{n}t)|\mathrm{d}t\leq\sqrt{\frac{\mathbb{E}Y_{j}^{2}}{\pi^{2}n\gamma_{n}^{2}}}\int|\psi(t)|\mathrm{d}t.$

Assumption 10(i) and $\int|\psi(t)|\mathrm{d}t=\int e^{-t^{2}/2}=2\pi$ for the
Gaussian kernel $K$ mean that the right side is of the order
$1/(\gamma_{n}\sqrt{n})=(\kappa_{0})^{-1}n^{\beta-1/2}=o\left(1\right)$ for
$\beta<1/4$. Next, as for $\hat{q}_{\mu}^{\prime}$,

$\displaystyle\mathbb{E}\left[\left.\hat{q}_{\mu}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]$
$\displaystyle=\frac{1}{\gamma_{n}}\int_{\underline{p}_{\hat{\theta}}}^{\bar{p}_{\hat{\theta}}}\mu(\hat{\theta},\tilde{p})K((\tilde{p}-p)/\gamma_{n})h(\hat{\theta},\tilde{p})\mathrm{d}\tilde{p}$
$\displaystyle=\frac{1}{\gamma_{n}}\int_{\underline{p}_{\hat{\theta}}}^{\bar{p}_{\hat{\theta}}}q_{\mu}(\hat{\theta},\tilde{p})K((\tilde{p}-p)/\gamma_{n})\mathrm{d}\tilde{p}=\int_{(\underline{p}_{\hat{\theta}}-p)/\gamma_{n}}^{(\bar{p}_{\hat{\theta}}-p)/\gamma_{n}}q_{\mu}(\hat{\theta},\gamma_{n}v+p)K(v)\mathrm{d}v.$

This converges uniformly to $q_{\mu}(\hat{\theta},p)$ in $p\in A_{n}$ by the
same arguments as (347) does to $q_{\mu}^{\prime}(\hat{\theta},p)$, given
Assumptions 6 and 7 ensuring the boundedness and continuity of
$q_{\mu}(\hat{\theta},\cdot)$ with probability tending to one as
$\hat{\theta}\overset{P}{\longrightarrow}\theta_{0}$.

Condition (336), part (343). Follows from (342) by setting $Y_{j}\coloneqq 1$
for all $j\in[n]$.

_Condition ( 337)._ To establish the uniform convergence of

$\displaystyle\widehat{\left(\frac{\partial\mu}{\partial\theta_{k}}\right)}(\hat{\theta},p)\coloneqq\frac{\hat{\dot{q}}_{\mu,k}(\hat{\theta},p)\hat{h}(\hat{\theta},p)-\hat{q}_{\mu}(\hat{\theta},p)\hat{\dot{h}}_{k}(\hat{\theta},p)}{(\hat{h}(\hat{\theta},p))^{2}}$
(352)

to

$\displaystyle\frac{\partial\mu}{\partial\theta_{k}}(\hat{\theta},p)\coloneqq\frac{\dot{q}_{\mu,k}(\hat{\theta},p)h(\hat{\theta},p)-\hat{q}_{\mu}(\hat{\theta},p)\dot{h}_{k}(\hat{\theta},p)}{(\hat{h}(\hat{\theta},p))^{2}}$
(353)

in $p\in A_{n}$, we can follow the same steps which led to (340)–(343) of
Condition (336), because Assumptions 6(iv) and 7 ensure that with probability
tending to one, $\lVert{h}\rVert_{A_{n}}$,
$\lVert{\dot{h}_{k}}\rVert_{A_{n}}$, $\lVert{q_{\mu}}\rVert_{A_{n}}$,
$\lVert{\dot{q}_{\mu,k}}\rVert_{A_{n}}$ are all bounded. As (342) and (343)
were proved above, it is sufficient to show both

$\displaystyle\lVert{\hat{\dot{q}}_{\mu,k}-\dot{q}_{\mu,k}}\rVert_{A_{n}}$
$\displaystyle\overset{P}{\longrightarrow}0\quad\text{ and }$ (354)
$\displaystyle\lVert{\hat{\dot{h}}_{k}-\dot{h}_{k}}\rVert_{A_{n}}$
$\displaystyle\overset{P}{\longrightarrow}0.$ (355)

_Condition ( 337), part (354)._ We proceed by showing

$\displaystyle\mathbb{E}\mathbb{E}\left[\left.\sup_{p\in
A_{n}}\bigg{|}\hat{\dot{q}}_{\mu,k}(\hat{\theta},p)-\mathbb{E}\left[\left.\hat{\dot{q}}_{\mu,k}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]\bigg{|}\,\right|\,\hat{\theta}\right]$
$\displaystyle\to 0\quad\text{ and }$ (356) $\displaystyle\sup_{p\in
A_{n}}\bigg{|}\mathbb{E}\left[\left.\hat{\dot{q}}_{\mu,k}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]-\dot{q}_{\mu,k}(\hat{\theta},p)\bigg{|}$
$\displaystyle\overset{P}{\longrightarrow}0.$ (357)

As for $\hat{q}_{\mu}^{\prime}$ above,

$\displaystyle\hat{\dot{q}}_{\mu,k}(\hat{\theta},p)$
$\displaystyle=\frac{(g^{-1})^{\prime}(p)}{n\gamma_{n}^{2}}\sum_{j\in[n]}Y_{j}X_{j,k}(2\pi)^{-1}\int(-it)e^{-it(\hat{\theta}^{\intercal}X_{j}-g^{-1}(p))/\gamma_{n}}\psi(t)\mathrm{d}t$
(358)
$\displaystyle=(g^{-1})^{\prime}(p)(2\pi)^{-1}\int\left(\frac{1}{n}\sum_{j\in[n]}Y_{j}X_{j,k}e^{-it\hat{\theta}^{\intercal}X_{j}}\right)e^{itg^{-1}(p)}(-it)\psi(\gamma_{n}t)\mathrm{d}t.$
(359)

Since $|-it|\leq|t|$ and $\mathcal{X}$ is bounded by Assumption 10, Euler’s
formula and Assumption 9(ii) imply that the bound of (344) also apply here up
to a constant, which bounds $\mathcal{X}$, times $\sup_{p\in
A_{n}}|(g^{-1})^{\prime}(p)|$. As $\sup_{p\in A_{n}}|(g^{-1})^{\prime}(p)|\leq
1/\left\lVert{g^{\prime}}\right\rVert_{\infty}<\infty$ by Assumption 11, (356)
holds.

_Condition ( 337), part (354), (357)._ We begin by deriving
$\dot{q}_{\mu,k}(\hat{\theta},p)=((\partial/\partial\theta_{k})(\mu
h))(\hat{\theta},p)$. For simplicity, assume that we only have two covariates,
both continuous, and we are interested in the derivative with respect to the
first coordinate of $\theta$ ($k\coloneqq 1$). It is straightforward to
generalise the arguments below for the general case. By the tower property of
expectation,

$\mu(\theta,p)=\mathbb{E}\left[\left.Y\,\right|\,\theta^{\intercal}X=g^{-1}(p)\right]=\mathbb{E}\left[\left.m(X)\,\right|\,\theta^{\intercal}X=g^{-1}(p)\right].$

In view of (38) of Proposition 6, we can write

$\displaystyle\mathbb{E}\left[\left.m(X)\,\right|\,\theta^{\intercal}X=t\right]$
$\displaystyle=\frac{1}{v(\theta,t)}\int_{\mathcal{X}_{1}}m\left(x_{1},\frac{t-\theta_{1}x_{1}}{\theta_{2}}\right)\Psi\left(x_{1},\frac{t-\theta_{1}x_{1}}{\theta_{2}}\right)\mathrm{d}x_{1}$
(360) $\displaystyle v(\theta,t)$
$\displaystyle\coloneqq\int_{\mathcal{X}_{1}}\Psi\left(x_{1},\frac{t-\theta_{1}x_{1}}{\theta_{2}}\right)\mathrm{d}x_{1},$
(361)

where $v(\theta,\cdot)$ is the density of $\theta^{\intercal}X$ satisfying
$v(\theta,\cdot)>0$ by Assumption 10. Thus, $h(\theta,\cdot)$, being the
density of $g(\theta^{\intercal}X)$, is equal to

$\displaystyle
h(\theta,p)=(g^{-1})^{\prime}(p)\int_{\mathcal{X}_{1}}\Psi\left(x_{1},\frac{g^{-1}(p)-\theta_{1}x_{1}}{\theta_{2}}\right)\mathrm{d}x_{1}.$
(362)

By Assumption 9(ii), (360) and (362) remain valid once we replace $\theta$
with $\hat{\theta}$. It then follows for continuously differentiable $\Psi$
and $m$ (Assumptions 11 and 10), that for $k=1$,

$\displaystyle q_{\mu}(\theta,p)$ $\displaystyle=\mu(\theta,p)h(\theta,p)$
$\displaystyle=(g^{-1})^{\prime}(p)\int_{\mathcal{X}_{1}}m\left(x_{1},\frac{g^{-1}(p)-\theta_{1}x_{1}}{\theta_{2}}\right)\Psi\left(x_{1},\frac{g^{-1}(p)-\theta_{1}x_{1}}{\theta_{2}}\right)\mathrm{d}x_{1}$
$\displaystyle\dot{q}_{\mu,k}(\hat{\theta},p)$
$\displaystyle=(g^{-1})^{\prime}(p)\int_{\mathcal{X}_{1}}\frac{\mathrm{d}}{\mathrm{d}\theta_{1}}\left[m\left(x_{1},\frac{g^{-1}(p)-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)\Psi\left(x_{1},\frac{g^{-1}(p)-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)\right]\mathrm{d}x_{1}$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}p}\int_{\mathcal{X}_{1}}-x_{1}m\left(x_{1},\frac{g^{-1}(p)-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)\Psi\left(x_{1},\frac{g^{-1}(p)-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)\mathrm{d}x_{1},$
(363)

where in the last step we used that $(g^{-1})^{\prime}>0$. We proceed by
showing the desired convergence. Let
$[\underline{t}_{\hat{\theta}},\bar{t}_{\hat{\theta}}]\coloneqq[g^{-1}(\underline{p}_{\hat{\theta}}),g^{-1}(\bar{p}_{\hat{\theta}})]$.
We have for $k=1$ by the tower property

$\displaystyle\mathbb{E}\left[\left.\hat{\dot{q}}_{\mu,k}(\hat{\theta},p)\,\right|\,\hat{\theta}\right]=\frac{g^{-1}(p)}{\gamma_{n}^{2}}\mathbb{E}\left[\left.Y_{j}X_{j,k}K^{\prime}((\hat{\theta}^{\intercal}X_{j}-g^{-1}(p)/\gamma_{n})\,\right|\,\hat{\theta}\right]$
$\displaystyle=\frac{(g^{-1})^{\prime}(p)}{\gamma_{n}^{2}}\mathbb{E}\left[\left.\mathbb{E}\left[\left.m(X_{j})X_{j,1}\,\right|\,\hat{\theta}^{\intercal}X_{j},\hat{\theta}\right]K^{\prime}((\hat{\theta}^{\intercal}X_{j}-g^{-1}(p))/\gamma_{n})\,\right|\,\hat{\theta}\right]$
$\displaystyle=\frac{(g^{-1})^{\prime}(p)}{\gamma_{n}^{2}}\int_{\underline{t}_{\hat{\theta}}}^{\bar{t}_{\hat{\theta}}}K^{\prime}\left(\frac{t-g^{-1}(p)}{\gamma_{n}}\right)\int_{\mathcal{X}_{1}}x_{1}m\left(x_{1},\frac{t-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)\Psi\left(x_{1},\frac{t-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)\mathrm{d}x_{1}\,\mathrm{d}t$
$\displaystyle=(g^{-1})^{\prime}(p)$
$\displaystyle\times\int_{\mathcal{X}_{1}}x_{1}\left[\frac{1}{\gamma_{n}^{2}}\int_{\underline{t}_{\hat{\theta}}}^{\bar{t}_{\hat{\theta}}}K^{\prime}\left(\frac{t-g^{-1}(p)}{\gamma_{n}}\right)m\left(x_{1},\frac{t-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)\Psi\left(x_{1},\frac{t-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)\mathrm{d}t\right]\mathrm{d}x_{1},$

where we used that (360) holds not only for $m$, but for any generic function
of $X$ by Proposition 6. Let $\lambda_{n}(x_{1},t)\coloneqq
m\left(x_{1},\frac{t-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)\Psi\left(x_{1},\frac{t-\hat{\theta}_{1}x_{1}}{\hat{\theta}_{2}}\right)$.
We show that the term in the square brackets converges to

$-\frac{\partial}{\partial
t}\lambda_{n}(x_{1},g^{-1}(p))=-\left(\frac{\mathrm{d}}{\mathrm{d}p}\lambda_{n}(x_{1},g^{-1}(p))\right)/[(g^{-1})^{\prime}(p)]$

uniformly in $x_{1}\in\mathcal{X}_{1}$ and $p\in A_{n}$, which completes the
proof for (357) in light of (363). Put
$\lambda_{n}^{\prime}(x_{1},t)\coloneqq\frac{\partial}{\partial
t}\lambda_{n}(x_{1},t)$. Integration by parts gives

$\displaystyle\frac{1}{\gamma_{n}^{2}}\int_{\underline{t}_{\hat{\theta}}}^{\bar{t}_{\hat{\theta}}}K^{\prime}\left(\frac{t-g^{-1}(p)}{\gamma_{n}}\right)\lambda_{n}(x_{1},t)\mathrm{d}t=\frac{1}{\gamma_{n}}\int_{(\underline{t}_{\hat{\theta}}-g^{-1}(p))/\gamma_{n}}^{(\bar{t}_{\hat{\theta}}-g^{-1}(p))/\gamma_{n}}K^{\prime}(v)\lambda_{n}(x_{1},\gamma_{n}v+g^{-1}(p))\mathrm{d}t$
$\displaystyle=\frac{1}{\gamma_{n}}\left\\{\lambda_{n}(x_{1},\bar{t}_{\hat{\theta}})K((\bar{t}_{\hat{\theta}}-g^{-1}(p))/\gamma_{n})-\lambda_{n}(x_{1},\underline{t}_{\hat{\theta}})K((\underline{t}_{\hat{\theta}}-g^{-1}(p))/\gamma_{n})\right\\}$
(364)
$\displaystyle-\int_{(\underline{t}_{\hat{\theta}}-g^{-1}(p))/\gamma_{n}}^{(\bar{t}_{\hat{\theta}}-g^{-1}(p))/\gamma_{n}}\lambda_{n}^{\prime}(x_{1},\gamma_{n}v+g^{-1}(p))K(v)\mathrm{d}v.$
(365)

Now $\lambda_{n}$ is bounded with probability tending to one by Assumption 10.
Recall that
$\underline{t}_{\hat{\theta}}=g^{-1}(\underline{p}_{\hat{\theta}}),\bar{t}_{\hat{\theta}}=g^{-1}(\bar{p}_{\hat{\theta}})$.
As $g^{-1}$ is increasing, and $K$ reaches its maximum at zero and satisfies
$\lim_{|u|\to\infty}K(u)\to 0$, (364) is of the order

$\displaystyle\gamma_{n}^{-1}\left[\sup_{p\in
A_{n}}K\left(\frac{g^{-1}(\bar{p}_{\hat{\theta}})-g^{-1}(p)}{\gamma_{n}}\right)+\sup_{p\in
A_{n}}K\left(\frac{g^{-1}(\underline{p}_{\hat{\theta}})-g^{-1}(p)}{\gamma_{n}}\right)\right]$
$\displaystyle=\gamma_{n}^{-1}\left[K\left(\frac{g^{-1}(\bar{p}_{\hat{\theta}})-g^{-1}(\bar{p}_{\hat{\theta}}-a_{n})}{\gamma_{n}}\right)+K\left(\frac{g^{-1}(\underline{p}_{\hat{\theta}})-g^{-1}(\underline{p}_{\hat{\theta}}+a_{n})}{\gamma_{n}}\right)\right].$
(366)

By a mean-value expansion, the first term in (366) is of the order

$\gamma_{n}^{-1}K\left(\left(\inf_{p\in[\underline{p}_{\hat{\theta}},\bar{p}_{\hat{\theta}}]}(g^{-1})^{\prime}(p)\right)a_{n}/\gamma_{n}\right).$

As
$\inf_{p\in[\underline{p}_{\hat{\theta}},\bar{p}_{\hat{\theta}}]}(g^{-1})^{\prime}(p)>0$,
this is $o_{P}\left(1\right)$ for $a_{n}/\gamma_{n}\to\infty$ and Gaussian
$K$. The same applies to the second term of (366), so (364) is
$o_{P}\left(1\right)$. Last, we show that (365) converges to
$-\lambda_{n}^{\prime}(x_{1},g^{-1}(p))$. Again, we can write

$\displaystyle\lambda_{n}^{\prime}(x_{1},g^{-1}(p))=\frac{\lambda_{n}^{\prime}(x_{1},g^{-1}(p))}{\gamma_{n}}$
$\displaystyle\times\left\\{\int_{-\infty}^{g^{-1}(p)-\bar{t}_{\hat{\theta}}}K(u/\gamma_{n})\mathrm{d}u+\int_{g^{-1}(p)-\bar{t}_{\hat{\theta}}}^{p-\underline{t}_{\hat{\theta}}}K(u/\gamma_{n})\mathrm{d}u+\int_{g^{-1}(p)-\underline{t}_{\hat{\theta}}}^{\infty}K(u/\gamma_{n})\mathrm{d}u\right\\}.$

A change of variables in (365) ($u\coloneqq-\gamma_{n}v$) and $K$ being
symmetric about zero give

$\displaystyle\sup_{x_{1}\in\mathcal{X}_{1},p\in
A_{n}}\bigg{|}\int_{(\underline{t}_{\hat{\theta}}-g^{-1}(p))/\gamma_{n}}^{(\bar{t}_{\hat{\theta}}-g^{-1}(p))/\gamma_{n}}\lambda_{n}^{\prime}(x_{1},\gamma_{n}v+g^{-1}(p))K(v)\mathrm{d}v-\lambda_{n}^{\prime}(x_{1},g^{-1}(p))\bigg{|}$
$\displaystyle\leq\sup_{x_{1}\in\mathcal{X}_{1},p\in
A_{n}}\bigg{|}\int_{-\infty}^{g^{-1}(p)-\bar{t}_{\hat{\theta}}}\lambda_{n}^{\prime}(x_{1},g^{-1}(p))\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}$
(367) $\displaystyle+\sup_{x_{1}\in\mathcal{X}_{1},p\in
A_{n}}\bigg{|}\int_{g^{-1}(p)-\underline{t}_{\hat{\theta}}}^{\infty}\lambda_{n}^{\prime}(x_{1},g^{-1}(p))\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}$
(368) $\displaystyle+\sup_{x_{1}\in\mathcal{X}_{1},p\in
A_{n}}\bigg{|}\int_{g^{-1}(p)-\bar{t}_{\hat{\theta}}}^{g^{-1}(p)-\underline{t}_{\hat{\theta}}}[\lambda_{n}^{\prime}(x_{1},g^{-1}(p)-u)-\lambda_{n}^{\prime}(x_{1},g^{-1}(p))]\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}.$
(369)

Since $\mathcal{X}_{1}$ is bounded by Assumption 10,
$\sup_{x_{1}\in\mathcal{X}_{1},p\in
A_{n}}|\lambda_{n}^{\prime}(x_{1},g^{-1}(p))|$ is bounded with probability
tending to one. Hence, as $g^{-1}$ is increasing, $K\geq 0$ and
$\bar{t}_{\hat{\theta}}=g^{-1}(\bar{p}_{\hat{\theta}})$, (367) is of the order
$\int_{-\infty}^{(g^{-1}(\bar{p}_{\hat{\theta}}-a_{n})-g^{-1}(p))/\gamma_{n}}K(v)\mathrm{d}v$.
Here,

$(g^{-1}(\bar{p}_{\hat{\theta}}-a_{n})-g^{-1}(p))/\gamma_{n}\leq(\inf_{p\in[0,1]}(g^{-1})^{\prime}(p))(-a_{n}/\gamma_{n})\leq(1/\left\lVert{g^{\prime}}\right\rVert_{\infty})(-a_{n}/\gamma_{n})$

by a mean-value expansion. As the infimum is positive, (367) is
$o_{P}\left(1\right)$ for $a_{n}/\gamma_{n}\to\infty$, and so is (368) by
similar arguments. The term (369) can be treated analogously to (349). First
note that taking the supremum over $p\in A_{n}$ is the same as taking the
supremum over
$t\in[g^{-1}(\underline{p}_{\hat{\theta}}+a_{n}),g^{-1}(\bar{p}_{\hat{\theta}}-a_{n})]\eqqcolon
T_{n}$ as $g^{-1}$ is increasing. Take a
$\gamma_{n}\ll\rho_{n}<(1/\left\lVert{g^{\prime}}\right\rVert_{\infty})a_{n}$
(e.g.
$\rho_{n}\coloneqq(1/\left\lVert{g^{\prime}}\right\rVert_{\infty})\kappa_{2}n^{-(\alpha+\beta)/2}$
for some $0<\kappa_{2}<\kappa_{1}$). Then (369) is bounded by

$\displaystyle\sup_{x_{1}\in\mathcal{X}_{1},t\in
T_{n}}\bigg{|}\int_{[t-\bar{t}_{\hat{\theta}},t-\underline{t}_{\hat{\theta}}]\cap\left\\{|u|\leq\rho_{n}\right\\}}[\lambda_{n}^{\prime}(x_{1},t-u)-\lambda_{n}^{\prime}(x_{1},t)]\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}$
(370) $\displaystyle\sup_{x_{1}\in\mathcal{X}_{1},t\in
T_{n}}\bigg{|}\int_{[t-\bar{t}_{\hat{\theta}},t-\underline{t}_{\hat{\theta}}]\cap\left\\{|u|>\rho_{n}\right\\}}[\lambda_{n}^{\prime}(x_{1},t-u)-\lambda_{n}^{\prime}(x_{1},t))]\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u\bigg{|}.$
(371)

Here, (370) is bounded by

$\displaystyle\sup_{x_{1}\in\mathcal{X}_{1},t\in
T_{n}}\sup_{|u|\leq\rho_{n}}|\lambda_{n}^{\prime}(x_{1},t-u)-\lambda_{n}^{\prime}(x_{1},t)|\sup_{x_{1}\in\mathcal{X}_{1},t\in
T_{n}}\int_{-\infty}^{\infty}\gamma_{n}^{-1}K(u/\gamma_{n})\mathrm{d}u$
$\displaystyle\leq\sup_{x_{1}\in\mathcal{X}_{1},t\in
T_{n}}\sup_{|u|\leq\rho_{n}}|\lambda_{n}^{\prime}(x_{1},t-u)-\lambda_{n}^{\prime}(x_{1},t)|.$

By Assumptions 10 and 11, $\lambda_{n}^{\prime}(x_{1},\cdot)$ is continuous
uniformly in $x_{1}$ and is therefore uniformly continuous on the compact set
$[g^{-1}(\underline{p}_{\hat{\theta}}),g^{-1}(\bar{p}_{\hat{\theta}})]\supset[g^{-1}(\underline{p}_{\hat{\theta}}+a_{n}),g^{-1}(\bar{p}_{\hat{\theta}}-a_{n})]$.
Thus, $|u|\leq\rho_{n}$ for a small enough $\rho_{n}$ implies that
$t-u,t\in[g^{-1}(\underline{p}_{\hat{\theta}}),g^{-1}(\bar{p}_{\hat{\theta}})]$
for all $t\in T_{n}$. (Note that $\rho_{n}$ is small enough if and only if
$|u|\leq\rho_{n}$ implies

$|u|\leq\min\left\\{g^{-1}(\underline{p}_{\hat{\theta}}+a_{n})-g^{-1}(\underline{p}_{\hat{\theta}}),g^{-1}(\bar{p}_{\hat{\theta}})-g^{-1}(\bar{p}_{\hat{\theta}}-a_{n})\right\\}.$

By a mean-value expansion, the right side is smaller than or equal to
$\inf_{p\in[0,1]}(g^{-1})^{\prime}(p)a_{n}\leq(1/\left\lVert{g^{\prime}}\right\rVert_{\infty})a_{n}$.
But then $\rho_{n}<(1/\left\lVert{g^{\prime}}\right\rVert_{\infty})a_{n}$ is
small enough.) As a consequence,

$\sup_{x\in\mathcal{X}_{1},t\in
T_{n}}\sup_{|u|\leq\rho_{n}}|\lambda_{n}^{\prime}(x_{1},t-u)-\lambda_{n}^{\prime}(x_{1},t)|=o_{P}\left(1\right),$

hence (370) is $o_{P}\left(1\right)$. As
$2\sup_{x_{\in}\mathcal{X}_{1},t\in[\underline{t}_{\hat{\theta}},\bar{t}_{\hat{\theta}}]}|\lambda_{n}(x_{1},t)|$
is bounded by probability tending to one as
$\hat{\theta}_{2}\overset{P}{\longrightarrow}\theta_{0,2}\neq 0$ by
Assumptions 10 and 11, (371) can be shown to be $o_{P}\left(1\right)$
similarly to (351). Thus, (369) is $o_{P}\left(1\right)$ which shows the
desired convergence of (365). Hence, (357) holds for $k=1$. The case for $k=2$
follows by writing

$\displaystyle\mathbb{E}\left[\left.m(X)\,\right|\,\theta^{\intercal}X=t\right]$
$\displaystyle=(1/v(\theta,t))\int_{\mathcal{X}_{2}}m\left(\frac{t-\theta_{2}x_{2}}{\theta_{1}},x_{2}\right)\Psi\left(\frac{t-\theta_{2}x_{2}}{\theta_{1}},x_{2}\right)\mathrm{d}x_{2}$
$\displaystyle v(\theta,t)$
$\displaystyle\coloneqq\int_{\mathcal{X}_{2}}\Psi\left(\frac{t-\theta_{2}x_{2}}{\theta_{1}},x_{2}\right)\mathrm{d}x_{2}$

instead of (360) and (361). By Assumption 10, these behave equally well.
Conclude that the desired convergence of $\hat{\dot{q}}_{\mu,k}$ (Condition
(337), part (354)) holds.

_Condition ( 337), part (355)._ Follows along the same lines as (354) by
setting $Y_{j}\coloneqq 1$ for all $j\in[n]$. Conclude that Condition (337)
holds.

_Remaining Uniform Consistency Results._ Showing

$\displaystyle\sup_{p\in
A_{n}}|\hat{\mu}(\hat{\theta},p)-\mu(\hat{\theta},p)|\overset{P}{\longrightarrow}0,$
(372) $\displaystyle\sup_{p\in
A_{n}}|\hat{\mu}_{2}(\hat{\theta},p)-\mu_{2}(\hat{\theta},p)|\overset{P}{\longrightarrow}0$
(373)

completes the proof of Proposition 7. As $h(\hat{\theta},\cdot)$ is bounded
away from zero with probability tending to one by Assumption 6, (372) is
implied by (342) and (343). Likewise, (373) is implied by
$\lVert{\hat{q}_{\mu_{2}}-q_{\mu_{2}}}\rVert_{A_{n}}\overset{P}{\longrightarrow}0$,
which can be shown as (342). ∎

## References

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