Adaptive estimation in symmetric location model under log-concavity constraint
Location estimation

Nilanjana Laha

N. Laha

Department of Biostatistics, Harvard University
Department of Biostatistics, Harvard University,

677 Huntington Ave,
Boston, MA 02115, U.S.A

Email<EMAIL_ADDRESS>
We revisit the problem of  estimating  the center of symmetry $\theta$ of an unknown symmetric density $f$.  Although [Stone, 1975], [Van Eeden, 1970], and [Sacks, 1975]  constructed adaptive estimators of $\th$ in this model, their estimators depend on  external tuning parameters. In an effort to reduce the burden of  tuning parameters,  we impose  an additional restriction of log-concavity on $f$.  We construct  truncated one-step estimators which are adaptive under the log-concavity assumption. Our simulations suggest that the untruncated version of the one step estimator, which is tuning parameter free, is also asymptotically efficient.
We also study the maximum likelihood estimator (MLE) of $\theta$  in the shape-restricted model.

[Primary ]62G99
[; secondary ]62G20, 62G07

shape constraint
symmetric location model
one step estimator

§ INTRODUCTION

In this paper, we  revisit the symmetric location model with an additional shape-restriction of log-concavity.
We  let $\mathcal{P}$  denote the class of  all densities  on the real line $\RR$.
For any $\th\in\RR$, denote by $\mathcal{S}_{\th}$ the class of all  densities  symmetric about $\theta$. Then the symmetric location model $\mathcal P_s$ is given by

\begin{equation}\label{definition: the symmetric model Ps}
\mathcal{P}_s =\bigg\{f\in\mathcal{P}\ \bl\ f(x;\th)=g(x-\th),\ \th\in\RR,\ g\in\mathcal{S}_0,\ \If<\infty\bigg\},
\end{equation}

where $\If$ is the Fisher information for location.
It is well-established that <cit.> $\If$ is finite if and only if $f$ is an absolutely continuous density satisfying

\[\edint\lb\dfrac{f'(x)}{f(x)}\rb^2 f(x)dx<\infty,\]

where $f'$ is an $L_1$-derivative of $f$. Also, in this case, $\If$ takes the form

\begin{equation*}
\If=\edint\lb\dfrac{f'(x)}{f(x)}\rb^2 f(x)dx.
\end{equation*}

Estimation of $\th$ in $\mathcal P_s$
is an old semi-parametric problem, dating back to [Stein, 1956]. From then on, the problem of estimating $\theta_0$ in $\mathcal P_s$  has been considered by many early authors including, but not limited to, [Stone, 1975], [Beran, 1974], [Sacks, 1975], and [Van Eeden, 1970].
There are two main reasons behind the assumption of symmetry in this model. First, as [Stone, 1975] has pointed out, if $f$ is totally unrestricted, $\th$ is not identifiable.  Second,

the definition of location becomes unclear in the absence of symmetry  [Takeuchi, 1975].
The appeal of the above  model lies in the fact that
adaptive estimation of $\th$ is possible in this model [Stone, 1975].
In other words,  there exist consistent estimators of $\th$ in $\mP_s$, whose asymptotic variance attains the parametric lower bound, which is $\If^{-1}$ in this case.  See Sections $3.2$, $3.3$, and $6.3$ of [Bickel et al., 1998] for more discussion on adaptive estimation in $\mathcal{P}_s$.

There are general classes of nonparametric estimators, which, following some clever reconstruction,  lead to adaptive estimators of $\theta$ in $\mathcal P_s$. Examples include the  one step estimator used by [Stone, 1975], and the Hodges-Lehmann rank estimator used by [Van Eeden, 1970]. [Beran, 1974] uses a linearized rank estimator introduced by [Kraft and Van Eeden, 1970], where [Sacks, 1975] uses a linear functional of order statistics. All these estimators involve various tuning parameters. The success of these type of nonparametric estimators generally depend crucially on the choice of the tuning parameters.  (cf. [Sacks, 1975]; see also   [Park, 1990] for a thorough empirical study of similar estimators in  a closely related nonparametric problem, i.e. the two-sample location problem.)  However, no data-dependent method has been prescribed to choose these tuning parameters. Therefore, despite attractive theoretical properties, the implementation of the nonparametric estimators of $\theta_0$ is not straightforward.

Although the  tuning parameters stemming from different nonparametric approaches appear to be different,  they generally fall in one of the following categories: (a) scaling parameter for approximating  derivatives by quotient (e.g. [Beran, 1974], [Sacks, 1975] and  [Van Eeden, 1970]), (b) bandwidth selection parameter if kernels are used (e.g. [Stone, 1975]), (c)  the number of basis functions  (e.g. [Beran, 1974]), (d) parameters arising due to truncation (e.g. the  estimators of  [Stone, 1975] and [Sacks, 1975]) or data-partitioning (e.g., [Van Eeden, 1970]).
We will elaborate a little bit on the first three type of tuning parameters.   They arise solely because the adaptive estimators of $\theta_0$  require estimating
$g$, $g'$ (e.g. [Stone, 1975], [Van Eeden, 1970], and [Beran, 1974]), and in some cases, higher derivatives (e.g. $g''$, [Sacks, 1975]). In fact, such tuning parameters are unavoidable in nonparametric estimation of the above quantities.  Moreover, [Hogg, 1974] points out that in practice,  nonparametric estimation of such functions  may be too slow.
This is precisely  where  semi-parammeric models can help because the additional structure  can  be exploited to construct  computationally efficient  estimators of $g$ and $g'$ without using tuning parameters.

If we impose an additional shape restriction of log-concavity on $g$, for instance,  the task of estimating $g'$   becomes much simple. The reason is, the class of  log-concave  densities is structurally rich enough  to admit a  maximum likelihood estimator (MLE) [Pal et al., 2007, Dümbgen and Rufibach, 2009]. Similar results hold for its subclasses, e.g. the class of all symmetric (about the origin) log-concave densities  as well [Doss and Wellner, 2019]. The log-concave MLE type density estimators allow for computationally efficient estimation of the scores without any tuning parameters. These score estimates can readily be used to construct a  one step estimator.

We show that under the log-concavity assumption, truncated versions of the above-mentioned one step estimator are adaptive provided the truncation parameter $\eta_n\to 0$ slowly enough.  This truncation parameter is our only tuning parameter,  which also is introduced purely due to technical reasons in the proof. Moreover, we empirically show that
the efficiency of our estimators monotonously increases as $\eta_n\to 0$. In fact, the  untruncated one step estimator attains the highest efficiency, and also performs reliably under varied settings.

Thus, for practical implementation, the proposed estimator of this paper is the untruncated one-step estimator, which is fully tuning parameter free.
We also touch upon another important tuning parameter free estimator of $\theta_0$, namely the MLE. In particular, we establish its existence under the  shape-constrained model. Our methods can be implemented using the R package  which can be accessed at <https://github.com/nilanjanalaha/log.location>.

The imposition of log-concavity on $\mathcal P_s$ may seem forced, but is not at all unnatural.  The class of log-concave densities, $\mathcal{LC}$, is an important subclass of the class of unimodal densities.   Many common symmetric unimodal  densities, e.g. Gaussian, logistic, and Laplace,   are log-concave.  Unimodality is a reasonable assumption  in  context of location estimation of symmetric densities.  As [Takeuchi, 1975] points out, in practice, multimodal densities generally result from unimodal mixtures. Separate procedures are available for the latter class.   The difficulty with  the  unimodality shape restriction, however, stems from the fact that the corresponding  density-class  is still large, especially it is  not structurally rich enough to admit an MLE [Birgé, 1997]. Therefore unlike the log-concavity assumption, the unimodality assumption does not provide  computational advantages. Hence, we impose the assumption of  log-concavity  on $\mP_s$ instead of just unimodality.

Finally, this paper is an attempt towards  bridging the gap between  the symmetric location model and log-concavity.
Although shape-constrained estimation has a rich history,  so far there has been  little to no use of shape-constraints in  one-sample symmetric location problem.

In fact, to  the best of our knowledge, [Van Eeden, 1970] is the only one to incorporate shape-constraints in treating  the problem considered here. Actually [Van Eeden, 1970] requires $f$ to be log-concave although her paper  does not mention log-concavity.  She requires the function $f'(F^{-1}(u))/f(F^{-1}(u))$ to be non-increasing in $u\in(0,1)$, which is equivalent to  $f$ being log-concave <cit.>. As made clear by our earlier discussion, [Van Eeden, 1970] does not use shape-restricted tools  tailored for log-concave densities because they were not available at that time.
We also want to mention  [Bhattacharyya and Bickel, 2013], who consider both location  and scale estimation in an  elliptical symmetry model, which albeit bearing some resemblance,  is  different from the model considered in this paper. Also, [Bhattacharyya and Bickel, 2013]'s estimation procedure is completely different from ours.

§.§ Notation and terminology

For a concave function $\ps:\RR\mapsto\RR$, the domain  $\dom(\ps)$ will be defined as in <cit.>, that is, $\dom(\ps)=\{x\in\RR\ :\ \ps(x)>-\infty\}$.
For any concave function $\ps$, we say $x\in\RR$ is a knot of $\ps$, if either $\ps'(x+)\neq\ps'(x-)$, or  $x$ is at the boundary of $\dom(\ps)$. We denote by $\mathcal{K}(\ps)$ the set of the knots of $\ps$. Unless otherwise mentioned, for a real valued function $h$, provided they exist,  $h'$ and $h'(\mathord{\cdot}-)$ will refer to the right and left derivatives of $h$, respectively.   We denote the support of any density $f$ by $\supp(f)=\{x\in\RR\ :\ f(x)>0\}$.

For a distribution function $F$, we let $J(F)$ denote the set $\{x\ :\ 0<F(x)<1\}$. For a sequence of distribution functions $\{F_n\}_{n\geq 1}$, we say  $F_n$ converges weakly to  $F$, and write $F_n\to_d F$, if  for all bounded continuous functions $h:\RR\mapsto\RR$, we have $\lim\limits_{n\to\infty}\int h dF_n=\int hdF$.
For  any real valued function $h:\RR\mapsto\RR$, we let $||h||_k$ denote its $L_k$ norm, i.e.

\[||h||_{k}=\lb\edint |h(x)|^k dx\rb^{1/k},\quad k\geq 1.\]

For densities $f_1$ and $f_2$, the Hellinger distance $H(f_1,f_2)$ is defined by

\[H^2(f_1,f_2)=\frac{1}{2}\edint (f_1(x)-f_2(x))^2 dx.\]

We denote the order statistics of a random sample  $(Y_1,\ldots,Y_n)$  by

As usual, we denote the set of natural numbers by $\mathbb{N}$. We  denote by $C$  an arbitrary constant which may vary from line to line. The expression $x\lesssim y$ will imply that there exists $C>0$ so that $x\leq Cy$.

§.§ Problem set up

To formalize the set up, first, let us define

\begin{equation}\label{def: class of concave functions}
\mathcal{C}:=\bigg\{\ph:\RR\mapsto[-\infty,\infty)\ \bl\ \ph \text{ is concave, closed, and proper} \bigg\}.
\end{equation}

We let $\mathcal{SC}_{\th}=\mathcal{S}_{\th}\cap\mathcal{C}$ denote the class of all closed and proper concave functions symmetric about $\theta\in\RR$. Here  a proper and closed concave function is as defined in [Rockafellar, 1970], page 24 and 50.
Letting $\mathcal{LC}$ denote the class of log-concave densities

\[\mathcal{LC}:= \bigg\{f\in\mathcal{P}\ \bl\ \phi=\log f\in\mathcal{C}\bigg\},\]

we set $\mathcal{SLC}_{\th}= \mathcal{LC}\cap\mathcal{S}_{\th}$.
Suppose we observe $n$  independent and identically distributed (i.i.d.) random variables  $X\equiv X_1,\ldots, X_n$ with density $f_0\equiv g_0(\cdot-\theta_0)\in\mP_0$, where

\begin{equation}
\label{definition: model}
\mP_0=\bigg\{f\in\mP \ \bl\ f(x;\th)=g(x-\th),\ \th\in\RR,\ g\in\mathcal{SLC}_0,\ \If<\infty \bigg\}
\end{equation}

is the symmetric log-concave location model. Our aim is to estimate the location parameter $\theta_0$.

Let us denote $\ph_0=\log f_0$, and $\ps_0=\log g_0$. We let $F_0$ and $G_0$ be the respective distribution functions of $f_0$ and $g_0$, and denote by $P_0$ the measure corresponding to $F_0$. We  denote the empirical distribution function of the $X_i$'s by $\Fm$, and write $\Pm$ for the corresponding empirical measure.

We use the following convention  throughout the paper while setting notations for the one step estimators and the MLE. We use a  on the quantities related to the MLE, e.g. the MLE of $\th_0$ and $g_0$ will be denoted by $\hthm$ and $\hgm$, respectively. The similar quantities in the one-step estimator context will use a , e.g. $\hth$, $\hn$ etc. Some quantities like $\hf$, the MLE in $\mathcal{LC}$,   or $\widehat{f}_\th$, the MLE in $\mathcal{SLC}_\theta$, will be introduced in context of the one step estimators, but their notations use the  instead of the  because they are  MLEs.

The article is organized as follows. In Section <ref>, we introduce the one step estimator, and  discuss its asymptotic properties. In Section <ref>,  we explore the MLE of $\th_0$ in $\mP_0$. We provide an empirical study  in Section <ref>.  The proofs are deferred to the appendix.

§ ONE STEP ESTIMATOR

Let  be a preliminary estimator of $\th_0$. Had $g_0$ been known, a valid estimator of $\theta_0$ would be readily given by the one step estimator <cit.>

\begin{equation}\label{def: one step: parametric}
\bth-\edint\dfrac{{\ps}'_0(x-\theta_0)}{\I}d\Fm(x).
\end{equation}

In fact, the above estimator is $\sqn$ consistent with asymptotic variance $\I^{-1}$ <cit.>.
Suppose $\hn\in\mathcal{S}_0$ is an estimator of $g_0$. Further suppose $\hln=\log \hn$ is directionally  differentiable on the support of $\hn$. The latter always holds if $\hn\in\mathcal{LC}$ <cit.>.
Suppose $\hln'$ is the right derivative of $\hln$.  Defining $\hln'$ to be zero outside $\supp(\hn)$, we can define an estimator of $\theta_0$ along the lines of (<ref>) as follows:

\begin{equation}\label{def: one-step estimator full}
\hthf=\bth-\edint\dfrac{\tilde{\ps}'_n(x-\bth)}{\hin}d\Fm(x),
\end{equation}

\begin{equation}\label{def: untruncated fisher info}
\hin=\edint \tilde{\ps}'_n(x-\bth)^2d\Fm(x)
\end{equation}

is an estimator of the Fisher information $\I$. We will refer to $\hthf$ as the untruncated one step estimator.

The asymptotic behavior of $\hln'$ can be hard to control in the tails, which creates technical difficulties in the asymptotic analysis of $\hthf$. As we already mentioned in the introduction, a common approach to tackle this problem is trimming the extreme observations, which leads to a truncated one step estimator similar to [Stone, 1975].

We let $\eta_n$  denote the truncation parameter, which is usually a small positive fraction.
Denote by $\hH$  the distribution function corresponding to .  Letting $\xin$ be the $(1-\eta_n)$-th quantile of $\hH$, we define the truncated one step estimator as follows:

\begin{equation}\label{def: one-step estimator: truncated}
\hth=\bth-\dint_{\bth-\xin}^{\bth+\xin}\dfrac{\hln'(x-\bth)}{\hin(\eta_n)}d\Fm(x).
\end{equation}

Here $\hin(\eta_n)$ is a truncated version of $\hin$, given by

\begin{align}\label{27indef}
&\hin(\eta_n)=\dint _{\bth-\xin}^{\bth+\xin}\tilde{\ps}'_n(x-\bth)^2d\Fm(x).
\end{align}

Note that the symmetry of $\hn$ about $0$ implies that $-\xin=\hH^{-1}(\eta_n)$. Ideally, we should denote the one step estimator in (<ref>) by $\hth(\eta_n)$   but here we suppress the dependence  on $\eta_n$ to avoid cumbersome notation.

$\I$ could also be estimated by a smoother version of $\hin(\eta_n)$, namely,

\begin{align*}
&\hin^{*}(\eta_n)=\dint _{\bth-\xin}^{\bth+\xin}\tilde{\ps}'_n(x-\bth)^2\hn(x-\bth)dx.
\end{align*}

However, our simulations indicate that the estimator $\hin(\eta)$ yields a more efficient one-step estimator. Therefore,  $\hin(\eta)$ is our preferred estimator for the Fisher information.

§.§ Main result

The first main result of this paper states that if $\eta_n\to 0$ at a sufficiently slow rate, then the truncated one step estimator defined in (<ref>) is adaptive for certain choices of $\hn$. However, we require a technical assumption on $\psi_0$ to prove this theorem.

Suppose $\psi_0'$ is any directional derivative of $\psi_0$. Then

\[|\psi_0'(x)-\psi'_0(y)|\leq \kappa|x-y|\quad\text{ for all }x,y\in\iint(\dom(\psi_0)).\]

Since $\psi_0$ is concave, it is directionally differentiable everywhere on $\iint(\dom(\psi_0))$. If $\psi_0$ is twice differentiable on $\iint(\dom(\psi_0))$, Assumption <ref> interprets as $|\psi_0''|\leq \kappa$.
Control on the second derivative is often required in the analysis of shape-restricted models <cit.>.
Simple algebra shows that common symmetric log-concave densities like Gaussian, Laplace, and Logistic  satisfy Assumption <ref>.
Later in Section <ref>, we consider an example where Assumption <ref> is violated. Whether Assumption <ref> is necessary is unknown to us, although Section <ref> hints that the truncated one step estimators may still be adaptive even under the  violation of Assumption <ref>.

Now we state the requirements for  $\hn$. Later in this section, we demonstrate how to build estimators which satisfy such conditions.

Let $y_n=o_p(1)$ be a random sequence.
The density estimator $\hn$ satisfies the followings:

$\|\hn-g_0\|_1\to_p 0$ and  $\sup_{x\in \RR}|\hn(x+y_n)-g_0(x)|\to_p 0$.

(B) For any $K\subset\iint(\dom(\psi_0))$, we have
$\sup_{x\in K}|\hln(x+y_n)-\ps_0(x)|\to_p 0.$

Suppose $x\in\iint(\dom(\ps_0))$ is a continuity point of $\ps_0'.$  Then

\[\hln'(x+y_n)\to_p\ps_0'(x).\]

Condition <ref> (A) implies $H(\hn,g_0)\to_p 0$ because $H(\hn,g_0)\lesssim \sqrt{\|\hn-g_0\|_1}$. However,
we require stronger control over the rate of decay of the Hellinger error $H(\hn,g_0)$.

There exists $p\in(0,1)$ so that $H(\hn, g_0)=O_p(n^{-p})$.

Now we are ready to state our main theorem. The proof of Theorem <ref> can be found in Appendix <ref>.

Suppose $f_0\in\mP_0$ satisfies Assumption <ref> and $\bth$ is a $\sqrt{n}$-consistent estimator of $\th_0$.
Let $\hn\in\mathcal{SLC}_0$ be an estimator of $g_0$ which satisfies Conditions <ref> and <ref>.  Suppose  $\eta_n=C n^{-2p'/5}$, where $C>0$ is any constant, and $p'\in(0,p]$, where $p$ is as in Condition <ref>.
Then the  estimator $\hth$  defined in (<ref>) satisfies

\begin{equation*}
\sqn(\hth-\th_0)\to_{d}N(0,\I^{-1}).
\end{equation*}

A couple of remarks  are in order. First, Theorem <ref> requires $\hn\in\mathcal{SLC}_0$. This automatically rules out most nonparametric density estimators including the symmetrized kernel density estimator of [Stone, 1975].
Second, Theorem <ref> requires $\bth$ to be  $\sqn$-consistent.  [Stone, 1975] and [Beran, 1974] impose  similar conditions on their preliminary estimators. The $Z$-estimator of the shift in the logistic location shift model is $\sqn$-consistent under minimal regularity conditions <cit.>. When $f_0\in\mP_0$,  the sample mean and the sample median  also satisfy  this  requirement.

Now we give example of two $\hn$'s, which  satisfy the conditions of Theorem <ref>.

Partial MLE estimator $\widehat{g}_{\bth}$:
For any $\theta\in\RR$, the density class $\mathcal{SLC}_{\theta}$ admits an MLE
<cit.>. When $\th=\bth$, the MLE  in the class $\mathcal{SLC}_{\bth}$ is a legitimate estimator of $f_0$. We denote the corresponding density by $\widehat{f}_{\bth}$. Then the centered density $\widehat{g}_{\bth}=\widehat{f}_{\bth}(\cdot+\bth)$ is a potential choice for $\hn$  because  $\widehat{g}_{\bth}\in\mathcal{SLC}_0$.
We call this estimator a Partial MLE estimator to distinguish it from the traditional MLE of $g_0$, which we will discuss in Section <ref>.
From  [Doss and Wellner, 2019] it follows that $\log\widehat{g}_{\bth}= \widehat{\ps}_{\bth}$ is a piecewise linear concave function with domain $[-a,a]$, where

Geometric mean type symmetrized estimator $\hn^{geo,sym}$:
We denote by $\hf$ the MLE of $f_0$ among the class of all log-concave densities, which exists by [Pal et al., 2007]. The finite sample and asymptotic properties of $\hf$ are well-established [Dümbgen and Rufibach, 2009, Cule and Samworth, 2010].  In particular,  $\log\hf$ is piecewise linear with domain $[X_{(1)},X_{(n)}]$. However,
the estimator $\hf$ need not be symmetric about any $\theta\in\RR$.
A symmetrized version of $\hf$ is given by

\begin{equation}\label{est 5}
\hn^{geo,sym}(z):=C_{n}^{geo}\lb \hf(\bth+z)\hf(\bth-z)\rb^{1/2},\quad z\in\RR
\end{equation}

where $C_{n}^{geo}$ is a random normalizing constant. Here “geo" refers to the mode of symmetrization, which is the geometric mean in this case. Since addition preserves concavity, $\log(\hf(\bth+z))+\log(\hf(\bth-z))$ is concave, which entails that $
\hn^{geo,sym}\in\mathcal{SLC}_0$.
The support of $\hn^{geo,sym}$ takes the form $[-a,a]$, where

\[a=\min(X_{(n)}-\bth,\bth-X_{(1)}).\]

Observe that the support of $\hn^{geo,sym}$  is smaller than that of $\widehat{g}_{\bth}$, and it may also exclude some data points. Simulations suggest that the performance of  $\hn^{geo,sym}$ can suffer, especially in small samples,  due to the exclusion of data points.

Proposition <ref> states that $\widehat{g}_{\bth}$ and $\hn^{geo,sym}$ satisfy Conditions <ref> and  <ref>, as postulated. The proof of Proposition <ref> can be found in Appendix <ref>.

Suppose $f_0\in\mP_0$. Then $\hn=\widehat{g}_{\bth}$ and $\hn^{geo,sym}$ satisfy Condition <ref> and Condition <ref> with $p=1/4$ and $2/5$, respectively.

The $\hln$ corresponding to $\hn=\widehat{g}_{\bth}$ and $\hn^{geo,sym}$ is non-smooth since $\hln$ is piecewise linear in both cases. Such an estimator may not be the best choice  in small samples. Although a smoothed version of $\hn$ may perform better in small samples,  tuning of the smoothing parameter in a data dependent way may be a non-trivial task. For the log-concave MLE $\hf$, however, [Chen and Samworth, 2013] construct a well-behaved smoothing parameter in a  completely data-dependent way.  This smoothing parameter is given by

\begin{equation}\label{definition bn}
\widehat{\lambda}_n:=\sqrt{\hs-\ts},
\end{equation}

where   $\hs$ is the sample variance and $\ts$ is the variance corresponding to  $\hf$, that is

\[\hs=\dfrac{1}{n-1}\si (X_i-\bX)^2\quad\text{and}\quad\ts=\edint z^2\hf(z)dz-\lb\edint z\hf(z)dz\rb^2.\]

That the right hand side of (<ref>) is positive follows from (2.1) of [Chen and Samworth, 2013].
In light of the above, we construct a smooth $\hn$ which is symmetric about zero although it is not log-concave.

Smoothed symmetrized  estimator $\hn^{sym,sm}$:
Let us define the smoothed version of $\hf$ by

\begin{equation}\label{def: smoothed log-concave MLE}
\hf^{sm}(z)=\dfrac{1}{\widehat{\lambda}_n}\edint \hf(z-t)\gd(t/\widehat{\lambda}_n)dt,\quad z\in\RR,
\end{equation}

where $\gd$ is the standard normal density and $\smbn$ is as defined in (<ref>).
We define the smoothed symmetrized  estimator by

\begin{align}\label{representation of htsm}
\htsm(z)=\dfrac{\hts(\bth+z)+\hts(\bth-z)}{2}.
\end{align}

It is natural to ask if similar data-dependent smoothing parameters exist for $\widehat{g}_{\bth}$ and $\hn^{geo,sym}$ as well. Although  a quantity analogous to $\widehat{\lambda}_n$ can be defined for these estimators, there is no guarantee that the former will be   positive. Nevertheless, data dependent smoothing of $\widehat{g}_{\bth}$ can be an interesting direction for future research.

It can be shown that $\hn^{sym, sm}$   satisfies Condition <ref> and Condition <ref> with $p=1/5$.
Moreover, although  $\hn^{sym, sm}$ is not log-concave, it leads to an
adaptive estimator of $\theta_0$ for suitably chosen $\eta_n$. The proof of Theorem <ref> can be found in Appendix <ref>.

Suppose $f_0\in\mP_0$ satisfies Assumption <ref>, and $\bth$ is a $\sqrt{n}$-consistent estimator of $\th_0$.
Let $\hn=\hnss$ and $\eta_n=C n^{-2p'/5}$, where $C>0$ and  $p'\in(0,1/5]$.
the  estimator $\hth$  defined in (<ref>) satisfies

\begin{equation*}
\sqn(\hth-\th_0)\to_{d}N(0,\I^{-1}).
\end{equation*}

We suspect that  the rate of  decay of the Hellinger error of the estimators $\widehat{g}_{\bth}$  and $\hnss$ is  faster than our obtained rates, which are $O_p(-1/4)$ and $O_p(-1/5)$, respectively.  Our guess is based on the fact that  the geometric symmetrized estimator $\hn^{geo, sym}$  and the full MLE    in $\mP_0$ (see Theorem <ref>) are Hellinger consistent at the rate $O_p(n^{-2/5})$. The latter indicates that   $H(\hn,g_0)$ is possibly $O_p(n^{-2/5})$ if  $\hn$ is an equally good estimator of $g_0$. However, the knowledge of $p$ does not contribute much in the tuning of $\eta_n$ for practical implementation.  Therefore, we do not pursue further theoretical investigation on  the best possible rate of $\eta_n$ in this paper.

For convenience, we list the key differences among our three main estimators of $g_0$  in Table <ref>.

Comparison of different $\hn$'s: here $\hf$ is the log-concave MLE, $\hts$ is the smoothed log-concave MLE as defined in (<ref>), and $C_n^{geo} $ is the normalizing constant in (<ref>).

Estimator ($\hn$)        $\hn^{sym,sm}$     $\widehat{g}_{\bth}$     $\hn^{geo,sym}$

Summary      Smmothed      Partial MLE     GM type

symmetrized             symmetrized

Formula     $2^{-1}(\hts(\bth+z)$     $\widehat{g}_{\bth}(z)$     $C_{n}^{geo}( \hf(\bth+z)$

of  $\hn(z)$      $+\hts(\bth-z))$         $\times \hf(\bth-z))^{1/2}$

Log-concave     No     Yes     Yes

Smooth      Yes     No       No

Support     $\RR$     $[-|X|_{(n)},|X|_{(n)}]$     $[-a,a],\ a=\min(X_{(n)}-\bth,$

We close this section with a conjecture. It has previously been mentioned that the lack of control on $\tp$ at the tails make asymptotic analysis of the untruncated estimator difficult. However, we conjecture that the untruncated estimator $\hthf$ is also adaptive, i.e. $\sqn(\hthf-\theta_0)\to_d N(0,\I^{-1})$. Our simulations in Section <ref> does not refute this conjecture.

§ MAXIMUM LIKELIHOOD ESTIMATOR (MLE)

In this section, we prove that the MLE of $(\th_0,g_0)$ exists, and explore some of its properties. Before going into further details, we introduce some new terminologies. Recall that by our definition of $\mathcal{SC}_\theta$, the class $\mathcal{SC}_0$ consists of all properd closed concave functions symmetric about the origin.
For $\ps\in\mathcal{SC}_0$ and $\th\in\RR$, following [Dümbgen et al., 2011] and [Xu and Samworth, 2019], we define the criterion function for  maximum likelihood estimation  by

\begin{equation}\label{criterion function: xu samworth}
\Psi(\th,\ps,F)=\edint \ps(x-\th)dF(x)-\edint e^{\ps(x-\th)}dx.
\end{equation}

Following [Silverman, 1982], we  included a Lagrange term  to get rid of the normalizing constant  involved in density estimation. This is a common device in log-concave density estimation literature <cit.>.

We use the notation $\Psi_n(\th,\ps)$ to denote the sample version  $\Psi(\th,\ps,\Fm)$ of $\Psi(\th,\ps,F)$. Thus,

\begin{equation}\label{criterion function: Doss}
\Psi_n(\th,\ps)=\edint \ps(x-\th)d\Fm(x)-\edint e^{\ps(x-\th)}dx.
\end{equation}

Let us denote  the MLE of $(\th_0,g_0)$ by $(\hthm,\hgm)$ when they exist.
We also denote $\hpm=\log\hgm$.  Observe that provided they exist,  $(\hthm,\hpm)$  satisfies

\[(\hthm,\hpm)=\argmax_{\th\in\RR,\ps\in\mathcal{SC}_0}\Psi_n(\th,\ps).\]

For fixed $\theta\in\RR$, denote by $\widehat \ps_{\th}$ the maximizer of $   \Psi_n(\th,\ps)$ in $\ps\in\mathcal{SC}_0$.  Theorem 2.1(C) of [Doss and Wellner, 2019] implies the maximizer $\widehat{\ps}_{\th}$ exists, is unique, and that it satisfies

\[\edint e^{\widehat{\ps}_{\th}(x)}dx=1.\]

It is not hard to see that if the MLE $(\hthm,\hpm)$ exists, then

\[\hthm=\argmax_{\th\in\RR}\Psi_n(\th,\pst)\quad \text{and}\quad\hpm=\widehat{\ps}_{\hthm}.\]

Note that $\hgm=e^{\hpm}$ is the MLE of $g_0$,  and $\hgf=\hgm(\mathord{\cdot}-\hthm)$ is the MLE of $f_0$.
Theorem <ref> implies that the   the MLE $(\hthm,\hpm)$ exists when $\Fm$ is non-degenerate. The proof of Theorem<ref> an be found in Appendix <ref>.

When $\Fm$ is non-degenerate, the MLE $(\widehat{\theta}_n , \widehat{\psi}_{n} )$ of $(\theta_0 , \psi_0 )$ exists. If $\hthm$ is unique, then $\hthm\in[X_{(1)},X_{(n)}]$. Otherwise, we can find at least one $\hthm\in[X_{(1)},X_{(n)}]$.

Observe that Theorem <ref>  does not inform whether  is unique. Since $\Psi_n(\th,\ps)$ may not be jointly concave in $\theta$ and $\ps$, existence of a maximizer does not lead automatically to its uniqueness. For a particular choice of $\hthm$ however, the estimator   $\hpm=\widehat{\ps}_{\hthm}$ is unique by Theorem $2.1(c)$ of [Doss and Wellner, 2019]. Therefore, if $(\th,\ps_1)$ and $(\th,\ps_2)$ both are MLEs of $(\th_0,\ps_0)$, we must have $\ps_1=\ps_2$.

Although we can not theoretically prove the uniqueness of $\hthm$, we are yet unaware of any set up which leads to non-unique MLE.
Moreover, in all our simulations,  turned out to be unique, even when the underlying density $f_0$ was skewed or non-log-concave. Considering this fact, in what follows, we  refer to  as “the MLE" instead of “an MLE". We must remark that even if $\hthm$ is not unique, all our theorems  still hold for each version of .

On the other hand, when $\Fm$ is degenerate, Lemma <ref> entails that the MLE  does not exist.
However,  for distributions with a density, probability of  $\Fm$ being degenerate is zero. Therefore  we will not worry about this  particular situation. The proof of Lemma <ref> is delegated to Appendix <ref>.

Suppose $\Fm$ is degenerate, i.e.  $\Fm\{x_0\}=1$ for some $x_0\in\RR$. Then the MLE of $(\theta_0,g_0)$ in $\mathcal P_0$ does not exist.

The following theorem sheds some light on the structure of $\hpm$.  This theorem is a direct consequence of Theorem $2.1(c)$ of [Doss and Wellner, 2019], and hence we skip the proof.

Suppose $(\hthm,\hpm)$ is the MLE.
For $\Fm$ non-degenerate,  $\hpm$  is piecewise linear with knots belonging to a subset of the set $\{0,\pm|X_1-\hthm|,\ldots,\pm|X_{n}-\hthm|\}$. Also,  for $x\notin[-|X-\hthm|_{(n)},|X-\hthm|_{(n)}]$, we have $\hpm(x)=-\infty$.
Moreover if $0\notin \{\pm|X_1-\hthm|,\ldots,\pm|X_{n}-\hthm|\}$, then $\hpm'(0\pm)=0.$

The MLE can be computed using our R package , which implements a grid search method to optimize $\Psi_n(\theta,\widehat \psi_{\theta})$ in $\theta$.

§.§ Asymptotic properties of the MLE

For $f_0\in\mP_0$,  we  showed that  the one-step estimators are consistent. Theorem <ref> (A) below shows that the MLE $\hthm$  enjoys similar consistency
property. In fact, $\hthm$ is strongly consistent for $\th_0$. Part A of Theorem <ref> also entails that $\hgm$ and $\hgf$ are strongly Hellinger consistent. Part B of Theorem <ref> concerns the rate of convergences.
The proof of Theorem <ref> is delegated to Appendix <ref>.

Suppose $f_0\in\mP_0$.
Then the following assertions hold:

*  As $n\to\infty$, $\hthm\as \theta_0$, $H(\hgf,f_0)\as 0$, and
$H(\hgm,g_0)\as 0$.

*  Furthermore, ${|\hthm-\th_0|=O_p(n^{-2/5})}$, $H(\hgf,f_0)=O_p(n^{-2/5})$, and

The rate of $H(\hgm,g_0)$ as given by Theorem <ref>  is standard for  log-concave density estimators. The MLEs in  $\mathcal{SLC}_{0}$ and $\mathcal{LC}$ have the same rate of  Hellinger error decay  <cit.>. Moreover, this rate
probably can not be improved by any other  estimator of $g_0$. To see why, first note that Theorem 1 of [Doss and Wellner, 2016] proves that the minimax rate of   Hellinger error decay in  $\mathcal{LC}$ is $O_p(n^{-2/5})$.
Remark 4.2 of [Doss and Wellner, 2019] conjectures that the minimax rate of estimation in the constrained class $\mathcal{SLC}_0$  stays the same. Since estimation of $g_0$ in $\mP_0$ can not be  easier than  estimation in the smaller class $\mathcal{SLC}_0$, it is likely  that the  minimax rate of estimating $g_0$ in $\mP_0$ is also $O_p(n^{-2/5})$.

However, the MLE $\hthm$ probably  convergences to $\th_0$ at a rate faster than  $O_p(n^{-2/5})$. Our simulations suggest that $\hthm$ is $\sqn$-consistent, based on which, we  conjecture that $\hthm$ is also an adaptive estimator of $\theta_0$. In our model, the low dimensional parameter of interest, i.e. $\theta_0$, is bundled with  the infinite dimensional nuisance parameter.  Obtaining the precise rate of convergence for the MLE  in such semiparametric models is typically difficult [Murphy and Vaart, 2000].  Nevertheless, since the MLE is tuning parameter free, finding its exact asymptotic distribution will be an interesting future research direction.

§ SIMULATION STUDY

(a) Plot of the standard Laplace, standard normal and standard logistic densities.
(b) Plot of the
symmetrized beta  density $f_{0,r}$,  defined in (<ref>), for  different values of $r$.
(c) Plot of Fisher information $\mathcal{I}_{f_{0,r}}$ vs $r$ where $f_{0,r}$ is the symmetrized beta density.

This section compares the efficiency of our estimators and the coverage of the resulting confidence intervals with  that of [Stone, 1975] and [Beran, 1974].
The general set-up of the simulation is as follows.
We consider as $g_0$ the standard normal, standard logistic, and standard Laplace density. We also  consider a fourth density, namely the symmetrized beta density,  which is defined as follows:

\begin{equation}\label{definition: Symmetrized beta}
f_0(x)\equiv f_{0,r}(x)=\dfrac{\Gamma\lb(3+r)/2\rb}{\sqrt{\pi r}\Gamma(1+r/2)}\lb 1-\dfrac{x^2}{r}\rb^{r/2}1_{[-\sqrt{r},\sqrt{r}]}(x),\quad r> 0.
\end{equation}

Here $\Gamma$ is the usual Gamma function.
It is straightforward to verify that in this case

\[\phi_0'(x)=\dfrac{-x}{1-x^2/r}1_{[-\sqrt{r},\sqrt{r}]}(x)\quad\text{and }\quad\phi_0^{\prime\prime}(x)=-\frac{(1+x^2/r)}{(1-x^2/r)^2}1_{[-\sqrt{r},\sqrt{r}]}(x).\]

Some computation shows  that $r\leq 2$ leads to $\mathcal{I}_{f_{0,r}}=\infty.$ However   for $r>2$, $\mathcal {I}_{f_{0,r}}<\infty$, and

$f_0\in\mP_0$. This is an example of a case where Assumption <ref> fails to hold because $\phi_0^{\prime\prime}$ is unbounded. We consider the symmetrized beta density with $r=2.1$ and $4.5$.

See  Figure <ref>a and <ref>b  for a pictorial representation of the above-mentioned densities.
Figure <ref>c  displays the plot of $\mathcal{I}_{f_{0,r}}$ versus $r$ for the symmetrized beta density, which depicts that $\mathcal{I}_{f_{0,r}}$  decreases steeply for $r>2$. This finding is consistent with $\I$ being $\infty$ when $f_0$ is the uniform density on $[-1,1].$

We set $\th_0=0$, and generate $3000$ samples of size $n=30$, $100$, $200$, and $500$ from each of the above-mentioned densities. We define the efficiency of an estimator $\theta_n$ by

\begin{equation}\label{Def: efficiency}
\text{Efficiency}(\th_n)=\dfrac{1/(n\I)}{ Var(\th_n)}.
\end{equation}

In practice, we replace $Var(\th_n)$ by its Monte Carlo estimate.

§.§.§ The shape-constrained estimators:

Along with the MLE and the untruncated one step estimator defined in (<ref>), we consider the truncated one step estimators with truncation level $\eta=10^{-2}$, $10^{-3}$, and $10^{-5}$.  We select the sample mean as the preliminary estimator $\bth$ because it exhibited slightly better overall performance than other potential choices of $\bth$, e.g. the median and the trimmed mean.   We choose the partial MLE estimator and the smoothed symmetrized estimator $\tilde g_n^{sym,sm}$ as the  estimator of $g_0$ because simulations suggest that they perform significantly better than  $\hn^{geo, sym}$.

§.§.§ Comparators: [Stone, 1975] and [Beran, 1974]'s estimators:

As mentioned earlier, [Stone, 1975]'s estimator is a truncated one step estimator which uses symmetrized Gaussian kernels  to estimate $g_0$. Similar to [Stone, 1975], we let
the corresponding truncation parameter  and the kernel bandwidth parameter  to be $d_n s_n$ and $t_n s_n$, respectively, where $s_n$ is the median absolute deviation (MAD), and $d_n>0$ and $t_n>0$ are  tuning parameters.
Following [Stone, 1975], we  take the preliminary estimator to be   the sample median.

As previously stated, [Beran, 1974]'s estimator is a  rank-based estimator which depends on the scores. [Beran, 1974] uses  Fourier series expansion to estimate the scores, which requires choosing (a) the number of  basis functions ($b_{c,n}$), and (b) a scaling parameter $\rho_n$, which is used to approximate a derivative term by quotients during the estimation of the Fourier coefficients of the score.  This  estimator uses a preliminary estimator of $\theta$, which we take to be the sample median following [Beran, 1974]'s suggestion.
In this case, the sum of squares of the estimated Fourier coefficients
is a  consistent estimator of $\I$ <cit.>.

For sample size $n=40$, [Stone, 1975] uses $d_n=20$ and $t_n=0.60$, but [Beran, 1974] does not give any demonstration on how to choose the tuning parameters.
To choose some reliable values for the associated tuning parameters, we start with some pre-selected grids, and employ a grid search procedure (see Appendix <ref> for more details). The selected tuning parameter is the maximizer of the estimated efficiency among the grid, where the   efficiency is estimated using  one hundred Monte Carlo replications. Of course, this procedure requires the knowledge of the unknown distribution, and hence, not implementable in practice. However, our procedure at least guarantees a reliable benchmark  to compare the performance of our estimators.
We refer to the resulting tuning parameters as “optimal" for the sake of simplicity. However, it should be kept in mind that these tuning parameters depend on the chosen grid, and therefore, may be different from the globally optimal tuning parameters if the grid selection is not accurate enough. This could have been overcome by an exhaustive search but that is beyond the scope of the current paper.

For each distribution and each sample size, we construct  two versions of the nonparametric estimators. The first version is based on the aforementioned optimal tuning parameter, and the other version
uses  tuning parameters slightly away from the optimal region. For convenience, we will refer to the second set of tuning parameters as “non-optimal".  See Appendix <ref> for more details on these tuning parameters.

We should mention that neither [Stone, 1975] nor [Beran, 1974] construct  confidence intervals. However,   both estimators rely on consistent estimators of $\I$, namely, the estimator $\widehat A_n(r_n,c_n)$   of [Stone, 1975] (see  (1.10) of [Stone, 1975]),  and the squared $L_2$ norm of the estimated score in  [Beran, 1974].
We use the above estimators of $\I$  to build the respective confidence  intervals of [Stone, 1975] and [Beran, 1974].

§.§ Results:

Figure <ref> implies that  [Stone, 1975] and [Beran, 1974]'s estimators have high  efficiency  when they are equipped with the optimal tuning parameters. In fact Stone's estimator has better efficiency than all other estimators in case of logistic and normal distribution. However, even with the optimal tuning parameter, the coverage of [Stone, 1975]'s confidence interval is quite low (see Figure <ref>). The coverage of [Beran, 1974]'s confidence interval is comparatively better but  still not as good as  the shape-constrained estimators (see Figure <ref>). The poor coverage of the nonparametric confidence intervals is probably  due to their smaller width, as shown by Figure <ref>. We suspect that for our tuning parameters, the nonparametric estimators of $\I$  overestimate $\I$, leading to narrow confidence intervals.
When the tuning parameters are non-optimal, the  nonparametric estimators suffer  in terms of both efficiency and the coverage. This is most evident in large samples because in this case,  their performance does not significantly improve  with the sample size. Figure <ref> and <ref> entail that
all estimators have markedly poor performance in the symmetrized beta case when $r=2.1$.

Let us turn our attention to the one step estimators now. Figure <ref> underscore that
the efficiency of the one step estimators  monotonously decreases with the truncation, with the highest efficiency being observed  at the truncation level zero. However, the difference becomes smaller as the truncation level decreases. In particular, at  truncation level $10^{-5}$, the difference almost vanishes. The one step estimators with lower truncation level, i.e.  $\eta\leq 0.001$, exhibit satisfactory performance in terms of both efficiency and coverage (see Figure <ref> and Figure <ref>). The  estimators with higher truncation level lag in terms of efficiency as expected, although they  exhibit superior coverage in some cases.

The additional gain in  coverage that sometimes accompany  higher levels of truncation is
probably due to slightly wider confidence intervals (see Figure <ref>). Wider confidence intervals are expected with high levels of truncation since the length of the confidence intervals, which is a constant multiple of $\hin(\eta)^{-1}$, increases in $\eta$.  However, higher level of truncation may not always lead to a better coverage, especially since high truncation level  can also result in significant loss of efficiency. See for instance the case of symmetrized beta with $r=2.1$, where the one step estimators with truncation $0.01$ lags behind the other one-step estimators   in terms of both efficiency and coverage. This case clearly demonstrates that the one step estimators with higher level of truncation are not always  reliable. In contrast, the one step estimators with low level of truncation, particularly the untruncated one step estimator, always exhibit  satisfactory performance. In view of above, we propose the untruncated estimator for practical implementation.

Close inspection shows  the  smoothed symmetrized estimators have better overall performance than the partial MLE estimators  with the obvious exception of Laplace distribution, which has a non-smooth density.
Finally,  we note that the one step estimators with lower truncation level  have better efficiency than the MLE under all distributions except  Laplace. However, when it comes to the coverage of the confidence intervals, the MLE can be competitive with the best one step estimators, especially in small samples.

In summary, the coverage of the nonparametric confidence intervals is not satisfactory for the tuning parameters considered here, and  the efficiency of the nonparametric estimators depends crucially on the tuning parameters.  For some choices of tuning parameter,  these estimators may exhibit excellent efficiency but for other choices, they severely underperform. In contrast, our untruncated one step estimator and the resulting confidence interval perform reasonably well under all scenarios.
The performance of the untruncated one-step estimator also speaks in favor of our conjecture that it is an adaptive estimator.
Although we do not show the plots of the mean squared error (MSE) here, they depict the same
patterns as the efficiency plots in Figure <ref>.

We close this section with a remark on the necessity of Assumption <ref>. The symmetrized beta distributions do not satisfy Assumption <ref>, but the one step estimators still seem to be efficient when $r=4.5$. Although the one step estimators
perform poorly in case of $r=2.1$, they still perform better than [Stone, 1975] and [Beran, 1974]'s estimators, whose  asymptotically efficiency under this distribution is theoretically validated. Thus,  our simulations do not refute the possibility that Assumption <ref> might be unnecessary.

Comparison plot when the one step estimators are the Partial MLE estimator

Comparison plot when the one step estimators are the smoothed symmetrized estimator

comparison of  efficiency: the data-generating distributions are normal (topleft), Laplace (topmiddle), Symmetrized beta with $r=2.1$ (topright), and $r=4.5$ (bottomleft), and logistic (bottommiddle).  For Stone's and Beran's estimators (in solid lines), (o) stands for the optimal tuning parameter, and (n) corresponds to the  non-optimal tuning parameter. Here $q$ stands for the truncation parameter $\eta$ in our one-step estimators (in dashed lines).

Comparison plot when the one step estimators are the Partial MLE estimator

Comparison plot when the one step estimators are the smoothed symmetrized estimator

Comparison of the coverage of the 95% confidence intervals:  the data-generating distributions are normal (topleft), Laplace (topmiddle), Symmetrized beta with $r=2.1$ (topright), and $r=4.5$ (bottomleft), and logistic (bottommiddle).
Here $q$ stands for the truncation parameter $\eta$ in our one-step estimators. The errorbars are given by $\pm 2$ standard deviation.

Comparison plot when the one step estimators are the Partial MLE estimator

Comparison plot when the one step estimators are the smoothed symmetrized estimator

Comparison of the average confidence interval length (averaged across the 3000 Monte Carlo samples): the data-generating distributions are normal (topleft), Laplace (topmiddle), Symmetrized beta with $r=2.1$ (topright), and $r=4.5$ (bottomleft), and logistic (bottommiddle).  For Stone's and Beran's estimators (in solid lines), (o) stands for the optimal tuning parameter, and (n) corresponds to the  non-optimal  tuning parameter. Here $q$ stands for the truncation parameter $\eta$ in our one-step estimators (in dashed lines). The errorbars are given by $\pm 2$ standard deviation.

Comparison of the coverage of the 95% confidence intervals for Beran's and Stone's estimators:  the data-generating distributions are normal (topleft), Laplace (topmiddle), Symmetrized beta with $r=2.1$ (topright), and $r=4.5$ (bottomleft), and logistic (bottommiddle). Here (o) stands for the optimal tuning parameter, and (n) corresponds to the non-optimal  tuning parameter. The errorbars are given by $\pm 2$ standard deviation.

§ DISCUSSION

In this paper, we show that under the additional assumption of log-concavity, adaptive estimation of $\theta_0$ is possible with only one tuning parameter.
Our simulations suggest that the tuning parameter-free untruncated one step estimator may also be adaptive.  This demonstrates the usefulness of  log-concavity  assumption in semiparametric models in facilitating a simplified  estimation procedure.
It is natural to ask what happens if the above shape restriction fails to hold. For functionals of log-concave MLE type estimators, this question can be partially answered building on the log-concave projection theory developed by [Dümbgen et al., 2011], [Cule and Samworth, 2010], [Xu and Samworth, 2019], and [Barber and Samworth, 2020]. See [Laha, 2019] for discussion of the case when the log-concavity assumption is violated in our model $\mP_0$.
In particular, it can be shown that, even if $f\notin\mP_0$, as long as $f$ is symmetric about  $\th_0$,  the MLE and the truncated one step estimators are still consistent under mild conditions.

§ ACKNOWLEDGEMENT

The author is  grateful to Jon Wellner for his help.

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§ APPENDIX

The appendix is organized as follows.
Appendices <ref>, <ref>, and <ref> contain the proofs for the one step estimators, where Appendices <ref>, <ref>, and <ref> contain the proofs for the MLE. The proof of the main theorem is presented first, followed by the auxiliary lemmas required for the proof. Some common technical facts, which are used repeatedly in the proofs, are listed at the end in
Appendix <ref>.
Appendix <ref> contains details on the selected tuning parameters for  [Stone, 1975] and [Beran, 1974]'s estimators.

Before proceeding any further, we introduce some new notations and terminologies.
For $i=1,\ldots,n$, consider  the pseudo-observations $Z_i=X_i-\theta_0$. Note that, if the $X_i$'s have density $ f_0$, then the $Z_i$'s have density $g_0$, and distribution function $G_0$. We will denote the log-densities corresponding to $\hts$, $\hnss$, $\widehat{g}_{\theta}$ and  ${\hn}^{geo,sym}$ by $\hlns$, $\hln^{sym, sm}$, $\widehat{\psi}_{\theta}$ and $\hln^{geo, sym}$, respectively.
As usual, $\hlnsp$, $(\hln^{sym, sm})'$, $\widehat{\psi}_{\theta}'$,  and $(\hln^{geo, sym})'$ will denote the corresponding right derivatives. We remark in passing that there is nothing special about the right derivative, and any $L_1$ derivative would have worked. However, we fix one specific version  to avoid future confusion. We denote the distribution functions of $\hts$, $\hnss$, and  ${\hn}^{geo,sym}$ by $\Hsm$, $\Gs$, and $\tilde G_n^{geo, sym}$, respectively.

The  empirical process of the $X_i$'s will be denoted by $\mathbb G_n=\sqn(\Fm-F_0)$.
For any  function $h:\RR\mapsto\RR$, and a measure $Q$ on $\RR$, we write
$Q h:=\int_{\RR} h dQ$ provided $h$ is integrable with respect to $Q$.
Suppose $\mathcal H$ is a  class of $Q$-measurable functions. We denote by
$ \|Q\|_{\mathcal H}$ the supremum $\sup_{h\in\mathcal H}\abs{Qh}$.
For the sake of simplicity, we will denote $\td=\theta_0-\bth$ and $\delta_n=\th_0-\hthm$ in our proofs.

For a measure $P$ on $\RR$, we define the $L_{P,k}$ norm of the function $h$ as

\[\|h\|_{P,k}=\lb\edint |h(x)|^k dP(x)\rb^{1/k},\quad k\geq 1.\]

For any class of functions $\mathcal H$, we will denote

\[\|\mathcal H\|_{P,k}=\sup_{h\in\mathcal H}\|h\|_{P,k}.\]

For two distribution functions $F_1$ and $F_2$ with densities $f_1$ and $f_2$, the total variation distance between $F_1$ and $F_2$ is given by
We define the
Wasserstein distance between two measures $\mu$ and $\nu$ on $\RR$ by

\begin{equation}\label{def: Wasserstein distance}
\end{equation}

where $F$ and $G$ are the distribution functions corresponding to $\mu$ and $\nu$ respectively.
This representation of $d_W(\mu,\nu)$ follows from  [Villani, 2003], page $75$. By an abuse of notation, sometime we will denote the above distance by $d_W(F,G)$ as well.

For two sets $A$ and $B$, $A\times B$ will represent the Cartesian product. For any set $A\subset \RR$, and $x\in\RR$, we use the usual notation $A+x$ to denote the translated set  $\{y+x\ :\ y\in A\}$.
The notation $\overline{A}$ will refer to the closure of the set $A$. For any function $h$,  $1_{[h(x)\leq C]}$ will denote the indicator function of the event $h(x)\leq C$. For any  set $A$, we let $1_A(x)$  be the indicator function of the event $x\in A$. As usual, we denote  by $\varphi$ the standard Gaussian density.

In some of our proofs, we will replace $\hnss$ by a more general mixture density which satisfies Condition <ref>.

[Condition for $\hnss$]
The density $\hn$ is symmetric about zero and satisfies
$\hn(x)=(g_{1n}(x)+g_{2n}(x))/2$, where $g_{1n}$ and $g_{2n}$ are log-concave densities. The densities $\hn$, $g_{1n}$, and $g_{2n}$ satisfy  Conditions <ref> and <ref>. Moreover, $\supp(g_{1n})=\supp(g_{2n})=\RR$, and the $p$ in Condition <ref> is the same for $\hn$, $g_{1n}$, and $g_{2n}$.

We will later show in Lemma <ref> that $\hts(\bth\pm\mathord{\cdot})$ and $\hnss$ satisfy Condition <ref>, and in Lemma <ref>, we will show that these densities satisfy Condition <ref> with $p=1/5$. Since $\hts(\bth\pm\mathord{\cdot})$ is the convolution of two log-concave densities, it is log-concave. That  $\hnss$ satisfies Condition <ref> follows immediately from the above results.

We will frequently use the fact that if $f$ is a log-concave density, then $f>0$ on $J(F)=\{0<F<1\}$ <cit.>. Therefore,  $\iint(\dom(\log f))=\iint(\supp(f))=\iint(J(F))$.  As a consequence,  $F^{-1}$ is  strictly increasing, and differentiable with derivative $1/f(F^{-1}(t))$ on $(0,1)$ by Fact <ref>. Also, a log-concave $f$ is thus  continuous on $\iint(J(F))$ by Theorem 10.1 of [Rockafellar, 1970]. When $f_0\in\mP_0$, furthermore, $f_0$ and $g_0$ are absolutely continuous  on $\RR$ by Theorem 3 of [Huber, 1964].
Now we state some more useful facts about log-concave densities.

[Lemma 1 of [Cule and Samworth, 2010]]

If $f$ is a univariate log-concave density, then there exists $\alpha>0$ and $\beta\in\RR$ so that  $f(x)\leq e^{-\alpha |x|+\beta} $.

If $f$ is log-concave, then $f/F$ is non-increasing on $J(F)$ and $f/(1-F)$ is non-decreasing on $J(F)$.

This is a well known fact about log-concave densities. See for reference Theorem 1 and Corollary 2 of [Bagnoli and Bergstrom, 2005].

The following two facts will be very useful to lower bound $\hn$ on $[-\xin,\xin]$.

Suppose $\hn$ is a log-concave density satisfying Condition <ref>.   Then for any $x\in\RR$,

\[\hn(x)\geq \omega_n\min(\tilde{G}_n(x), 1-\tilde{G}_n(x)), \]

where $\omega_n\geq 0$   satisfies $\omega_n\to_p \omega_0>0$.  Here $\omega_0>0$ is a constant depending only on $g_0$.

If $\tilde G_n(x)$ is zero or one, then the statement  trivially holds. Therefore, we assume $x\in J(\tilde G_n)$, i.e. $0<\tilde G_n(x)<1$. For $q\in(0,1/2)$, Fact <ref> implies

\[\hn(\tilde G_n^{-1}(q))\geq 2q\slb\hn(\tilde G_n^{-1}(1/2))\srb\geq 2\min(q,1-q)\hn(\tilde G_n^{-1}(1/2)).\]

For $q\in(1/2,1)$, on the other hand, Fact <ref> implies

\[\hn(\tilde G_n^{-1}(q))\geq 2(1-q)\hn(\tilde G_n^{-1}(1/2))\geq 2\min(q,1-q)\hn(\tilde G_n^{-1}(1/2)).\]
Because $\tilde G_n(x)\in(0,1)$, replacing $q$ by $\tilde G_n(x)$  we obtain that

\begin{align}\label{infact: f grtr than F}
\hn(x)\stackrel{(a)}{=}\hn(\tilde G_n^{-1}(\tilde G_n(x)))\geq  2\hn(\tilde G_n^{-1}(1/2))\min\slb\tilde G_n(x),1-\tilde G_n(x)\srb.
\end{align}

Here (a) uses the fact that $\hn(x)>0$ which follows since $x\in\iint(J(\tilde G_n))=\iint(\dom(\hln))$.
The rest of the proof follows setting $\omega_n=2\hn(\tilde G_n^{-1}(1/2))$, which converges in probability to $\omega_0=2g_0(0)$ by Condition <ref> and Fact <ref>.

Suppose either $\hn$ satisfies  Condition <ref>, or $\hn$ is a log-concave density satisfying Condition <ref>. Then the assertions of Fact <ref>  hold.

If $\hn\in\mathcal{LC}$, the proof follows from Fact <ref>.

Therefore we consider the case when $\hn$ satisfies Condition <ref>. Since the component densities $g_{1n}$ and $g_{2n}$ in  Condition <ref> are log-concave, Fact <ref> applies to them. Denote by $G_{1n}$ and $G_{2n}$ the corresponding distribution functions. Equation <ref> in the proof of Fact <ref> implies

\[g_{1n}(x)+g_{2n}(x)\geq b_n \lbs  G_{1n}(x)\slb 1- G_{1n}(x)\srb+ G_{2n}(x)\slb 1- G_{2n}(x)\srb\rbs,\]

\[b_n=2\min\lb g_{1n}(G_{1n}^{-1}(1/2)),  g_{2n}(G_{2n}^{-1}(1/2))\rb\to_p 2g_0(0)\]

by the fact that $g_{1n}$ and $g_{2n}$ satisfy Condition <ref> and Fact <ref>.
Since the function $(x,y)\mapsto xy$ is convex on $\{(x,y):x,y\geq 0\}$, it follows that

\[\hn(x)\geq b_n \frac{G_{1n}(x)+ G_{2n}(x)}{2}\lb 1-\frac{ G_{1n}(x)+ G_{2n}(x)}{2}\rb=b_n\tilde G_n(x)(1-\tilde G_n(x)).\]

The  proof  follows noting
$x(1-x)\geq \min(x,1-x)/2$ for $x\in(0,1)$.

§ PROOF OF THEOREM <REF>

We first argue that it suffices to prove the theorem only for the case when $\eta_n$ equals $Cn^{-2p/5}$.
In the latter case, we would show $\sqn(\hth-\th_0)\to_d N(0,\I^{-1})$ for $\eta_n=Cn^{-2p/5}$ when $H(\hn, g_0)=O_p(n^{-p})$. Note that for any $p'\in(0,p]$,  $H(\hn, g_0)=O_p(n^{-p'})$ trivially holds since
$H(\hn, g_0)=O_p(n^{-p})$. Therefore, replacing $p$ by $p'$ in what we  just proved,  $\sqn(\hth-\th_0)\to_d N(0,\I^{-1})$  would follow identically for $\eta_n=Cn^{-2p'/5}$. Thus, it is enough to consider the case when $\eta_n=Cn^{-2p/5}$.

From  (<ref>) we obtain that

\begin{align*}
-(\hth-\bth)= \dint_{\bth-\xin}^{\bth+\xin}\dfrac{\hln'(x-\bth)}{\hin(\eta_n)}d\Fm(x)=\dint_{-\xin}^{\xin}\dfrac{\hln'(z)}{\hin(\eta_n)}d\Fm(z+\bth).
\end{align*}

Denoting $\td=\th_0-\bth$, we observe that the above expression  writes as

\begin{align}\label{theorem: main: one-step: first split}
\MoveEqLeft\underbrace{\dint_{-\xin}^{\xin}\dfrac{\tp(z)-\psp'(z-\td)}{\hin(\eta_n)}d(\Fm(z+\bth)-F_0(z+\bth))}_{T_{1n}}\nonumber\\
&\ +\underbrace{\dint_{-\xin}^{\xin}\dfrac{\tp(z)}{\hin(\eta_n)}\lb f_0(z+\bth)-g_0(z)\rb dz
&\ +  \underbrace{\dint_{-\xin}^{\xin}\dfrac{\psp'(z)}{\hin(\eta_n)}g_0(z)dz}_{T_{4n}} + \underbrace{\dint_{-\xin}^{\xin}\dfrac{\psp'(z-\td)}{\hin(\eta_n)}d(\Fm(z+\bth)-F_0(z+\bth))}_{T_{5n}}
\end{align}

Observe that  $T_{3n}$ and $T_{4n}$ vanish since $\tp$ and $\psi'_0$ are odd functions while $g_0$ is an even function.

The proof of Theorem <ref> has three main steps. The first step uses Donsker Theorem to show that the empirical process term  $T_{1n}$ is $o_p(n^{-1/2})$. The term $T_{2n}$ accounts for the bias due to the use of $\bth$ instead of the true center $\theta_0$ in the construction of the scores. The second step of the proof shows that the order of $T_{2n}$ is same as $\td=\theta_0-\bth$. In particular, we will show that $T_{2n}=-\td(1+o_p(1))$.
Since $\td=O_p(n^{-1/2})$, the above two steps lead to

\[\sqn(\bth-\hth)=o_p(1)+\sqn(\bth-\th_0)+\sqn T_{5n}.\]

The third  step of the proof shows that the term $\sqn T_{5n}$ is asymptotically normal
with variance $\I^{-1}$.
A rearranging of the terms in the above display then establishes the desired asymptotic convergence of $\sqn(\hth-\theta_0)$. The rest of the proof is devoted towards proving the above-mentioned three steps.

§.§.§ First step: asymptotic negligibility of $\sqn T_{1n}$:

First, let us denote $\mathcal T_n= [\bth-\xin,\bth+\xin]$.
Recall that in Section <ref> we denoted the empirical process $\sqn(\Fm-F_0)$ by $\mathbb{G}_n$.
Note that $\sqn T_{1n}$ also writes as

\begin{align}\label{intheorem: T1n: representation}
\sqn  T_{1n} = \sqn\dint_{\bth-\xi_n}^{\bth+\xi_n}\dfrac{\tp(x-\bth)-\php'(x)}{\hin(\eta_n)}d(\Fm-F_0)(x)=\edint\dfrac{ h_n(x)}{\hin(\eta_n)}d\mathbb{G}_n(x),
\end{align}

where by $h_n$ we denote the function

\begin{equation}\label{inlemma: def: main: hn}
h_n(x)=(\tp(x-\bth)-\phi_0'(x))1_{\mathcal T_n}(x),\quad x\in\RR.
\end{equation}

Because $\eta_n=O( n^{-2p/5})$, Lemma <ref> implies

\begin{equation}\label{inlemma: t1: bounding hln prime}
\sup_{x\in\mathcal T_n}|\hln'(x-\bth)|=O_p(n^{p/5}).
\end{equation}

Thus $\hln'$ restricted to the compact set $\mathcal T_n$ is  bounded. We can extend the function $x\mapsto \hln'(x) 1_{\mathcal T_n}(x)$ to $\RR$ in a way such that the resulting function $\widehat u_n$ is still monotone and has the same bound. This can be done by setting $\widehat u_n$ to be $\hln'(\bth-\xin)$ and  $\hln'(\bth+\xin)$ on the intervals  $(-\infty,\bth-\xin]$ and $[\bth+\xin,\infty)$, respectively. Note also that we can replace $\hln'$ by $\widehat u_n$ in the definition of $h_n$, i.e.

\[h_n(x)=(\widehat u_n(x-\bth)-{\phi}_0'(x))1_{\mathcal T_n}(x).\]

Let us denote  $M_n=Cn^{p/5}$ for some $C>0$ and define

\begin{equation}\label{intheorem: def: U n}
\mathcal{U}_{n}(M_n)=\bigg\{u:\RR\mapsto[-M_{n},M_{n}]\ \bl\ \ u\text{ is non-increasing}\bigg\}.
\end{equation}

Since $\|\widehat u_n\|_{\infty}=O_p(n^{p/5})$, for sufficiently large $C$,  $\hat u_n(\mathord{\cdot}-\bth)\in \mathcal{U}_n(M_n)$ with high probability.
Now  define the class $\mathcal{H}_n(C)$ by

\begin{align*}
\mathcal{H}_n(C)=  \bigg\{h:\RR\mapsto  \RR\ \bl \ &\ h(x)=(u(x)-{\phi}_0'(x))1_{[r_1,r_2]}(x),\ u\in\mathcal{U}_{n}(M_n), \\
\|h\|_{P_0,2}\leq Cn^{-2p/5}(\log n)^{3/2},\ \ \|h\|_\infty\leq M_n,\\
&\ [r_1,r_2]\subset [\th_0-C\log n,\th_0+C\log n]\cap\iint(\dom(\phi_0)) \bigg \}.
\end{align*}

The notation $\mathcal H_n(C)$ does not depend on $M_n$ because $M_n=Cn^{p/5}$ is also a function of $C$.

We want to show that $h_n\in\mathcal H_n(C)$ with high probability for large $n$.
Note that

\[\sup_{x\in\mathcal T_n}|\phi'_0(x)|=\sup_{x\in [-\d-\xi_n,-\d+\xi_n]}|\psi'_0(x)|.\]

Lemma <ref> in conjunction with the fact that $\eta_n=O(n^{-2p/5})$ implies

\begin{equation}\label{intheorem: boundedness of psp}
\sup_{x\in \mathcal T_n}|\phi_0'(x)|=O_p(\log n).
\end{equation}

Thus (<ref>) and (<ref>) imply $\|h_n\|_{\infty}=O_p(n^{p/5})$. Lemma <ref> bounds the $L_{P_0,2}$ norm of $h_n$ entailing $\|h_n\|_{P_0,2}=O_p(n^{-2p/5}(\log n)^{3/2})$.
Lemma <ref> implies, on the other hand,

\[
\lim_{n\to\infty}  P\slb  \mathcal T_n\subset [\th_0-C\log n,\th_0+C\log n]\cap \iint(\dom(\phi_0))\srb= 1.
\]

Therefore, we conclude that given $t>0$, we can choose $C>0$ so large such that $P(h_n\in\mathcal{H}_n(C))>1-t$.

Theorem 2.7.5 of [Van der Vaart and Wellner, 1996] (pp. $159$) states that there exists an absolute constant $C'>0$ so that for any $\e>0$ and any probability measure $\RR$ on the real line,

\begin{equation}\label{inlemma: finite entropy increasing}
\log N_{[\ ]}(\e,\mathcal{U}_{n}(M_n),L_2(Q))\leq C'M_n\e^{-1}.
\end{equation}

On the other hand, using Theorem 2.7.5 of [Van der Vaart and Wellner, 1996], it can also be shown that the
class $\mathcal{F}_I$ of all indicator functions of  the form $1_{[z_1,z_2]}$, where $z_1\leq z_2$ with $z_1,z_2\in\RR$,  satisfies

\begin{equation}\label{inlemma: finite entropy indicator functions}
\log N_{[\  ]}(\e,\mathcal{F}_I,L_2(Q))\leq C' 2\e^{-1}.
\end{equation}

Using (<ref>) and (<ref>) we derive that

\[\log N_{[\ ]}(\e,\mathcal{H}_n(C),L_2(P_0))\lesssim {M_n}{\e}^{-1}.\]

For $x<1$, the bracketing integral

\begin{align*}
\mathcal{J}_{[\ ]}(x,\mathcal{H}_n(C),L_2(P_0))=&\ \dint_{0}^{x}\sqrt{1+\log N_{[\ ]}(\e,\mathcal{H}_n(C),L_2(P_0))}d\e\\
\lesssim &\  2M_n\dint_{0}^{x/M_n}\e^{-1/2}d\e,
\end{align*}

which equals $\sqrt{x M_n}$. Let us also denote $K_n=Cn^{-2p/5}(\log n)^{3/2}$. Note that

\[\|\mathcal{H}_n(C)\|_{P_0,2}=\sup_{h\in\mathcal{H}_n(C)}\|h\|_{P_0,2}=K_n.\]

Then from Fact <ref>
it follows that

\begin{align*}
E\|\mathbb{G}_n\|_{\mathcal{H}_n(C)}\lesssim &\   \mathcal{J}_{[\ ]}(K_n,\mathcal{H}_n(C),L_2(P_0))\slb 1+\frac{  \mathcal{J}_{[\ ]}(K_n,\mathcal{H}_n(C),L_2(P_0))}{K_n^2\sqn}M_n\srb
\end{align*}

which is bounded by a constant multiple of

\[
\sqrt{K_nM_n}+K_n^{-1}M_n^2n^{-1/2} \lesssim (\log n)^{3/4}n^{-p/10}+(\log n)^{-3/2}n^{-1/2}.
\]

Now fix $t'>0$  and $\xi>0$. We can choose $C$ so large such that $P( h_n\notin \mathcal{H}_n(C))<\xi/2$. Therefore

\begin{align*}
\MoveEqLeft  P\lb \edint h_n(x)d\mathbb{G}_n(x)>t' \rb\\
\leq &\  P\lb \edint h_n(x)d\mathbb{G}_n(x)>t' , h_n\in \mathcal{H}_n(C)\rb+P\lb h_n\notin \mathcal{H}_n(C)\rb\\
\stackrel{(a)}{\leq} &\  E\lbt\sup_{h\in\mathcal{H}_n(C)}\bl\edint h(x)d\mathbb{G}_n(x)\bl\rbt/t'+\xi/2\\
=&\ O_p((\log n)^{3/4}n^{-p/10})/t'+\xi/2,
\end{align*}

which is less than $\xi$ for sufficiently large $n$. Here (a) follows from
Markov inequality. Since $t'$ and $\xi$ are arbitrary, we conclude that $\int h_n d\mathbb{G}_n$ is $o_p(1)$.

an application of Lemma <ref> leads to $\hin(\eta_n)\to_p\I$, and thus from (<ref>),  $\sqn T_{1n}=o_p(1)$ follows.

§.§.§ Second step: asymptotic limit of $T_{2n}/\td$:

Let us define $\mathcal A_n=[-\xin,\xin-\td]$,
Observe that  $T_{2n}/\td$ can be written as

\begin{align}\label{intheorem:t2n:representation}
\dint_{-\xin}^{\xin}\dfrac{\tp(z)}{\hin(\eta_n)}\dfrac{\lb g_0(z-\td)-g_0(z)\rb}{\td} dz\ =&\ \dint_{-\xin}^{\xin}\dfrac{\tp(z)}{\hin(\eta_n)}\dfrac{\dint_{z}^{z-\td}g_0'(t)dt}{\td} dz\\
= &\ -\dint_{\RR}\underbrace{1_{\mathcal A_n}(t)g_0'(t)\dfrac{\dint_{t}^{t+\td}\tp(z)dz}{\td\hin(\eta_n)}}_{b_n(t)}dt,\nn
\end{align}

where the last equality  follows by Fubini's Theorem since $g_0$ is absolutely continuous.

Note that
(<ref>) implies $T_{2n}=-\int_{\RR} b_n(t)dt$. The following lemma, which is proved in Appendix <ref>, establishes $T_{2n}\to_p -1$, thus completing the proof of the second step.

Under the set up of Theorem <ref>,
$\mathbb Y_n\equiv\edint b_n(t)dt\to_p 1$ where $b_n(t)$ is as defined in (<ref>).

§.§.§ Third step: showing the asymptotic normality of $T_{5n}$:

A change of variable leads to

\begin{align}\label{intheorem: T3n split new}
\sqn T_{5n}=&\ \sqn\int_{\bth-\xin}^{\bth+\xin}\dfrac{\psp'(x-\th_0)}{\hin(\eta_n)}d(\mathbb{F}_n-F_0)(x)\nonumber\nn\\
=&\  \int_{-\infty}^{\infty}\dfrac{\phi'_0(x)}{\hin(\eta_n)}d\mathbb{G}_n(x) -\dint_{C_n} \dfrac{\phi_0'(x)}{\hin(\eta_n)}d\mathbb{G}_n(x),
\end{align}

The central limit theorem yields

\[\edint\phi'_0(x)d\mathbb{G}_n(x)=\sum_{i=1}^n\frac{\phi_0'(X_i)-E[\phi_0'(X_i)]}{\sqn}\to_d N(0,\I).\]

Then from Lemma <ref> and  Slutsky's theorem it follows that

\[ \int_{-\infty}^{\infty}\dfrac{\phi'_0(x)}{\hin(\eta_n)}d\mathbb{G}_n(x)\to_d N(0,\I^{-1}).\]

Thus it suffices to show that
the second term on the right hand side of (<ref>) is $o_p(1)$. To that end, observe that  $1-1_{C_n}=1_{C_n^c}$ belongs to  the class of all indicator functions of  the form $1_{[z_1,z_2]}$,  where $z_1\leq z_2$ with $z_1,z_2\in\RR$. Since the latter class is Donsker by (<ref>),  Theorem 2.1 of [van der Vaart and Wellner, 2007] entails that  the second term on the right hand side of (<ref>) is of order $o_p(1)$ provided

\[\hin(\eta_n)^{-2}\edint 1_{C_n}(x)\phi'_0(x)^2f_0(x)dx\to_p 0.\]

Since $\hin(\eta_n)\to_p\I>0$ by Lemma <ref>, we only need to show that the integral in the last display is $o_p(1)$. Because $\I<\infty$, Fact <ref>  implies that given any $\e>0$, there exists $\sigma>0$ so that
$P_0(\mathcal B)<\sigma$ implies $\int_{\mathcal B}\phi_0'^2(x)f_0(x)dx<\e$ for any $P_0$-measurable set $\mathcal{B}\subset\RR$. Thus the proof follows if we can  show that $\int_{\mathcal C_n}f_0(x)dx=o_p(1)$. To that end, observe that

\begin{align*}
\dint_{C_n}f_0(x)dx=&\ 1-F_0(\bth+\xin)+F_0(\bth-\xin) \\
\to_p &\  1-F_0(\th_0+G_0^{-1}(1))+F_0(\th_0+G_0^{-1}(0))
\end{align*}

by continuous mapping theorem because (a) $\bth\to_p\theta_0$, (b) $\xin\to_p G_0^{-1}(1)$ by Lemma <ref>, and (c) $F_0$ is continuous. Since
$\theta_0+G_0^{-1}(1)=F_0^{-1}(1)$ and $\theta_0+G_0^{-1}(0)=F_0^{-1}(0)$, the proof follows.

§.§ Proof of key lemmas for Theorem <ref>

Recall that we defined $\mathcal A_n=[-\xin,\xin-\td]$ in the proof of Theorem <ref>. Let us define
$\mathcal A'_n=[-\xin-2|\td|,\xin+2|\td|]$.
We also denote

\[\mathcal I_{1n}=\dint_{\mathcal A_n}\tilde \psi'_{n_k}(t)^2\tilde g_{n_k}(t)dt\quad\text{and}\quad \mathcal I_{2n}=\dint_{\mathcal A_n+\td}\tilde \psi'_{n_k}(t)^2\tilde g_{n_k}(t)dt.\]

First we will show that it suffices to consider almost sure convergence of $ Y_n$ along some suitably chosen subsequence. We claim that given any subsequence of $\{n\}$, we can always obtain a further subsequence $\{n_k\}_{k\geq 1}$ so that  the set

\begin{align}\label{inlemma: def: mathcal M}
\mathcal M=\lbs &\ \overline \theta_{n_k}\to_k\theta_0,\ \widehat I_{n_k}(\eta_{n_k})\to_k\I,\ \xi_{n_k}\to_k G_0^{-1}(1),\ \omega_{n_k}\to_k \omega_0, \nn\\
&\ \frac{(\log {n_k})^2 H(\tilde g_{n_k},g_0)^2}{\inf_{x\in\mathcal A'_n}\tilde g_{n_k}(x)}\to_k 0,\ \lim_{k\to\infty}\mathcal I_{in_k }=\I\ \text{for}\ i=1,2,\\
&\ \|\tilde{g}_{n_k}-g_0\|_\infty\to_k 0,\ \mathcal{A}_{n_k}' \subset \iint(\dom(\psi_0))\text{ for all sufficiently large }n_k \rbs\nn
\end{align}

has probability one, where $\omega_n$ and $\omega_0$ are as in Fact <ref>.
The claim follows directly by  Fact <ref>  noting

* $\bth\to_p\th_0$.

* $\hin(\eta_n)\to_p\I$ by Lemma <ref>.

* $\xin\to_p G_0^{-1}(1)$ by Lemma <ref>.

* $\omega_n\to_p\omega_0$ by Fact <ref>.

* Suppose $\tilde \xi_n=\tilde G_n^{-1}(1-\eta_n/2)$. Then Lemma <ref> implies $\sup_{x\in\mathcal [-\tilde\xi_n,\tilde\xi_n]}(\tilde g_{n_k}(x))^{-1}$ is $O_p(n^{2p/5})$. However, $\mathcal A_n'\subset [-\tilde\xi_n,\tilde\xi_n] $ by Lemma <ref> with probability tending to one since $\td=O_p(n^{-1/2})$. Since $H(\hn,g_0)=O_p(n^{-p})$ and $p\in(0,1)$, it  follows that

\[\frac{(\log n)^2H(\hn,g_0)^2}{\inf_{x\in\mathcal A_n'}\hn(x)}\to_p 0.\]

* Suppose $\tilde \xi_n=(\tilde G_n)^{-1}(1-\eta_n/2)$.
Lemma <ref> implies that with probability tending to one, $\xin+2|\td|\leq \tilde\xi_n$, which implies

\[\mathcal{I}_{1n}\leq \mathcal{I}_{2n}\leq \dint_{-\tilde\xi_n}^{\tilde\xi_n}\tp(z)^2\hn(z)dz\stackrel{(a)}{\to}_p\I,\]

where (a) follows
from Lemma <ref> noting $\hn\in\mathcal{SLC}_0$.
On the other hand, Fatou's lemma and Condition <ref> indicates that

\[\liminf_n\mathcal{I}_{2n}\geq \mathcal{I}_{1n}\geq \edint \liminf_{n}\slb 1_{ \mathcal A_n}(z)\tp(z)^2\hn(z)\srb dz=\I.\]

Therefore, $\mathcal{I}_{1n}, \mathcal{I}_{2n}\to_p\I$.

* $\|\hn-g_0\|_\infty\to_p 0$ by Condition <ref>.

* Since $\td=O_p(n^{-1/2})$, Lemma <ref> yields  $P(\mathcal{A}_{n}' \subset \iint(\dom(\psi_0)))\to_n 1$.

Suppose we can show that as $k\to\infty$,
$\mathbb Y_{n_k}\to 1$  on $\mathcal  M$. Then it would establish that every subsequence
of $n$ has a further subsequence $n_k$ along which $\mathbb Y_{n_k}\as  1$. Then  Fact <ref> would yield $\mathbb Y_n\to_p 1$, as desired.

For the sake of simplicity, we will drop  $k$ from the subscript from  the definitions of  $\mathcal M$ and $\mathbb Y_{n_k}$.

Now we  derive some useful inequalities which hold on $\mathcal M$.
First, Lemma <ref> implies that

\[\xin\leq \frac{-5\log 2+2p\log n}{5w_n}.\]

Because $w_n\to_n w_0>0$,
there exists $C>0$ so that $\xin\leq C\log n$ for all sufficiently large $n$ on $\M$. Equation <ref>, on the other hand, implies that $|\psi_0'(\xin+|\td|)|$ is of the order of $\xin$. Therefore for large enough $C$,

\begin{equation}\label{intheorem: bound on psi knot prime}
\limsup_n\sup_{t\in\mathcal A_n'}\psi_0'(t)=|\psi_0'(\xin+|\td|)|\leq C\log n\quad \text{on }\M.
\end{equation}

Here the monotonicity of $\ps_0'$ was used to obtain the last equality.
Also note that because $\mathcal A_n'\subset \iint(\dom(\psi_0))$ for all sufficiently large $n$ on $\M$,
we can apply Lemma <ref>
on $g_0$ to obtain

\begin{align}\label{intheorem: bound on g/g(-)}
\limsup_n\sup_{t\in\mathcal A_n}\frac{g_0(t)}{g_0(t+\td)}\leq  \limsup_n e^{O(|\td|\xi_n)}\stackrel{(a)}{=} 1\quad \text{on }\M,
\end{align}

where (a) follows because $\td\xin\to_n 0$ on $\M$.

Next we will establish the pointwise convergence of $b_n(t)$ on $\M$. Since $\|\hn-g_0\|_\infty\to 0$ on $\M$,  Lemma <ref>(B) holds on $\M$.
Using Lemma <ref>(B) and the mean value Theorem,  we can show that on $\M$, $(\hln(t+\td)-\hln(t))/\td\to \psi'_0(t)$ for any $t\in\iint(\dom(\psi_0))$ that is a continuity point of $\psi_0'$. Because $\psi_0$ is concave, $\psi_0'$ is continuous Lebesgue almost everywhere on $\iint(\dom(\psi_0))$ <cit.>.
Also noting $\widehat{\mathcal I}_{n}(\eta_{n})\to_n\I$ on $\M$,  we obtain that

\[g_0'(t)\dfrac{\dint_{t}^{t+\td}\tp(z)dz}{\td\hin(\eta_n)}\to_n \frac{g_0'(t)\psi_0'(t)}{\I},\quad \text{Lebesgue a.e. } t\in \iint(\dom(\psi_0))\ \text{on }\M.\]

Since  $\mathcal A_n\subset\iint(\dom(\psi_0))$ for sufficiently large $n$, and $\xin\to G^{-1}_0(1)$ on $\M$, it follows that $1_{\mathcal A_n}(t)$ converges to $1_{\iint(\dom(\psi_0))}(t)$ pointwise on $\M$ as well. Noting   $g_0'(t)=\psi_0'(t)g_0(t)$ for all  $t\in\iint(\dom(\psi_0))$, we then obtain that on $\M$,

\begin{align}\label{intheorem: T2: pointwise in dct}
b_n(t)=1_{\mathcal A_n}(t)g_0'(t)\dfrac{\dint_{t}^{t+\td}\tp(z)dz}{\td\hin(\eta_n)}\to_n 1_{\iint(\dom(\psi_0))}(t)\frac{\psi_0'(t)^2g_0(t)}{\I}
\end{align}

for all $t\in\RR$ except a set of Lebesgue measure zero.

The concavity of $\hln$ implies that its right derivative
$\tp$ is non-increasing. Hence, for any $t\in\iint(\dom(\psi_0))$, we have,

\begin{equation}\label{inlemma: T2: DCT}
\min\bigg\{\tp(t),\tp(t+\td)\bigg\}\leq \dfrac{\dint_{t}^{t+\td}\tp(z)dz}{\td}\leq \max\bigg\{\tp(t),\tp(t+\td)\bigg\},
\end{equation}

yielding         _t^t+(z)dz||    ≤|(t)|+|(t+)|.
Using (<ref>), we can bound $|b_n(t)|$ noting

\begin{align*}
|b_n(t)|\leq &\ \underbrace{1[t\in\mathcal A_n]|\psi_0'(t)|g_0(t) \frac{|\tp(t)|}{\hin(\eta_n)}}_{b_{1n}(t)} + \underbrace{1[t\in\mathcal A_n]|\psi_0'(t)|g_0(t) \frac{|\tp(t+\td)|}{\hin(\eta_n)}}_{b_{2n}(t)}.
\end{align*}

Now defining

\[T_{21,n}(t)=1[t\in\mathcal A_n]|\psi_0'(t)|\sqrt{g_0(t)}\lb\sqrt{g_0(t)}-\sqrt{\hn(t)}\rb \frac{|\tp(t)|}{\hin(\eta_n)},\]

\[T_{22,n}(t)=1[t\in\mathcal A_n]|\psi_0'(t)|\sqrt{g_0(t)} \frac{|\tp(t)|\sqrt{\hn(t)}}{\hin(\eta_n)},\]

\[T_{23,n}(t)=1[t\in\mathcal A_n]|\psi_0'(t)|\frac{{g_0(t)}}{\sqrt{g_0(t+\td)}}\lb\sqrt{g_0(t+\td)}-\sqrt{\hn(t+\td)}\rb \frac{|\tp(t+\td)|}{\hin(\eta_n)},\]

\[T_{24,n}(t)=1[t\in\mathcal A_n]|\psi_0'(t)|\frac{{g_0(t)}}{\sqrt{g_0(t+\td)}} \frac{|\tp(t+\td)|\sqrt{\hn(t+\td)}}{\hin(\eta_n)},\]

we note that

\[b_{1n}(t)= T_{21,n}(t)+T_{22,n}(t)\quad\text{and}\quad b_{2n}(t)= T_{23,n}(t)+T_{24,n}(t).\]

Thus we can upper bound $|b_n(t)|$ by $c_n(t)$ where

\[c_n(t)=T_{21,n}(t)+T_{22,n}(t)+T_{23,n}(t)+T_{24,n}(t).\]

Our aim is to apply Fact <ref> (Pratt's Lemma) with $a_n=0$ to prove the current Lemma.
To this end, we first show that the following assertions hold on $\M$:

A1.   $T_{21,n}\to_n 0$ and $T_{23,n}\to_n 0$ Lebesgue almost everywhere on $\RR$.

A2. $\int_{\RR} T_{21,n}(t)dt\to_n 0$ and $\int_{\RR} T_{23,n}(t)dt\to_n 0$.

A3. There are   functions $t_{22}:\RR\mapsto\RR$ and $t_{24}:\RR\mapsto\RR$ so that $T_{22}\to_n t_{22}$ and $T_{24}\to_n t_{24}(t)$ Lebesgue almost everywhere on $\RR$.

A4. The functions $t_{22}$ and $t_{24}$ in A3 are integrable. Moreover, $\int_{\RR} T_{22,n}(t)dt\to_n \int_{\RR} t_{22}(t) dt $ and $\int_{\RR} T_{24,n}(t)dt\to_n \int_{\RR} t_{24}(t) dt $.

Let us denote $c(t)=t_{22}(t)+t_{24}(t)$. Then
A1-A4 imply that on $\M$, $c_n(t)\to_n c(t) $ Lebesgue almost everywhere, $c$ is integrable, and
$\int_{\RR} c_n(t)dt \to_n \int_{\RR} c(t)dt $. Since $|b_n(t)|\leq c_n(t)$ and (<ref>) holds,  Pratt's Lemma (see Fact <ref>) yields

\[\edint b_n(t)dt\to_n \dint_{G_0^{-1}(0)}^{G_0^{-1}(1)}\frac{g_0'(t)^2\psi_0'(t)dt}{\I}=1\quad \text{on }\M ,\]

which completes the proof of Lemma <ref>.

§.§.§ Proof of A1 and A3:

Since $\|\hn-g_0\|_\infty\to_n 0$ on $\mathcal M$, Lemma <ref>  implies that on $\M$, the functions $\hn,\hn(\mathord{\cdot}+\td)$ converge  pointwise to $ g_0$, and $\tp,\tp(\mathord{\cdot}+\td)$ converge to $\psi_0'$ Lebesgue almost everywhere on $\iint(\dom(\psi_0))$. Continuity of $g_0$ implies $g_0(t+\td)\to_n g_0(t)$ for all $t\in\RR$.  Using the above, it can be shown that

\[ T_{21,n}, T_{23,n}\to_n 0,\quad T_{22,n},T_{24,n}\to_n 1_{\iint(\dom(\psi_0))}{\psi_0'^2g_0}/{\I} \quad \text{a.e. Lebesgue}\ \text{on }\M.\]

§.§.§ Proof of A2:

Using Cauchy-Schwarz inequality, the bound on $g_0$ from Fact <ref>, and the bound on $\psi_0'$ from (<ref>), we can show that there exists $C>0$ such that the following holds for all sufficiently large $n$ on $\M$:

\begin{align*}
\MoveEqLeft \edint |T_{21,n}(t)|dt\
\\
\leq &\ C \log n\left(\dint_{-\xin-\td}^{\xin}\frac{\slb\sqrt{g_0(t)}-\sqrt{\hn(t)}\srb^2}{{\hn(t)}} \right)^{1/2}\lb \dint_{\mathcal A_n}\frac{|\tp(t)|^2\hn(t) dt}{\hin(\eta_n)^2}\rb^{1/2}\\
\leq &\ C \log n\frac{H(\hn, g_0)}{\slb \inf_{x\in\mathcal A'_n} \hn(x) \srb^{1/2}}\frac{\sqrt{\mathcal I_{1n}}}{\hin(\eta_n)},
\end{align*}

which approaches  zero as $n\to\infty$  because

\begin{equation}\label{inlemma: key lemma: A2 proof}
\frac{ C(\log n) H(g_0,\hn)}{ \slb\inf_{x\in\mathcal A_n'}\hn(x)\srb^{1/2}}\to_n 0,\quad \mathcal I_{1n},\ \hin(\eta_n)\to_n \I\quad\text{on }\M.
\end{equation}

by (<ref>).

The proof for $T_{23,n}$ is similar. An application of the Cauchy-Schwarz inequality, the bound on $\psi_0'$ by (<ref>) and the bound in (<ref>) imply that the following holds for all large $n$ on $\M$:

\begin{align*}
\edint T_{23,n}(t)dt\leq &\
\frac{ C(\log n)H(g_0,\hn)}{ \slb\inf_{x\in\mathcal A_n'}\hn(x)\srb^{1/2}}\frac{\lb\dint_{-\xin+\td}^{\xin}\tp(t)^2\hn(t)dt\rb^{1/2}}{\hin(\eta_n)}\\
=&\ \frac{ C(\log n) H(g_0,\hn)}{ \slb\inf_{x\in\mathcal A_n'}\hn(x)\srb^{1/2}}\frac{\sqrt{\mathcal I_{2n}}}{\hin(\eta_n)},
\end{align*}

which converges to zero  by (<ref>) and the fact that $\mathcal{I}_{2n}\to_n\I$ on $\M$,
thus completing the proof of A2.

§.§.§ Proof of A4:

Let us define

\[\mathcal T_{22,n}(t)= |T_{22,n}(t)|/g_0(t)\quad\text{ and }\quad \mathcal T_{24,n}(t)= |T_{24,n}(t)|/g_0(t).\]

We will show that on $\mathcal M$,
for each $\e>0$, there exists $\sigma>0$ so that the following bounds are true for any $G_0$-measurable set $\mathcal B\subset\RR$ satisfying $\int_{\mathcal B} g_0(t)dt<\sigma$:

\begin{align}\label{intheorem: UI: step 1}
\limsup_n\dint_{\mathcal B} \mathcal T_{22,n}(t)g_0(t)dt<\e\quad\text{and}\quad \limsup_n\dint_{\mathcal B} \mathcal T_{24,n}(t)g_0(t)dt<\e.
\end{align}

Next, we will show that

\begin{align}\label{intheorem: UI: step 2}
\limsup_n\edint \mathcal T_{22,n}(t)g_0(t)dt<\infty\ \text{and}\  \limsup_n\edint \mathcal T_{24,n}(t)g_0(t)dt<\infty.
\end{align}

If (<ref>) and (<ref>) hold,  Fact <ref>
underscores that  the sequences $(\mathcal T_{22,n})_{n\geq 1}$ and $(\mathcal T_{24,n})_{n\geq 1}$ are uniform integrable with respect to the measure  induced by $G_0$. Then A4 follows from A3 and Theorem 16.13 of [Billingsley, 2012] (Vitali convergence Theorem).
Thus it suffices to show that (<ref>) and (<ref>) hold.

Note that since $\I<\infty$, by Fact <ref>, given any $\e>0$, we can choose $\sigma>0$ so that for any $G_0$-measurable set $\mathcal B\subset\RR$ satisfying $\int_{\mathcal B}g_0(t)dt<\sigma$, the integral
$\int_{\mathcal B}\psi_0(x)^2g_0(x)<\e^2\I=\e'$ (say).
It will soon be clear why this choice of $\e'$ works. Using the Cauchy-Schwarz inequality in the third step, we calculate

\begin{align*}
\dint_{\mathcal B}\mathcal T_{22,n}(t)g_0(t)dt=  &\  \dint_{\mathcal B}T_{22,n}(t)dt= \dint_{\mathcal B\cap \mathcal A_n}|\psi_0'(t)|\sqrt{g_0(t)}\frac{|\tp(t)|\sqrt{\hn(t)}}{\hin(\eta_n)}\\
\leq &\ \lb\dint_{\mathcal B}\psi_0'(t)^2g_0(t)dt \rb^{1/2}\lb\dint_{\mathcal A_n}\frac{\tp(t)^2\hn(t)dt}{\hin(\eta_n)^2}\rb^{1/2},
\end{align*}

which is bounded by $\sqrt{\e'\mathcal{I}_{1n}}/\hin(\eta_n)$. Noting $\mathcal{I}_{1n}\to_n \I$ and  $\hin(\eta_n)\to_n\I$ on $\M$, we obtain

\[\limsup_n \dint_{\mathcal B}\mathcal T_{22,n}(t)g_0(t)dt\leq \sqrt{\e'/\I}=\e\quad\text{on }\M.\]

Letting $\mathcal B=\RR$, and repeating the above steps, we can show that

\[ \limsup_n\int_{\RR}\mathcal{T}_{22,n}(t)g_0(t)dt<1\quad\text{ on }\M.\]

For $\mathcal T_{24,n}$, the Cauchy-Schwarz inequality yields

\begin{align}\label{inlemma: bound: T24}
\dint_{\mathcal B}\mathcal{T}_{24,n}(t)g_0(t)dt\leq &\ \lb\dint_{\mathcal B\cap \mathcal{A}_n} \frac{g_0(t)^2\psi_0'(t)^2}{g_0(t+\td)}dt\rb^{1/2}
\lb \dint_{-\xin+\td}^{\xin}\frac{\tp(t)^2\hn(t) dt}{\hin(\eta_n)^2}\rb^{1/2}\nn\\
\stackrel{(a)}{\leq} &\
\sqrt{\e'}\sup_{t\in\mathcal A_n}\lb\frac{g_0(t)}{g_0(t+\td)}\rb^{1/2}\frac{\mathcal{I}_{2n}^{1/2}}{\hin(\eta_n)}.
\end{align}

Here (a) follows because $\int_{\mathcal B}\psi'_0(t)^2g_0(t)dt<\e'$.
The fact that $\mathcal{I}_{1n}$, $\hin(\eta_n)\to_n\I$ on $\M$, in conjunction with the bound in (<ref>),  implies

\[\limsup_n  \dint_{\mathcal B}\mathcal{T}_{24,n}(t)g_0(t)dt\leq \sqrt{\e'/\I}=\e\quad\text{on }\M.\]

Thus (<ref>)  is proved. Letting $\mathcal B=\RR$ leads to  $ \limsup_n\int_{\RR}\mathcal{T}_{24,n}(t)g_0(t)dt<1$ on $\M$, thus finishing the proof of (<ref>).

§.§ Auxilliary lemmas for the proof of Theorem <ref>

§.§.§ Lemmas on $\xi_n$:

Unless otherwise mentioned, for all the lemmas on $\xi_n$,  $\xi_n$ will denote $\tilde G_n^{-1}(1-\eta_n)$, where the choice of $\hn$ should be clear from the context.

Suppose $\hn\in\mathcal{S}_0$ satisfies Conditions <ref> and <ref>.
Let $\eta_n=C n^{-2p/5}$, where $C>0$ and $p$ is as in Condition <ref>.
Then for $\xi_n=\tilde G_n^{-1}(1-\eta_n)$, we have

\[P\slb [-\xi_n,\xi_n]\subset\iint(\dom(\psi_0))\srb\to 1.\]

Using Fact <ref> in step (a) we obtain that

\[|G_0(-\xin)-\tilde G_n(-\xin)|\leq d_{TV}(G_0,\tilde G_n)\stackrel{(a)}{\leq} \sqrt{2}H(\hn,g_0)=O_p(n^{-p})\]

by Condition <ref>.
Therefore $G_0(-\xin)\geq \tilde G_n(-\xin)+o_p(n^{-p})\geq \eta_n+o_p(n^{-p})$ because $F(F^{-1}(q))\geq q$ for any distribution function $F$, and $q\in(0,1)$. Since $\eta_n=Cn ^{-2p/5}\gg n^{-p}$, it follows that $P(G_0(-\xin)\geq \eta_n/2)\to 1$. Thus $P(-\xin\in \iint(dom(\psi_0)))\to_n 1 $. Since $\psi_0\in \mathcal {SC}_0$,  $\iint(\dom(\psi_0))$ is an interval of the form $(-a,a)$ for some $a>0$. Noting $-\xin\in(-a,a)$ implies $[-\xin,\xin]\subset(-a,a)$, the proof follows.

Consider the set up of Lemma <ref>. Then $\xi_n\to_p G_0^{-1}(1)$ as $\eta_n\to 0$.

Note that $G_0^{-1}(1)$ takes value in $\RR \cup\{\infty\}$.
Lemma <ref> implies
$P(\xi_n\leq G_0^{-1}(1))\to 1$. Suppose, if possible,
$\xi_n\to_p G_0^{-1}(1)$ does not hold. Then, since $G_0^{-1}$ is continuous on $(0,1)$,
there exists $t\in(0,1)$ so that

\[\limsup_n P(\xi_n< G_0^{-1}(t))>0.\]

However, because $\eta_n\to 0$,  $1-\eta_n\geq 2t$ for sufficiently large $n$, which yields
$\xi_n\geq\tilde G_n^{-1}(2t)$.  Now by Fact <ref> and Condition <ref>, $G_n^{-1}(2t)\to_pG_0^{-1}(2t)$. However, $G_0^{-1}(2t)>G_0^{-1}(t)$, where the strict inequality follows because $g_0$ being log-concave, is positive  $J(G_0)$. Therefore the proof follows by contradiction.

Suppose either $\hn\in\mathcal{SLC}_0$ is a density  satisfying Condition <ref> and Condition <ref>, or $\hn$ satisfies Condition <ref>.  Let $\eta_n=Cn^{-2p/5}$ where $p$ is as in Condition <ref> (or  Condition <ref>) and $C>0$. Then

\[\xi_n\leq\frac{-\log 2+2p(\log n)/5}{\omega_n}\]

where $\omega_n$ is as in Fact <ref>. In fact,
$|\xi_n|=O_p(\log n)$.

Observe  that if  $\hn\in\mathcal{SLC}_0$, then $\hn(\tilde G_n^{-1}(z))>0$ for $z\in(0,1)$. If $\hn$ satisfies Condition <ref>, then also the above holds because by Condition <ref>, $\hn>0$ on $\RR$. Since $\hn$ is symmetric about zero,  $0=\tilde G_n^{-1}(1/2)$.  Noting $\xi_n=-\tilde{G}_n^{-1}(\eta_n)$,  we therefore derive that

\begin{align*}
\xi_n=\tilde{G}_n^{-1}(1/2)-\tilde{G}_n^{-1}(\eta_n)=\dint_{\eta_n}^{1/2}\frac{dz}{\hn(\tilde G_n^{-1}(z))}\leq \dint_{\eta_n}^{1/2}\frac{dz}{\omega_n\tilde G_n(\tilde G_n^{-1}(z))},
\end{align*}

where $\omega_n$ is as in Fact <ref>.
Because $\hn>0$ on $J(\tilde G_n)$, it follows that  $\tilde G_n$ is continuous on $J(\tilde G_n)$. Therefore, we have
$\tilde G_n(\tilde G_n^{-1}(z))=z$, implying
\[\xi_n\leq \frac{\log(1/2)-\log(\eta_n)}{\omega_n}=\frac{-\log 2+2p(\log n)/5}{\omega_n}.\]

Since $\omega_n\to_p\omega_0>0$ by Fact <ref>, the proof follows.

Consider the set up of Lemma <ref>. Let $\tilde \xi_n=\tilde G_n^{-1}(\eta_n/2)$.
Suppose $y_n$ is a sequence of non-negative random variables so that $P(y_n< \eta_n/(2g_0(0)))\to 1$.  Then

\begin{align}\label{inlemma: yn set inclusion}
P([-\xi_n-y_n,\xi_n+y_n]\subset [-\tilde{\xi}_n,\tilde{\xi}_n])\to 1.
\end{align}

Under our set up, $\hn$ is positive on the set $J(\tilde G_n)$. Therefore  the function $\tilde G_n^{-1}$ is continuous on $(0,1)$. Hence the mean value theorem implies

\[\tilde G_n^{-1}(\eta_n)-\tilde G_n^{-1}(\eta_n/2)= \frac{\eta_n}{2\hn(\tilde G_n^{-1}(t))}\geq \frac{\eta_n}{2\|\hn\|_\infty}\]

for some $t\in[\eta_n/2,\eta_n]$.
Condition <ref>  implies that $\|\hn\|_\infty\to_p\|g_0\|_\infty= g_0(0)$.
Therefore, as $n\to\infty$,

\[P\lb \liminf_n \frac{\tilde G^{-1}_n(\eta_n)-\tilde G^{-1}_n(\eta_n/2)}{\eta_n}\geq \frac{1}{2g_0(0)}\rb\to 1.\]

Hence if $y_n< \eta_n/(2g_0(0))$, then
$ \tilde{G}^{-1}_n(\eta_n)-y_n \geq  \tilde{G}^{-1}_n(\eta_n/2)$ with probability tending to one.
Since $\hn$ is symmetric about zero, we obtain that
$ \tilde{G}^{-1}_n(1-\eta_n)+y_n < \tilde{G}^{-1}_n(1-\eta_n/2)$
with probability tending to one.
Since $\xi_n=\tilde{G}^{-1}_n(1-\eta_n)$ and $\tilde{\xi}_n=\tilde{G}^{-1}_n(1-\eta_n/2)$, the proof  follows.

Consider  the set up of Lemma <ref>.  Then for $y_n=o_p(\eta_n)$, we have

\[P\slb[-\xin-|y_n|,\xin+|y_n|]\subset \iint(\dom(\psi_0))\srb\to 1.\]

Letting $\tilde{\xi}_n=-\tilde G_n^{-1}(\eta_n/2)$, and applying Lemma <ref>, we obtain $P([-\tilde{\xi}_n,\tilde{\xi}_n]\subset \iint(\dom(\psi_0)))\to 1$. Then the result follows from Lemma <ref>.

§.§.§ lemmas on  $\hn$ and $g_0$:

Suppose $\xin=\tilde G^{-1}_n(1-\eta_n)$ where $\hn\in\mathcal {SLC}_0$ satisfies Condition <ref>. Then

A. $\sup_{x\in[-\xi_n,\xi_n]}\hn(x)=O_p(1)$ and $\sup_{x\in[-\xi_n,\xi_n]}\hn(x)^{-1}=O_p(\eta^{-1}_n)$.

B. $\hln=\log\hn$ satisfies $\sup_{x\in[-\xi_n,\xi_n]}\hln(x)=O_p(1)$. For $\eta_n=Cn^{-2p/5}$ with $p\in(0,1)$ and $C>0$, we have $\sup_{x\in[-\xi_n,\xi_n]}(-\hln(x))=O_p(\log n)$.

The upper bound on $\hn$ follows from Fact <ref> and  Condition <ref>. For the upper bound on $\hn^{-1}$, note that  Fact <ref> implies that

\begin{align*}
\hn(x)\geq &\  w_n \min(\tilde G_n(x), 1-\tilde G_n(x)), \quad\text{for all }x\in\RR.
\end{align*}

Since $\tilde G_n$ is a non-decreasing and $1-\tilde G_n$ is a non-increasing function, any $x\in[-\xin,\xin]$ satisfies

\[\hn(x)\geq {\omega_n}{\min(\tilde G_n(-\xin),1-\tilde G_n(\xin))}=\omega_n\eta_n\]

because $\tilde G_n(-\xin)=\eta_n$.
Since the random variable $\omega_n\to_p\omega_0>0$ by Fact <ref>,
part A of the current lemma follows.
Part B follows directly from Part A.

Assume $f_0\in\mathcal{P}_0$. Suppose  $\{\hn\}_{n\geq 1}$ is a sequence of log-concave densities satisfying $\|\hn-g_0\|_\infty\to_n 0$.   Then the followings  hold for any $y_n\to_n 0$:

(A) Let $\hln=\log \hn$. Then
$\hln(\mathord{\cdot} +y_n)\to_n \ps_0$  everywhere on $\iint(\dom(\psi_0))$.

$\hln'(\mathord{\cdot} +y_n)\to_n\ps_0'$  Lebesgue almost everywhere on $\iint(\dom(\psi_0))$. In particular, if $x$ is a continuity point of $\ps_0'$, then $\hln'(x +y_n)\to_n\ps_0'$.

Since $g_0$ is continuous, it follows that $\sup_{x\in \RR}|\hn(x+y_n)-g_0(x)|\to_n 0$. Since for each  $x\in\iint(\dom(\psi_0))$, there exists an open neighborhood around $x$ where $|\psi_0|<\infty$, $\hln(x+y_n)\to_n \psi_0(x)$ for each $x\in\iint(\dom(\psi_0))$. Therefore part (A) follows. For part (B), first note that if $x\in\iint(\dom(\psi_0))$ is a continuity point of $\psi_0'$, then $\tp(x+y_n)\to_n \tp(x)$  by Theorem 25.7 of [Rockafellar, 1970]. Now since $\psi_0$ is concave, $\psi_0$ is continuously differentiable at $x$ if it is differentiable at  $x$ <cit.>.
However, a concave  $\psi_0$ is differentiable Lebesgue almost everywhere on $\dom(\phi_0)$ <cit.>. Therefore, the lemma follows.

§.§.§ Lemmas on $g_0$:

Suppose $g_0$ satisfies Assumption <ref>. Let $\kappa$ be as in Assumption <ref>. Then for any $x\in\dom(\psi_0)$,  $\psi_0$ satisfies

\[|\psi_0(x)|\leq |\psi_0(0)|+\kappa x^2.\]

In particular, if  $\eta_n=Cn^{-p}$ for $p\in(0,1)$ and $C>0$,   then under the set up of Lemma <ref>, $\xin=\tilde G_n^{-1}(1-\eta_n)$ satisfies

\[\sup_{x\in[-\xi_n,\xi_n]}|\psi_0(x)|=O_p((\log n)^2).\]

Because $\psi_0\in\mathcal{SC}_0$, zero is the mode of $\psi_0$. Therefore, the upper bound on $\psi_0$ follows since $\psi_0(x)<\psi_0(0)$.
For the lower bound, first note that the  concavity of $\psi_0$  indicates that if $x\geq 0$ and $x\in\dom(\psi_0)$, then

\[\psi_0(x)\geq \psi_0(0)+\psi'_0(x-)x.\]

By our notation, $\psi_0'(x+)=\psi_0'(x)$.
Noting Assumption <ref> implies $\psi_0'(x-)\geq \psi_0'(0-)-\kappa x$, we derive

\[\psi_0(x)\geq \psi_0(0)+\psi_0'(0-)x-\kappa x^2.\]

Since $\psi_0'(0-)\geq 0$, the above yields $\psi_0(x)\geq \psi_0(0)-\kappa x^2$ for all $x\geq 0$.
Since $\psi_0$ is symmetric about zero, we derive that $\psi_0(x)\geq \psi_0(0)-\kappa x^2$ for all $x\in\RR$. In conjunction with the fact that $\psi_0(x)\leq \psi_0(0)$, the latter implies $|\psi_0(x)|\leq |\psi_0(0)|+\kappa x^2$ for all $x\in\RR$. Since  $P([-\xi_n,\xi_n]\subset\dom(\psi_0))\to 1$ by Lemma <ref>, the rest of the proof follows noting
$\xi_n=O_p(\log n)$ for $\eta_n=n^{-2p/5}$ by Lemma <ref>.

Suppose $\psi_0\in\mathcal{SC}_0$ satisfies Assumption <ref>. Further suppose   $\eta_n$ is as in Lemma <ref> and $y_n>0$ satisfies $y_n=o_p(\eta_n)$. Then

\[\sup_{x\in[-\xi_n-y_n,\xi_n+y_n]}|\psi_0'(x)|=O_p(-\log(\eta_n)).\]

Since $\psi_0\in\mathcal{SC}_0$, $\psi_0'$ attains its maxima on any interval at the endpoints.
Lemma <ref> implies $[-\xi_n-y_n,\xi_n+y_n]\subset\dom(\psi_0)$ with probability approaching one. Therefore
Assumption <ref> implies

\begin{equation}\label{inlemma: lemma bound psi knot prime}
\psi_0'(-\xin-y_n)\leq |\psi_0'(0)|+\kappa(|\xin|+y_n).
\end{equation}

Rest of the proof follows from Lemma <ref> and the fact that $y_n=o_p(1)$.

Under the set up of Theorem <ref>, there exists $C>0$ so that if $b>0$ satisfies $[-b-|\td|,b+|\td|]\subset\iint(\dom(\psi_0))$, then

\[\sup_{t\in[-b,b]}\frac{g_0(t)}{g_0(t+\td)}\leq  e^{|\td|(C+\kappa b+\kappa|\td|)}.\]

Recalling that we use $\psi_0'$ to denote the right derivative of $\psi_0$, we obtain

\begin{align*}
\frac{g_0(t)}{g_0(t+\td)}= &\ \exp(\psi_0(t)-\psi_0(t+\td))\\
\stackrel{(a)}{\leq} &\ \exp(|\td|\max\{|\psi_0'(t)|,|\psi_0'(t+\td)|\})
\end{align*}

where (a) follows from (<ref>).
If $t,\ t-\td\in \iint(\dom(\psi_0))$, by
Assummption <ref>, it also holds that  $|\psi_0'(t)|\leq \psi_0'(0-)+\kappa|t|$ and $|\psi_0'(t+\td)|\leq \psi_0'(0-)+\kappa|t|+\kappa|\td|$. Thus for $C=\psi'_0(0-)$, we obtain that

\[\frac{g_0(t)}{g_0(t+\td)}\leq e^{|\td|(C+\kappa |t|+\kappa|\td|)},\]

from which, the result follows.

§.§.§ Lemmas regarding $\hln$:

Suppose $\hn$ satisfies Condition <ref> and Condition <ref>  with $p\in(0,1)$. Further suppose $a_n$ and $\hn$ satisfies

\begin{equation}\label{inlemma: statement: os: nilanjana 31}
\sup_{x\in[-a_n, a_n]} (|\psi_0(x)|+|\hln(x)|)=O_p((\log n)^2). \end{equation}

\begin{align*}
\dint_{-a_n}^{a_n} (\hln(x)-\psi_0(x))^2 g_0(x)dx=&O_p((\log n)^2n^{-2p}),\\
\dint_{-a_n}^{a_n} (\hln(x)-\psi_0(x))^2 \hn(x)dx=&O_p((\log n)^2n^{-2p}).
\end{align*}

In particular, if  $\hn\in\mathcal{SLC}_0$, then (<ref>) holds with $a_n=\xin(\tilde G_n)=\tilde G_n^{-1}(1-\eta_n)$, where $\eta_n=Cn^{-2p/5}$ for some $C>0$.

We first invoke an algebraic fact. For  any $x,y>0$,

\[(\sqrt x-\sqrt y)^2=\min(x,y)\lb \sqrt{\frac{\max(x,y)}{\min(x,y)}}-1\rb^2=\min(x,y)\slb e^{|\log x-\log y|/2}-1\srb^2.\]

Since  for any $z>0$, $z$ and $z^2/2$ are bounded above by $e^z-1$, it follows that

\[\slb e^{|\log x-\log y|/2}-1\srb^2\geq (\log x-\log y)^2/4, (\log x-\log y)^4/8^2.\]

\begin{align}\label{inlemma: Nilanjana 31}
4\dint_{-a_n}^{a_n}(\sqrt{\hn(x)}-\sqrt{g_0(x)})^2dx\geq &\  \dint_{-a_n}^{a_n}\min(\hn(x), g_0(x))(\hln(x)-\psi_0(x))^2dx\nn\\
8^2\dint_{-a_n}^{a_n}(\sqrt{\hn(x)}-\sqrt{g_0(x)})^2dx\geq &\  \dint_{-a_n}^{a_n}\min(\hn(x), g_0(x))(\hln(x)-\psi_0(x))^4dx.
\end{align}

\begin{align*}
\MoveEqLeft \dint_{-a_n}^{a_n}(\hln(x)-\psi_0(x))^2g_0(x)dx \\
= &\ \dint_{-a_n}^{a_n}(\hln(x)-\psi_0(x))^2\slb g_0(x)-\min(g_0(x),\hn(x))\srb dx\\
&\ +\dint_{-a_n}^{a_n}(\hln(x)-\psi_0(x))^2\min(g_0(x),\hn(x)) dx\\
\stackrel{(a)}{=}&\  \dint_{-a_n}^{a_n}(\hln(x)-\psi_0(x))^2( g_0(x)-\hn(x))1_{[\hn<g_0]} dx+O_p(n^{-2p})\\
=&\  \underbrace{\dint_{-a_n}^{a_n}(\hln(x)-\psi_0(x))^2( \sqrt{g_0(x)}-\sqrt{\hn(x)})^21_{[\hn<g_0]}dx}_{T_1}\\
-2&\ \underbrace{\dint_{-a_n}^{a_n}(\hln(x)-\psi_0(x))^2 \sqrt{\hn(x)}(\sqrt{\hn(x)}-\sqrt{g_0(x)})1_{[\hn<g_0]}dx}_{T_2}+O_p(n^{-2p})
\end{align*}

where (a) follows from (<ref>) and Condition <ref>.
We can upper bound $|\psi_0(x)-\hln(x)|$ noting

\[\sup_{x\in[-a_n,a_n]}|\psi_0(x)-\hln(x)|\leq \sup_{x\in[-a_n, a_n]} (|\psi_0(x)|+|\hln(x)|)=O_p((\log n)^2) \]

by (<ref>).
Therefore $T_1\leq O_p((\log n)^2)H(\hn,g_0)^2 $, which is $O_p(n^{-2p})$. On the other hand, noting $T_2$ can be written as

\begin{align*}
T_2=&\ \dint_{-a_n}^{a_n}(\hln(x)-\psi_0(x))^2 \sqrt{\min(\hn(x),g_0(x))}(\sqrt{\hn(x)}-\sqrt{g_0(x)})1_{[\hn<g_0]}dx,
\end{align*}

by an application of the Cauchy-Schwarz inequality, we derive

\[|T_2|\leq \lb \dint_{-a_n}^{a_n}(\hln(x)-\psi_0(x))^4 \min(\hn(x),g_0(x))dx\rb^{1/2}H(\hn,g_0),\]

which, by (<ref>) and Condition <ref>, is $O_p(n^{-2p})$, thus completing the proof of the first part.

It remains to show that (<ref>) holds when $\hn\in\mathcal{SLC}_0$ and $a_n=\tilde G_n^{-1}(1-\eta_n)$.
Lemma <ref> entails that this $a_n$ satisfies

\begin{align}\label{inlemma: os: Nilanjana 31}
\sup_{x\in[-a_n, a_n]} |\psi_0(x)|=O_p((\log n)^2).
\end{align}

The proof of the current lemma then follows noting  Lemma <ref> implies

\[\sup_{x\in[-\xin,\xin]}|\hln(x)| =O_p(\log n).\]

§.§.§ Lemmas regarding  $\hln'$:

Let $\rho_n=\eta_n/\log n$.  Suppose $\hn$ is a log-concave density satisfying  Condition <ref> and Condition <ref>. Let $a_n$ be a positive sequence  satisfying (<ref>) such that $a_n=O_p(\log n)$,

\begin{equation}\label{inlemma: statement: os: suport inclusion}
[-a_n-\rho_n,a_n+\rho_n]\subset \iint(\dom(\psi_0))\cap\iint(\dom(\hln)),
\end{equation}

P(G̃_n(-a_n)>η_n/4, 1-G̃_n(a_n)>η_n/4)→1.

\[ \dint_{-a_n}^{a_n}\slb \tp(z)-\psi_0'(z)\srb^2dz=O_p( (\log n)^3n^{-4p/5}),\]

\[\dint_{-a_n}^{a_n} \slb \tp(z)-\psi'_0(z)\srb^2 \mu_n(z)dz=O_p( (\log n)^3n^{-4p/5})),\]

for any density $\mu_n$ such that $\|\mu_n\|_\infty=O_p(1)$, where $p$ is as in Condition <ref>.  In particular, the lemma holds if
$\hn\in\mathcal{SLC}_0$ and $a_n=\xin(\tilde G_n)=\tilde G_n^{-1}(1-\eta_n)$, where $\eta_n=Cn^{-2p/5}$ for some $C>0$.

Since $\hln$ is concave and $\rho_n>0$, any $z\in\dom(\hln)$ satisfies

\begin{equation}\label{inlemma: os: FI lemma: tp inequality}
\frac{\hln(z+\rho_n)-\hln(z)}{\rho_n}\leq \tp(z+)\leq \tp(z-)\leq \frac{\hln(z)-\hln(z-\rho_n)}{\rho_n}.
\end{equation}

Now suppose (<ref>) holds.
the quantities

\[\Delta_n^{+}(z)=\frac{\hln(z+\rho_n)-\hln(z)}{\rho_n}-\frac{\psi_0(z+\rho_n)-\psi_0(z)}{\rho_n},\]

\[\Delta_n^{-}(z)=\frac{\hln(z)-\hln(z-\rho_n)}{\rho_n}-\frac{\psi_0(z)-\psi_0(z-\rho_n)}{\rho_n}\]

are well defined for all $z\in[-a_n,a_n]$.
Recalling   $\hln'(z)=\hln'(z+)$ and $\psi_0'(z)=\psi_0'(z+)$
by our notation, we can then show that  under (<ref>),

\begin{align*}
\hln'(z)-\psi_0'(z)\stackrel{(a)}\leq &\ \frac{\hln(z)-\hln(z-\rho_n)}{\rho_n}-\frac{\psi_0(z)-\psi_0(z-\rho_n)}{\rho_n}\\
&\ +\lb\frac{\psi_0(z)-\psi_0(z-\rho_n)}{\rho_n} -\psi_0'(z)\rb
\stackrel{(b)}{\leq} \Delta_n^{-}(z)+\kappa\rho_n
\end{align*}

for all $z\in[-a_n,a_n]$, where (a) follows by (<ref>), and (b) follows because

\[\frac{\psi_0(z)-\psi_0(z-\rho_n)}{\rho_n} -\psi_0'(z)\leq \rho_n^{-1}\bl\dint_{z-\rho_n}^{z} \slb \psi_0'(t)-\psi_0'(z)\srb dt\bl\leq \kappa\rho_n/2\]

since Assumption <ref> applies on the set $[z-\rho_n,z]\subset\iint(\dom(\psi_0))$.
Similarly, we can show that

\[ \Delta_n^{+}(z)-\kappa\rho_n\leq\hln'(z)-\psi_0'(z)\quad\text{for all } z\in[-a_n,a_n],\]

provided (<ref>) holds. Thus we have established that

\begin{equation}\label{inlemma: os : FI lemma: delta inequalities}
|\tp(z)-\psi_0'(z)|\leq \max\{\Delta_n^{+}(z), \Delta_n^{-}(z)\}+\kappa \rho_n
\end{equation}

whenever (<ref>) holds.
Now observe that  the integral $\int_{-a_n}^{a_n}\Delta^{+}_n(z)^2dz$ is well defined under (<ref>), and equals

\begin{align*}
\MoveEqLeft   \dint_{-a_n}^{a_n}\lb\frac{\hln(z+\rho_n)-\hln(z)}{\rho_n}-\frac{\psi_0(z+\rho_n)-\psi_0(z)}{\rho_n} \rb^2dz\\
\leq &\ 2\dint_{-a_n}^{a_n}\lb\frac{\hln(z+\rho_n)-\psi_0(z+\rho_n)}{\rho_n}\rb^2dz+2\dint_{-a_n}^{a_n}\lb\frac{\hln(z)-\psi_0(z)}{\rho_n}\rb^2dz\\
\stackrel{(a)}{\leq} &\  \frac{2}{\rho_n^2\min(\hn(a_n+\rho_n),\hn(-a_n+\rho_n))}\dint_{-a_n}^{a_n}(\hln(z+\rho_n)-\psi_0(z+\rho_n))^2\hn(z+\rho_n)dz\\
&\ +\frac{2}{\rho_n^2\min(\hn(a_n),\hn(-a_n))}\dint_{-a_n}^{a_n}(\hln(z)-\psi_0(z))^2\hn(z)dz\\
\stackrel{(b)}{=} &\  \frac{O_p((\log n)^2n^{-2 p})}{\rho^{2}_n \min(\hn(a_n+\rho_n),\hn(-a_n-\rho_n))},
\end{align*}

where (a) follows because $\hn$ being log-concave, and hence unimodal, attains minimum over an interval at either of the endpoints; and (b) follows from Lemma <ref> and the fact that $a_n$ and $\rho_n$ are positive.
Let us define

\begin{align}\label{inlemma: def: os: myen}
\e_n(\rho_n)=\frac{(\log n)^2n^{-2 p}}{\rho^{2}_n \min(\hn(a_n+\rho_n),\hn(-a_n-\rho_n))
\end{align}

Since (<ref>) holds with probability tending to one by our assumption, we can write
Similarly, we can show that
$\int_{-a_n}^{a_n}\Delta^{-}_n(z)dz$ is $O_p(\e_n(\rho_n))$.
The above, combined with (<ref>), leads to

\begin{align*}
\dint_{-a_n}^{a_n}(\tp(z)-\psi_0'(z))^2dz\leq &\ 2\dint_{-a_n}^{a_n}\Delta_n^{-}(z)^2dz+2\dint_{-a_n}^{a_n}\Delta_n^{+}(z)^2dz +4\kappa^2\rho_n^2a_n\\
=&\ O_p(\e_n(\rho_n))+O_p(\rho_n^2a_n).
\end{align*}

Note that $\rho_n^2a_n=O(n^{-4p/5}/\log n)$ because $\rho_n=\eta_n/\log n$ and $a_n=O_p(\log n)$ by our assumption. Also, $\int_{-a_n}^{a_n}(\tp(z)-\psi_0'(z))^2\mu_n(z)dz$ can be bounded by

2f_∞ _-a_n^a_nΔ_n^-(z)^2dz+_-a_n^a_nΔ_n^+(z)^2dz+2κ^2ρ^2_n

which is $O_p(\e_n(\rho_n))+O_p(n^{-4/5}/(\log n)^2)$ because $\|\mu_n\|_\infty$ is $O_p(1)$ and $\rho_n$ equals $\eta_n/\log n$.

To prove the first part of the lemma, it only remains to show that

\begin{equation}\label{inlemma: os: rate of e n}
\e_n(\rho_n)=O_p( (\log n)^3n^{-4p/5}).
\end{equation}

Since $a_n-\rho_n\in\iint(\dom(\hln))$ under (<ref>),
Fact <ref> implies

\[\hn(-a_n-\rho_n)\geq \omega_n \tilde{G}_n (-a_n-\rho_n)=\omega_n\tilde{G}_n(-a_n)-\omega_n\dint_{-a_n-\rho_n}^{-a_n}\hn(z)dz\]

under (<ref>),
where $\omega_n$ is as in fact <ref>.
Note that

\[\bl\dint_{-a_n-\rho_n}^{-a_n}\hn(z)dz\bl\leq \rho_n\|\hn\|_{\infty}.\]

Also since $P(\tilde G_n(-a_n)>\eta_n/4)\to 1$ by our assumption, the followings hold with probability tending to one,

\[\hn(-a_n-\rho_n)\geq  \omega_n(\eta_n/4-\rho_n\|\hn\|_\infty)\stackrel{(a)}{\geq}\omega_n\eta_n/8 ,\]

where (a) follows because $\rho_n=o(\eta_n)$ and  $\|\hn\|_\infty=O_p(1)$ by Condition <ref> and Fact <ref>.
However, since $\omega_n\to_p\omega_0$ by Fact <ref>,  the last display implies
$\hn(-a_n-\rho_n)^{-1}$ is $O_p(1/\eta_n)$.
Similarly,   we can show that $\hn(a_n+\rho_n)^{-1}$ is $O_p(1/\eta_n)$.

\[\epsilon_n(\rho_n)=\frac{O_p((\log n)^2n^{-2p})}{\eta_n\rho_n^2}\]

Since $\eta_n=n^{-2p/5}$ and $\rho_n=\eta_n/\log n$,  (<ref>) follows, thus completing the proof of the first part of Lemma <ref>.

Now suppose $\hn\in\mathcal{SLC}_0$ and  $a_n=\xi_n(\tilde G_n)=\tilde G_n^{-1}(1-\eta_n)$. They satisfy (<ref>) by Lemma <ref>. Also $\xin=O_p(\log n)$ by Lemma <ref>. Noting $\rho_n=o(\eta_n)$, (<ref>) follows from  Lemma <ref>
and Lemma <ref>.

Since (<ref>) trivially holds,  second part of Lemma <ref> also follows.

Suppose $\hn$ satisfies Condition <ref> and Condition <ref>. Let $a_n$ be a sequence of positive random variables  satisfying

\begin{equation}\label{inlemma: statement: FI: an}
a_n=O_p(\log n),\quad
P\slb a_n\in\iint(\dom(\psi_0))\srb\to 1,\quad\text{and}\quad
a_n\to_p G_0^{-1}(1).
\end{equation}

Further suppose that $\hn$ and $a_n$ satisfy

\begin{align}\label{inlemma: statement: FI}
\dint_{-a_n}^{a_n}(\tp(z)-\psi_0'(z))^2\hn(z)dz=O_p((\log n)^3n^{-4p/5}),
\end{align}

where $p$ is as in Condition <ref>.
$\int_{-a_n}^{a_n}\tp(z)^2\hn(z)dz\to_p \I$. In addition, if $\hn\in\mathcal{SLC}_0$, then  $a_n=\xin(\tilde G_n)=(\tilde G_n)^{-1}(1-\eta_n)$ where $\eta_n=Cn^{-2p/5}$ for some $C>0$.

Note that

\begin{align*}
\MoveEqLeft    \dint_{-a_n}^{a_n}\slb\tp(z)^2\hn(z)-\psi_0'(z)^2g_0(z)\srb dz\\
= &\ \dint_{-a_n}^{a_n}\slb(\tp(z)-\psi_0'(z))^2\hn(z)+2\psi_0'(z)\tp(z)\hn(z)-\psi_0'(z)^2\hn(z)-\psi_0'(z)^2g_0(z)\srb dz\\
=&\ \underbrace{ \dint_{-a_n}^{a_n}(\tp(z)-\psi_0'(z))^2\hn(z)dz}_{T_1}+ 2\underbrace{\dint_{-a_n}^{a_n}\psi_0'(z)(\tp(z)-\psi_0'(z))\hn(z)dz}_{T_2}\\
&\ +\underbrace{\dint_{-a_n}^{a_n}\psi_0'(z)^2(\hn(z)-g_0(z))dz}_{T_3},
\end{align*}

It is clear that by our assumption,
$T_1=O_p((\log n)^3n^{-4p/5})$, which is $o_p(1)$.

Because $\psi_0'$ is a non-increasing odd function, on any interval, $|\psi_0'|$ attains its maximum at both end points. Therefore,

\begin{align*}
\leq &\ |\psi_0'(a_n)|\dint_{-a_n}^{a_n}|\tp(z)-\psi_0'(z)|\hn(z)dz \\
\stackrel{(a)}{\leq} &\ |\psi_0'(a_n)|\lb\dint_{-a_n}^{a_n}(\tp(z)-\psi_0'(z))^2\hn(z)dz \rb^{1/2},
\end{align*}

where (a) follows by the Cauchy-Schwarz inequality. Thus, $|T_2|\leq |\psi_0'(a_n)|\sqrt{T_1}$.
However,  Assumption <ref> implies that $|\psi_0'(a_n)|\leq |\psi_0'(0)|+\kappa a_n=O_p(\log n)$ provided $a_n\in \iint(\dom(\psi_0))$. By our assumption on $a_n$, the latter holds with probability tending to one.  Hence,

\[T_2=O_p(\log n) \sqrt{T_1}=O_p((\log n)^{5/2}n^{-2p/5})=o_p(1).\]

Finally,  using the fact  $|\psi_0'(a_n)|=O_p(\log n)$ again, we bound $|T_3|$ by

\begin{align*}
O_p((\log n)^2) d_{TV}(\hn, g_0)
\stackrel{(a)}{\leq}  & O_p((\log n)^2)H(\hn,g_0)\stackrel{(b)}{=} O_p((\log n)^2 n^{-p})=o_p(1),
\end{align*}

where (a) follows from Fact <ref> and (b) follows noting  $H(\hn,g_0)=O_p(n^{-p})$ by Condition <ref>. Thus we have shown that

\[\dint_{-a_n}^{a_n}\slb\tp(z)^2\hn(z)-\psi_0'(z)^2g_0(z)\srb dz=o_p(1).\]

Since $a_n\to_p G_0^{-1}(1)$ by our assumption, noting $G_0^{-1}(0)=-G_0^{-1}(1)$, we also obtain $-a_n\to_p G_0^{-1}(0)$. Hence,

\[\dint_{-a_n}^{a_n}\psi_0'(z)^2g_0(z)dz\to_p\dint_{G_0^{-1}(0)}^{G_0^{-1}(1)}\psi_0'(z)^2g_0(z)dz=\I,\]

which completes the proof of the first part of Lemma <ref>.
Second part of Lemma <ref> follows noting $\xin=O_p(\log n)$ by Lemma <ref>, $\xin\in\iint(\dom(\psi_0))$ with probability tending to one by Lemma <ref>,  $\xin\to_p G_0^{-1}(1)$ by Lemma <ref>, and $\hn\in\mathcal{SLC}_0$ satisfies (<ref>)  by Lemma <ref>.

Consider the set up of Theorem <ref>.
Suppose $\eta_n=Cn^{-2p/5}$, where $C>0$ and $p$ is as in Condition <ref>. Let $y_n$ be a sequence of positive random variables such that
$P(|y_n|\leq \eta_n/(2g_0(0)))\to 1$. Then

\[ \sup_{x\in[-\xi_n-y_n,\xi_n+y_n]}|\tp(x)|=O_p(\eta_n^{-1/2})=O_p(n^{p/5}).\]

Let $q\in(0,1/2)$.
Since $\hn$, being log-concave, is positive on $\iint(J(\tilde G_n))$, using Fact <ref> we obtain that

\[
\dint_{\tilde{G}^{-1}_n(q/2) }^{\tilde{G}^{-1}_n(q)} \tp(x)^2\hn(x)dx=\dint_{q/2}^q\tp(\tilde{G}^{-1}_n(z))^2dz.
\]

Note that $\tp$ is non-increasing and positive on $(-\infty, -x]$, and $\hn(-x)$ is positive and non-decreasing on $(-\infty, -x]$. Thus $\tp\circ \tilde{G}^{-1}_n$ is non-increasing. Therefore

\[ q\tp(\tilde{G}^{-1}_n(q))^2/2 \leq \dint_{\tilde{G}^{-1}_n(q/2) }^{\tilde{G}^{-1}_n(q)} \tp(x)^2\hn(x)dx \stackrel{(a)}{\leq} \dint_{\tilde{G}^{-1}_n(q/2) }^{\tilde{G}^{-1}_n(1-q/2)} \tp(x)^2\hn(x)dx \]

where (a) follows because $q<1-q/2$ for all $q\in(0,1/2)$.
Suppose $q=\eta_n/2$. Note that $\tilde{\xi}_n=-\tilde{G}^{-1}_n(\eta_n/2)$ and $|\tp|$ is symmetric about zero. Then the last display leads to

\[\tp(\tilde{\xi}_n)^2\leq \frac{4\dint_{\tilde{G}^{-1}_n(\eta_n/4) }^{\tilde{G}^{-1}_n(1-\eta_n/4)} \tp(x)^2\hn(x)dx}{\eta_n}.\]

From Lemma <ref> it follows that the integral converges in probability to $\I$. Therefore, $|\tp(\tilde{\xi}_n)|=O_p(\eta_n^{-1/2})$ which implies

\begin{equation}\label{inlemma: tp bound}
\sup_{x\in[-\tilde{\xi}_n,\tilde{\xi}_n]}|\tp(x)|=O_p(\eta_n^{-1/2}).
\end{equation}

The rest of the proof follows from (<ref>) and Lemma <ref>.

Consider the set up of Theorem <ref>.
Then  $\|h_n\|_{P_0, 2}^2=O_p(n^{-4p/5}(\log n)^{3})$, where $h_n$ is as defined in (<ref>).

Note that $\|h_n\|_{P_0,2}^2$ equals

\begin{align*}
\MoveEqLeft \dint_{-\xi_n}^{\xi_n}(\hln'(z)-\psi_0'(z-\td))^2 f_0(z+\bth)dz\\
\leq &\  2\dint_{-\xi_n}^{\xi_n}\slb\hln'(z)-\psi_0'(z)\srb^2f_0(z+\bth)dz+2\dint_{-\xi_n}^{\xi_n}\slb \psi_0'(z-\td)-\psi_0'(z)\srb^2f_0(z+\bth)dz\\
\leq &\ 2\underbrace{\dint_{-\xi_n}^{\xi_n}\slb\hln'(z)-\psi_0'(z)\srb^2g_0(z-\td)dz}_{T_1}+2\underbrace{\dint_{-\xi_n}^{\xi_n}\slb \psi_0'(z-\td)-\psi_0'(z)\srb^2g_0(z-\td)dz}_{T_2}.
\end{align*}

$T_1$ is $O_p(n^{-4p/5}(\log n)^3)$ by Lemma <ref>. Since Assumption <ref> implies $\psi_0'$ is Lipschitz with constant $\kappa$ on its domain, Lemma <ref> entails that  $T_2$ is bounded by

\[2\kappa^2\td^2\dint_{-\xi_n}^{\xi_n}g_0(z-\td)dz\]

which is $O_p(n^{-1})$ since $\td=O_p(n^{-1/2})$. Hence, the proof follows.

§.§.§ Lemma on consistency of Fisher information $\hin(\eta_n)$:

Under the set up of Theorem <ref>, $\hin(\eta_n)\to_p\I$ where $\hin(\eta_n)$ is as defined in (<ref>).

Denoting $\td=\theta_0-\bth$, we observe that

\begin{align}\label{inlemma: consistency of FI}
\MoveEqLeft|\hin(\eta_n)-\Ig(\eta_n)|\nn\\
\leq &\ \underbrace{\bl\dint_{\bth-\xi_n}^{\bth+\xi_n}\hln'(x-\bth)^2d(\Fm-F_0)(x)\bl}_{T_1}
+\underbrace{\bl\dint_{-\xi_n}^{\xi_n} \hln'(x)^2 \slb g_0(x-\td)-g_0(x)\srb dx\bl }_{T_2}\nn\\
&\ + \underbrace{\bl\dint_{-\xi_n}^{\xi_n} \hln'(x)^2 \slb g_0(x)-\hn(x)\srb dx\bl}_{T_3}+\underbrace{\bl\dint_{-\xi_n}^{\xi_n} \hln'(x)^2 \hn(x) dx-\I\bl}_{T_4}
\end{align}

Let us consider the term $T_1$ first. Denoting  $M_n=C n^{p/5}$ as in the
the proof of the first step of Theorem <ref>, we recall the class of functions $\mathcal U_n(M_n)$  defined in (<ref>).

In the same way we showed that $h_n\in\mathcal{H}_n(C)$ with high probability in the proof of the first step of Theorem <ref>,
we can show that the function defined by

\[\tilde {h}_n(x)=\tp(x-\bth)^21_{[\bth-\xin,\bth+\xin]}(x)\]

is a member of the class

\begin{align*}
\mathcal{V}_n(C)=\bigg\{h:\RR\mapsto & \RR\ \bl \ h(x)=u(x)^2 1_{[r_1,r_2]}(x),\ u\in\mathcal{U}_{n}(M_n),\\
[r_1,r_2]\subset [\th_0-C\log n,\th_0+C\log n]\cap\iint(\dom(\phi_0)) \bigg \}
\end{align*}

with high probability for all  large $n$ provided $C>0$ is sufficiently large. Using (<ref>), (<ref>) and following some standard calculations,
we can show that

\[\sup_Q\log N_{[\ ]}(\e,\mathcal{V}_n(C),L_2(Q)\lesssim M_n^2\e^{-1},\]

where the supremum is over all probability measures on $\RR$.

Because bracketing number is larger than covering number, it also follows that

\[\sup_Q\log N(\e,\mathcal{V}_{n}(C),L_2(Q))\lesssim M_n^2\e^{-1}.\]

The definition of $\mathcal U_n(M_n)$ in (<ref>) implies that the functions in $\mathcal{V}_n(C)$ are uniformly bounded by $M_n^2$.
Since for any fixed $\e>0$,

\[M_n^2\sup_Q\log N(\e,\mathcal{V}_{n}(C),L_2(Q))\lesssim M_n^4\e^{-1}=O(n^{-4p/5}),\]

Fact <ref> leads to $E\|\mathbb F_n-F_0\|_{\mathcal{V}_n(C)}=o(1)$. Thus  Markov inequality yields that $\|\mathbb F_n-F_0\|_{\mathcal{V}_n(C)}=o_p(1)$. Since for large $C$, $P(\tilde{h}_n\in\mathcal{V}_n(C))$ with high probability,  it can be shown that $\int \tilde {h}_n d(\mathbb F_n-F_0)\to_p 0$, which establishes $T_1=o_p(1)$.

Since the supremum of $|\tp|$ over  $[-\xin,\xin]$ is $O_p(n^{p/5})$  by Lemma <ref>,
we obtain that

\[T_2\leq O_p(n^{2p/5})d_{TV}(g_0(\cdot-\td),g_0)\stackrel{(a)}{\leq }O_p(n^{2p/5}) H(g_0(\cdot-\td),g_0)\stackrel{(b)}{=}O_p(\td n^{2p/5}),\]

which is $o_p(1)$ because $\td=O_p(n^{-1/2})$ and $p\in(0,1)$. Here (a) and (b) follow from Fact <ref> and Fact <ref>, respectively.

In a similar way we can show that

\[T_3\leq O_p(n^{2p/5})d_{TV}(\hn,g_0)\leq O_p(n^{2p/5})H(\hn,g_0),\]

which is $O_p(n^{-3p/5})$ by Condition <ref>.

Finally, noting $T_4$ is also $o_p(1)$ by Lemma <ref>, the proof follows from (<ref>).

§ PROOF OF PROPOSITION <REF>

We will first show that $\widehat{g}_{\bth}$ and $\hn^{geo, sym}$ satisfy Condition <ref>. Then using this result, we will show in  Lemma <ref> and Lemma <ref> that $\widehat{g}_{\bth}$ and $\hn^{geo, sym}$ satisfy Condition <ref>, respectively. To show that Condition <ref> holds for these two densities, we prove a general Proposition which states that Condition <ref> holds for all the density estimators of $g_0$ we have discussed so far.

Suppose $f_0\in\mathcal{P}_1$ and $\bth$ is a  consistent estimator of $\th_0$. Then   $\hn=$ $\widehat{g}_{\bth}$, $\hf(\bth\pm\mathord{\cdot})$, $\hn^{geo,sym}$,  $\hts(\bth\pm\mathord{\cdot})$,  and $\hnss$
satisfy Condition <ref>.

The key step in proving Proposition <ref> is showing that the $L_1$ consistency  in Condition <ref>(A) holds, which is established by Lemma <ref>. The proof of
Lemma <ref> can be found in  Appendix <ref>.

Suppose $\bth\to_p \th_0$, and $\hn$ is one among
$\hts(\bth\pm\mathord{\cdot})$, $\hf(\bth\pm\mathord{\cdot})$, $\hnss$, $\hn^{geo,sym}$, and $\widehat{g}_{\bth}$. Then $\|\hn-g_0\|_1\to_p 0$.

Now we are ready to prove  Proposition <ref>.

As in the proof of Lemma <ref>, one can show that it suffices to prove   Proposition <ref> when $\bth\as\th_0$, and $y_n\as 0$.  Hence, in what follows,  we assume that $\bth\as\th_0$, and $y_n\as 0$.
First we will verify Condition <ref> when $\hn\in\mathcal{LC}$. Note that, this covers the case of $\widehat{g}_{\bth}$, $\hf(\bth\pm\mathord{\cdot})$, $\hn^{geo,sym}$, and $\hts(\bth\pm\mathord{\cdot})$. We will consider the case of $\hnss$ separately because the latter is not log-concave.

Assuming $\hn\in\mathcal{LC}$, to verify part A of Condition <ref>, we first note that

\begin{align*}
\|\hn(\cdot+y_n)-\q\|_1\leq  \|\hn(\cdot +y_n)-\q(\cdot +y_n)\|_1+
\|\q(\cdot+y_n)-\q\|_1,
\end{align*}

whose first term converges to zero almost surely by
Lemma <ref>.
Also, since $\q$ is continuous, $\q(x+y_n)$ converges to $\q(x)$ for each $x\in\RR$.
Therefore the second term also converges to zero almost surely by Glick's Theorem <cit.>.
Thus we obtain that $\|\hn(\cdot+y_n)-\q\|_1\as 0$.
Since $\hn(\mathord{\cdot}+y_n)$ is log-concave, the above, combined with  Proposition 2(c) of [Cule and Samworth, 2010], yields that
$\|\hn(\cdot+y_n)-\q\|_{\infty}\as 0$ which completes the verification of part A of Condition <ref>.
As a consequence,

\[\hln(x+y_n)=\log(\hn(x+y_n))\as\psp(x),\quad \text{ for each }x\in\iint(\dom(\psp)).\]

Since  $\hln$ is concave for $\hn\in\mathcal{LC}$, Theorem 10.8 of [Rockafellar, 1970] entails that the above
pointwise convergence translates to uniform convergence on all compact sets inside $\iint(\dom(\psp))$, which leads to

$$\sup_{x\in K}|\hln(x+y_n)-\psp(x)|\as 0,$$

proving part B of Condition <ref>.
Since $\hln$ is concave, Part C follows directly  from Part B  by Theorem 25.7 of [Rockafellar, 1970].
Thus we  have established Condition <ref> for $\hn=\widehat{g}_{\bth}$, $\hf(\bth\pm\mathord{\cdot})$, $\hn^{geo,sym}$,
and $\hts(\bth\pm\mathord{\cdot})$.

Now we verify Condition <ref> for $\hnss$.
Part A of Condition <ref> can be verified  noting  (<ref>)  implies

\begin{align*}
\MoveEqLeft 2\sup_{x\in \RR}|\htsm(x+y_n)-\qsm(x)|\\
\leq  &\  \sup_{x\in \RR}|\hf^{sm}(\bth+x+y_n)-\qsm(x)| +\sup_{x\in \RR}|\hf^{sm}(\bth-x+y_n)-\qsm(x)|,
\end{align*}

which converges to zero almost surely because, as we have already shown, the log-concave density $\hts$ satisfies Condition <ref>(A).

To prove part B,

we observe that

\begin{align*}
\sup_{x\in K}\bl\hlnsm(x+y_n)-\psp(x)\bl\leq \dfrac{\sup_{x\in K}|\htsm(x+y_n)-\q(x)|}{\min\lb\inf\limits_{x\in K}\htsm(x+y_n),\inf\limits_{x\in K}\q(x)\rb},
\end{align*}

whose numerator converges to zero almost surely by part A of Condition <ref>.  Thus, to verify part B of Condition <ref> for $\htsm$, we only need to show that the denominator of the term on the right hand side of last display is bounded away from zero. To this end, notice that

\[\inf_{x\in K}\htsm(x+y_n)\geq \inf_{x\in K}\min(\hts(\bth+x+y_n),\hts(\bth-x-y_n))\stackrel{(a)}{\as} \inf_{x\in K}g_0(x),\]

where (a) follows because we just showed that  $\hts(\bth\pm\mathord{\cdot})$ satisfy  Condition <ref>. Now $ \inf_{x\in K}g_0(x)>0$ because  $K$ is a  subset of  $\iint(\dom(\psp))$. Thus we have verified  part B of Condition <ref>  for $\htsm$.

Next note that  $\hts$ is a smooth function, and it is also positive on  $\RR$. Therefore $\hlnsm$ and  $\hlns$ are differentiable on $\RR$. Therefore, for any $x\in\RR$,

\begin{equation}\label{definition of tpz by tpc}
\end{equation}

where  $\varrho_n(x)=\hts(\bth+x)/2\hnss(x)<1$.

\begin{align*}
\MoveEqLeft|\hlnspm(x+y_n)-\psp'(x)| \leq  \varrho_n(x+y_n)|\hlnsp(\bth+x+y_n)-\psp'(x)|\nonumber\\
&\ +\ (1-\varrho_n(x+y_n))|\hlnsp(\bth-x-y_n)-\psp'(-x)|.
\end{align*}

Since $ \varrho_n$ is uniformly bounded by one,  Condition <ref>(C)  applied on the concave function $\hlns(\bth\pm\mathord{\cdot})$ completes the verification of part C for $\hlnsm$.

§.§ Auxiliary lemmas for the proof of proposition <ref>

First we show that  it suffices to prove the current lemma when $\bth\as \th_0$.  Suppose we are able to prove $\|\hn-g_0\|_1$ is strongly consistent when $\bth\as 0$. Since $\bth\to_p 0$, Fact <ref> implies
given any subsequence of $\{\bth\}_{n\geq 1}$, there exists a further subsequence $\{\bar{\th}_{n_k}\}_{k\geq 1}$ such that $\bar{\th}_{n_k}\as \th_0$ as $k\to\infty$. Therefore, along this subsequence $\{n_k\}_{k\geq 1}$, the $L_1$ distance between $\hn$ and $g_0$  approaches zero almost surely.  In that case,  Fact <ref> implies  that $\|\hn-g_0\|_1$ converges in probability to zero. Therefore, in what follows, we assume that $\bth\as \th_0$.

We begin with the case of $\hts$.
Theorem 1 of  [Chen and Samworth, 2013] implies that when $f_0$ has finite second central moment, we have

\begin{equation}\label{convergence: L1: hts}
\edint|\hts(x)-\psm(x)|dx\as 0.
\end{equation}

That $f_0$ has second central moment is immediate by Fact <ref>.
Note that

\[\|\hts(\bth+\cdot)-g_0\|_1\leq \|\hts-f_0\|_1+\|f_0(\bth+\cdot)-g_0\|_1,\]

whose first term converges to zero almost surely by (<ref>), and the second term

\[\|f_0(\bth+\cdot)-g_0\|_1=\|g_0(-\td+\cdot)-g_0\|_1\stackrel{(a)}{\leq} \sqrt 2 H(g_0(-\td+\cdot),g_0)\stackrel{(b)}{\lesssim}\td\as 0\]

where (a) and (b) follow from Fact <ref> and Fact <ref>, respectively. Thus we have established that $\|\hts(\bth+\cdot)-g_0\|_1\as 0$. Since $g_0$ is symmetric about zero,   $\|\hts(\bth-\cdot)-g_0\|_1\as 0$ follows.
Because $\|\hf-f_0\|_1\as 0$ by Theorem $4$ of [Cule and Samworth, 2010], the proof of
$\|\hf(\bth\pm\mathord{\cdot})-g_0\|_1\as 0$ follows in the same way.

The  $L_1$ consistency of $\htsm$  also follows noting  (<ref>) implies

\begin{align}\label{inlemma: L1 convergence: smoothed}
2 \|\htsm-\qsm\|_1\leq \|\hts(\bth+\cdot)-\qsm\|_1+
\|\hts(\bth-\cdot)-\qsm\|_1\as 0.
\end{align}

Next, we consider the  geometric mean estimator $\hg^{geo,sym}$.
We have already established

\begin{equation}\label{L1 convergence: MLE: centered}
\edint|\hf(\bth\pm x)-\q( x)|dx\as 0,
\end{equation}

which entails that  the distribution functions of $\hf(\bth\pm\mathord{\cdot})$ converge weakly to  $G_0$. The above, combined with
Proposition $2$(b) of [Cule and Samworth, 2010] shows that (<ref>) leads to almost sure convergence of
$\hf(\bth\pm x)$ to  $\q( x)$ almost everywhere on $\RR$ with respect to the Lebesgue measure. As a consequence, it follows that

\[\hg^{geo,sym}(x)C^{geo}_{n}\as\sqrt{\q(x)\q(-x)}=\q(x)\quad a.e.\ \ x.\]

Recall from (<ref>) that

\[C^{geo}_{n}=\edint\sqrt{\hf(\bth+x)\hf(\bth-x)}dx.\]

From Scheffé's Lemma it follows that $C^{geo}_{n}\as \int d\tilde{G}_0=1$.
We have thus established that $\hg^{geo,sym}$ converges almost everywhere to $\q$ almost surely. Therefore, $\Gs$ converges weakly to $G_0$ almost surely. The desired strong $L_1$ consistency then follows from Proposition 2(c) of [Cule and Samworth, 2010].

To establish the $L_1$ consistency of the partial MLE estimator $\widehat{g}_{\bth}$, we appeal to  the  projection  theory developed in  [Xu and Samworth, 2019]. According to this theory, $\widehat{G}_{\bth}$ can be interpreted  as the  the projection (w.r.t. Kullback-Leibler divergence)  of ,  the empirical distribution function of the $X_i-\bth$'s, onto the space of the distribution functions with density in $\mathcal{SLC}_0$. This projection operator has some continuity properties. In particular, if we can show that

\begin{equation}\label{convergence of empirical: wasserstein metric : partial mle estimator}
d_W(\Fn, G_0)\as 0,
\end{equation}

the desired $L_1$ consistency $\|\widehat{g}_{\bth}-\q\|_1\as 0$ follows from Proposition $6$ of [Xu and Samworth, 2019] provided $G_0$ is non-degenerate and it has first finite moment. The non-degeneracy is trivial and the existence of first moment follows from Fact <ref>. Hence, it is enough to prove (<ref>) holds,  for which, by Theorem $6.9$ of [Villani, 2009],
it suffices to show

\begin{equation}\label{condition: dw: first condition}
\edint|x|d\Fn(x)\as \edint|x|dG_0(x),
\end{equation}

and that  converges to $G_0$ weakly with probability one.
Since $\bth$ is strongly consistent for $\th_0$, and

\[
\edint|x|d\Fn(x)=\edint |x-\bth|d\Fm(x),
\]

for any $d>0$,  an application of Glivenko-Cantelli Theorem  <cit.> yields

\begin{align*}
\sup\limits_{\bth\in[\th_0-d,\th+d]}\bl\edint |x-\bth|d(\Fm-F_0)(x)\bl\as 0.
\end{align*}

On the other hand,  strong consistency of $\bth$ implies  $|x-\bth|\leq|x-\th_0|+1$  with probability one for all sufficiently large $n$,  where the latter is integrable with respect to $f_0$. Therefore, the dominated convergence theorem  leads to

\[\edint |x-\bth|dF_0(x)\as \edint |x-\th_0|dF_{0}(x)=\edint |x|dG_0(x),\]

which proves (<ref>).

Our next step is to prove the weak convergence of to $G_0$. To this end, we note that

\[
\Fn(x)=\Fm(x+\bth)=\edint 1_{(-\infty,x+\bth]}(z)d\Fm(z),
\]

which converges almost surely to

\begin{align*}
\edint 1_{(-\infty,x+\th_0]}(z)dF_0(z)=G_0(x)
\end{align*}

by an application of basic Glivenko-Cantelli Theorem <cit.>, and the fact that $F_0(x+\bth)\as F_0(x+\th_0)$ for all $x\in\RR$. This establishes (<ref>), which proves the strong $L_1$ consistency of $\widehat{g}_{\bth}$, thus finishing the proof of the current lemma.

§.§.§ Lemmas on Hellinger error of $\widehat g_{\bth}$ and $\tilde{g}_n^{geo, sym}$:

Suppose $f_0=g_0(\mathord{\cdot}-\theta_0)$where $g_0\in\mathcal{SLC}_0$ and $\bth-\theta_0=O_p(n^{-1/2})$. Then
$H(\widehat g_{\bth},g_0)=O_p(n^{-1/4})$.

From Theorem 4.1 of [Doss and Wellner, 2019] it follows that $H(\widehat g_{\th_0},g_0)=O_p(n^{-2/5})$. The result will therefore follow by triangle inequality if we can show that
$H(\widehat g_{\th_0},\widehat g_{\bth})=O_p(n^{-1/4})$.
To that end, for any function $\ph:\RR\mapsto\RR$, and distribution function $G$, we define the functional $\om:(\ph,G)\mapsto\RR$ by

\begin{equation}\label{definition: new criterion function }
\om(\ph,G)=\edint \ph(x)dG(x)-\edint e^{\ph(x)}dx.
\end{equation}

Recall that we defined $\widehat \psi_{\th}$ to be $\log \widehat g_{\th}$ for any $\th>0$.
Denoting $\mathbb F_{n,Y}$ to be the empirical distribution function of random variables $Y_1,\ldots,Y_n$,  we observe that for any $\th>0$, $\widehat\psi_{\th}$ writes as <cit.>

\[ \widehat \psi_{\th}= \argmax_{\phi\in\mathcal{SC}_0} \Phi(\phi, \mathbb F_{n,X-\th})
= \argmax_{\phi\in\mathcal{SC}_0} \Phi(\phi, \mathbb F_{n,Z+\th_0-\th}),\]

where $Z=X-\theta_0$. Let us denote $\dd=\th_0-\th$.
Using Lemma <ref> we obtain that

\[\argmax_{\phi\in\mathcal{SC}_0} \Phi(\phi, \mathbb F_{n,Z+\dd})=\argmax_{\phi\in\mathcal{C}} \Phi(\phi, \mathbb F^{sym}_{n, Z+\dd}),\quad\text{ where }\]

\[\mathbb F^{sym}_{n, Z+\dd}(x)=\frac{\mathbb F_{n,Z+\dd}(x)+1-\mathbb F_{n,Z+\dd}(-x)}{2}\stackrel{(a)}{=}\frac{\mathbb F_{n,Z}(x-\dd)+1-\mathbb F_{n,Z}(-x-\dd)}{2}\]

is the symmetrized version of $\mathbb F_{n, Z+\dd}$. Here (a) follows because $\mathbb F_{n,Z+\dd}(x)$ equals $\mathbb F_{n, Z}(x-\dd)$.
In particular, the choice  $\th=\th_0$ yields $\dd=0$, which leads to

$\widehat\psi_{\th_0}=\argmax_{\phi\in\mathcal{C}} \Phi(\phi, \mathbb F^{sym}_{n, Z})$,
$\mathbb F^{sym}_{n, Z}(x)=(\mathbb F_{n, Z}(x)+1-\mathbb F_{n, Z}(-x))/{2}.$
When $\th=\bth$, on the other hand, $\dd=\td$, which yields

\[ \widehat \psi_{\bth}
= \argmax_{\phi\in\mathcal{C}} \Phi(\phi, \mathbb F^{sym}_{n,Z+\td}),\ \mathbb F^{sym}_{n, Z+\td}(x)=\frac{\mathbb F_{n, Z}(x-\td)+1-\mathbb F_{n, Z}(-x-\td)}{2}.\]

If we can show that $\mathbb F^{sym}_{n, Z}(x)$ and $\mathbb F^{sym}_{n, Z+\td}$ are non-degenerate with finite first moment, then  Theorem 1 of [Barber and Samworth, 2020] yields

\begin{equation}\label{intheorem: Rina barbara statement}
E[H(\widehat g_{\bth},\widehat{g}_{\th_0})]\lesssim \lb\frac{d_W(\mathbb F^{sym}_{n, Z},\mathbb F^{sym}_{n, Z+\td})}{\epsilon_{\mathbb F^{sym}_{n, Z}}}\rb^{1/2} ,
\end{equation}

where for any distribution function $F$, $\e_F$ is defined by

\[\epsilon_{F}=E_{ F}[|Y-E_{F}[Y]|].\]

Here $E_F$ is the expectation with respect to $F$.
Since $G_0$ is non-degenerate, $\mathbb F_{n, Z}$ is non-degenerate with probability one. Therefore both $\mathbb F^{sym}_{n, Z}(x)$ and $\mathbb F^{sym}_{n, Z+\td}$ are non-degenerate with probabilty one. Also, because $\mathbb F_{n, Z}$ has finite first moment for all $n\geq 1$, both $\mathbb F^{sym}_{n, Z}(x)$ and $\mathbb F^{sym}_{n, Z+\td}$ have finite first moment for all $n\geq 1$. Therefore (<ref>) holds.

Next, we show that $\epsilon_{\mathbb F^{sym}_{n, Z}}$ is bounded away from zero almost surely. To that end, we first prove the side result that $ d_W(\mathbb F_{n,Z}^{sym},G_0)\as 0$.
By (<ref>),

\begin{align*}
d_W(\mathbb F_{n,Z}^{sym},G_0)=&\ \edint |\mathbb F_{n,Z}^{sym}(x)-G_0(x)|dx  \\
= &\ \edint \left | \frac{\mathbb F_{n, Z}(x)+1-\mathbb F_{n, Z}(-x)}{2}-G_0(x)\right |,
\end{align*}

which, due to the symmetry of $g_0$  about the origin,  equals

\begin{align*}
\MoveEqLeft \edint \left | \frac{\mathbb F_{n, Z}(x)+1-\mathbb F_{n, Z}(-x)}{2}-\frac{G_0(x)+1-G_0(-x)}{2}\right |\\
\leq &\ \frac{1}{2}\lb\edint\left | \mathbb F_{n, Z}(x)-G_0(x)\right |dx+\edint\left | \mathbb F_{n, Z}(-x)-G_0(-x)\right |dx\rb\\
\leq &\ \edint\left | \mathbb F_{n, Z}(x)-G_0(x)\right |dx,
\end{align*}

which equals $d_W(\mathbb F_{n, Z}, G_0)$. The latter converges to zero almost surely by Varadarajan's Theorem <cit.> and the strong law of large numbers. Therefore,  $ d_W(\mathbb F_{n,Z}^{sym},G_0)\as 0$ follows.

Proposition 1 of [Barber and Samworth, 2020] implies that if $F$ and $F'$ are  distribution functions  with finite first moment, then $\epsilon_F>0$, and  $|\epsilon_{F}-\epsilon_{F'}|$ is bounded by $ 2d_W(F,F')$. Now $G_0$ being log-concave, has finite first moment. Therefore we have $\epsilon_{G_0}>0$. Also since $G_0$ and $\mathbb F_{n,Z}^{sym}$ are non-degenerate,  it follows that

\[\abs{\epsilon_{\mathbb F_{n,Z}^{sym}}-\epsilon_{G_0}}\leq d_W({\mathbb F_{n,Z}^{sym}},G_0),\]

which implies