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A household term in relational database management, SQL is the first choice of any user or organization when it comes to effective communication with a database. The popularity and use of SQL is courtesy its query set to retrieve large sets of data swiftly, established standards which are being followed and implemented by the likes of ANSI and bare-minimum coding to manage database systems. SQL is considered an important tool to this day being used everywhere by the second while consultants and professionals with a vice grip on SQL are still a hot commodity in the industry.
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Marlin Magazine recently ran a feature article following three marlin fishing families, and two of them were Jarrett Bay owners, the Richardsons of Whoo Dat and Huddles of Builder's Choice!
History shows that if we splash enough salt and scales on our kids over the years, they're likely to grow into avid fishermen. Stick with it long enough, and maybe they'll even become world-class anglers, captains or mates. We recently caught up with three accomplished marlin fishing families — the Richardsons, the Huddles and the Ingrams — to see what makes them tick and how their fishing legacies have evolved.


This is from my road trip files.
I had the opportunity to drive from St. Louis to Wakefield, MA back in 2007. It was by far the best trip I have done.
Routing: I-70 to MO 370. 370 to I-270. I-270 to I-55. I-55 to I-80. I-80 through Indiana to the point in Ohio where 90 splits off. I-90 to I-95. I took the "southern" route home. The southern route was I-90 to 84 to 380 to 80 to 57. 57 to 72. 72 to 55. 55-270.
I left the house at 2 AM and headed out on the road. I stopped at QT and got a BFC of Monster Energy. The moon was setting and I was heading away from it. 70 was active with truckers but other than that traffic was light. 370 was a ghost town. I had some trouble on 270 as they were doing some work and the St. Louis County Police were everywhere. I crossed the Chain of Rocks and headed into Madison Co. Illinois. 270 seems to be pretty dangerous in Illinois as it was only two lanes each way. I do not know if it's still like that. The truck fencing in the middle helps.
At about 2:45 am I made it to 55 and headed north. The moon was to my left and it was bright orange. By 3:50 I made it to Springfield and pondered if I should take 72 to 57 to 80. I stayed on 55 and made good time. I passed a Greyhound bus. By 6 AM I was at the 80 interchange. I headed eastward yet again.
Nobody told me how boring the Indiana Tollway is. The scenery was all pretty much the same. Except for the towns and large cities, farms are all you get. Ohio was a little better.
Then I got to Cleveland.
Cleveland was brutal. Traffic everywhere. Gridlock seemed to extend into Lake County, but once past 271 it dropped off. Pennsylvania was quick.
Then came the Thruway. How deceiving it is. Initially I thought Utica and Albany were at most 50-55 miles apart. Nope. It's 100 miles. I didn't hit Albany until early in the morning. The terrain on the way to Albany is gorgeous (as I found out in 2012.) The area by Herkimer slowed me down somewhat.
Crossing the Hudson and ultimately into MA was awesome. I was back in my neck of the woods. made excellent time on the Mass Pike. 128 had some slowdowns, but I exited off at MA 129 and headed to my grandfather's house.
On the way back, I left at 2 AM and headed down 128 to the Pike and got on. Seemed to take no time in getting to 84. The stretch of 84 in Hartford and the surrounding areas scares me, as the road twists and turns. Going across NYS was fun. The Shawangunks were glowing in the rising sun. I ate breakfast at Port Jervis. I then crossed into PA and headed to 380. I took that south and headed onto 80. I stayed on 80 until I-57 in Illinois. I got to see Kankakee. Question: why is the interchange between 72 and 57 a damned cloverleaf? I would think it should be a rural stack.
72 seemed to fly by. I closed the loop at Springfield and headed onto 55 south. They need to do something about the cloverleaf at 55 and 72. Someone could get killed there. There was a bad accident on 270 in Glen Carbon, so I exited and went down 157 to 55-70. Crossed the PSB and headed on 70 west.
While I was in MA I clinched quite a few routes.
Love your writing style. You seem to remember this one in great detail for it being a decade ago.
What is so significant about Kankakee? Seems to be a lot of focus on it on this forum.
I-72 is pretty underused so I wouldn't expect an upgrade to the I-57/72 interchange anytime soon. They are looking at upgrading the junction of I-57/74 just to the north, though.
I believe I-270 in Illinois is still only 2 lanes each way.
Loved this post, I would like to see more road trip anecdotes/stories shared on here.
Yeah, the Indiana Toll Road can be a bit dull. It doesn't help that there are two stretches with more than 20 miles between exits on it.
Those damn sequential exit numbers are deceiving in and of themselves. I don't have any clue how northeasterners can calculate distances and thus travel times.
Question: why is the interchange between 72 and 57 a damned cloverleaf? I would think it should be a rural stack.
Because IDOT. IDOT, for some reason, seems to have avoided stacks of any kind for the most part. Most of the stack-type interchanges in Illinois either fully belong to or involve ISTHA.
Its the way up there with the sequential numbering. Especially I-90 being reversed (E to W) instead of the usual W to E. Then I-87, has three schemes due to three different jurisdictions as you have the NYC Deegan (though maybe NYSDOT, but influenced by the city), the Thruway, and the Northway all with their own exit numbering each. New Yorkers refer to all three named roads as a different one and not as the whole I-87.
VT is another one that loves its sequential numbering and probably don't want to give it up as they are so used to it all. NH is another, as MA is planning to convert, and CT will change the numbers when the signs need replacing. RI, I have no clue what their opinion is on it, but being its such a small state having milebased numbers won't change things much. DE started with kilometer based on DE 1, but still both I-95 and I-495 have not been changed, and the Turnpike in NJ you can ask JeffandNicole the answer to that one.
NHDOT wants to convert, and had even secured Federal funding for it and prepared contract documents. But the Governor's Executive Council, which has to personally approve ALL expenditures proposed by state agencies, decided it was an unnecessary use of funds and rejected the proposal.


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text:
\section{Introduction}

Estimation of the conditional mode of an outcome variable given regressors, called \textit{modal regression}, is an active research area in the recent statistics literature. In particular, if the conditional distribution is highly skewed or has fat tails,
then one would be more interested in the conditional mode  than the conditional mean or median since in such cases the mean or median may fail to capture a major trend of  the conditional distribution.
As such, modal regression has a wide variety of applications including the analysis of traffic and forest fire data \cite{EinbecTutz2006,YaoLi2014}, econometrics \cite{Lee1989,Lee1993,KempSantos2012,Ho2017}, and machine learning \cite{Sasaki2016,Feng2017}. For example, \cite{KempSantos2012} argue that the mode is the most intuitive measure of central tendency for positively skewed data found in many econometric applications such as  wages, prices, and expenditures (\cite{KempSantos2012}, p. 93). See also \cite{Chen2017} and \cite{Chacon2018} for recent reviews on modal regression.

Existing approaches to estimation of the conditional mode includes nonparametric kernel estimation \cite{Chen2016} and linear modal regression \cite{Lee1989,Lee1993,KempSantos2012,YaoLi2014}, among others.
The nonparametric estimation is able to avoid model misspecification but has slow rates of convergence that deteriorate as the number of regressors increases. Namely, if the number of continuous regressors is $p$, then the rate of convergence of the kernel density based estimator in \cite{Chen2016} is at best $n^{-2/(p+7)}$ under four times differentiability of the joint density.  On the other hand, the linear modal regression is able to avoid such ``curse of dimensionality'' but requires to solve a multi-dimensional non-convex optimization problem.

In this paper, we propose a new estimator for the conditional mode that is able to avoid  the curse of dimensionality and at the same time is computationally scalable, thereby complementing the above existing methods.
The proposed method is based on the observation that the derivative of the conditional quantile function with respect to the quantile index is the reciprocal of the conditional density evaluated at the conditional quantile function and hence the conditional mode is obtained by minimizing the derivative of the conditional quantile function.
Specifically, we assume a linear quantile regression model to estimate the conditional quantile function as in \cite{KoenkerBassett1978} (see also \cite{Koenker2005}), and estimate its derivative by a numerical differentiation of the estimated conditional quantile function.
The proposed estimator is then obtained by minimizing the estimated derivative.
Notably, the proposed method  is computationally attractive since the computation of the quantile regression estimate can be formulated as a linear programming problem and so is highly scalable (cf. Chapter 6 in \cite{Koenker2005}), and the minimization of the estimated derivative is a one-dimensional optimization problem and so can be carried out by a grid search.

We develop asymptotic theory for the proposed estimator, which turns out to be non-standard.
Specifically, we find that the proposed estimator has convergence rate $(nh^{2})^{-1/3}$ where $n$ is the sample size and $h=h_n \to 0$  is a sequence of bandwidths, and the limiting distribution is a scale transformation of Chernoff's distribution \cite{Chernoff1964}. Chernoff's distribution is defined as the distribution of a maximizer of a two-sided Brownian motion with a negative quadratic drift, and appears as e.g. limiting distributions of estimators for monotone functions; see \cite{GroeneboomWellner2001}.
Our result on the limiting distribution would be of interest from theoretical and practical perspectives. First, the proposed estimator provides a new example of estimators having Chernoff's distribution as limiting distributions, which would be of theoretical interest.
Second, the fact that the limiting distribution is a scale transformation of Chernoff's distribution makes inference for our estimator relatively simple.
This is in contrast to e.g. Manski's maximum score \cite{Manski1975} whose limiting distribution is a maximizer of a Gaussian process with its covariance function depending on the distribution of regressors; see \cite{KimPollard1990}.
Building upon the limiting distribution, we develop inference methods for our estimator.
The one is an analytical confidence interval based on consistently estimating the scaling constant, and the other is based on the subsampling \cite{PolitisRomano1994,Politis1999}.
We also derive a multivariate limit theorem for the proposed estimator, which can be used to construct  simultaneous confidence intervals for the modal function over finite design points.

In addition to the theoretical results, we conduct Monte Carlo simulations to assess the finite sample performance of the proposed estimator together with the analytical and subsampling confidence intervals.
We  suggest a practical method to choose the bandwidth based upon the idea suggested in \cite{KoenkerMachado1999}. We compare the performance of the proposed estimator with the linear modal regression estimator of \cite{KempSantos2012,YaoLi2014} via the root mean square error for the two data generating processes  where the true modal function is linear or nonlinear.
Finally, we apply the proposed estimator to predicting the net hourly electrical energy output using Combined Cycle Power Plant Data \cite{Kaya2012,Tufekci2014}.
These numerical results show evidence that the proposed estimator works well in the finite sample.

The literature related to this paper is broad. Nonparametric estimation of the unconditional mode goes back to Parzen \cite{Parzen1962} and Chernoff \cite{Chernoff1964} in 1960s; see also \cite{Romano1988}.
Modal regression originates from \cite{SagerThisted1982} and the literature has flourished since then \cite{Lee1989,Lee1993,EinbecTutz2006,KempSantos2012,Yao2012,YaoLi2014,Chen2016,ZhouHuang2016,Sasaki2016,Ho2017,Krief2017,KhardaniYao2017,Feng2017}.
However, none of these papers do not consider a quantile regression based estimator for the conditional mode.
\cite{Lee1989,Lee1993,KempSantos2012,YaoLi2014} consider linear modal regression; \cite{Lee1989,Lee1993} assume a restrictive condition that the conditional distribution is symmetric around the origin to derive limiting distributions of the estimators. The symmetry of the conditional distribution implies that  the conditional mean, median, and mode are all identical. Subsequently, \cite{KempSantos2012,YaoLi2014} relax the symmetry assumption and propose estimators that enjoy asymptotic normality.
In the present paper, instead of linearity of the conditional mode, we assume a linear quantile regression model. Importantly, the linear quantile regression model does not imply linearity of the conditional mode, and so there are no strict inclusion relations between the two assumptions; see Remark \ref{rem: linearity} ahead.
The recent work of \cite{Chen2016} studies nonparametric kernel estimation of the conditional mode. To be precise, \cite{Chen2016} do not assume the existence of the unique global mode and  allow for multiple local modes. Extension of our approach to multiple local modes would be of interest but is beyond the scope of the present paper. \cite{Yao2012} propose a local modal regression (LMR) estimator that can be seen as a local linear estimator for the conditional mode, and establish asymptotic results analogous to those of a local linear estimator for the conditional mean. In particular, the rate of convergence of the LMR estimator is faster than that of the kernel density based estimator of \cite{Chen2016}. This is, however, due to Condition (A6) in \cite{Yao2012} that is essentially the conditional symmetry assumption on the error term (note that $h_2$ in \cite{Yao2012} is fixed) and under which the conditional mode and mean coincide. In the present paper, we assume no symmetry assumptions on the conditional distribution.

From a technical point of view, derivation of the limiting distribution of the proposed estimator is by no means trivial.
First of all, it is not a priori straightforward to foresee that the convergence rate is $(nh^{2})^{-1/3}$ and the limiting distribution is a scale transformation of Chernoff's distribution. Second, because our objective function depends on the bandwidth tending to zero as the sample size increases, our result does not follow from the general theorem, Theorem 1.1, in \cite{KimPollard1990}, which is a pioneering work on cube root asymptotic theory. The recent work of \cite{SeoOtsu2018} extends \cite{KimPollard1990} to allow the objective function to depend on the bandwidth, but some of their regularity conditions are severely restrictive or difficult to verify in our problem. Hence, we provide a separate and self-contained proof of the main theorem, Theorem \ref{thm: limiting distributions} ahead, which requires  a substantial work. See also the discussion after Theorem \ref{thm: limiting distributions}.

The rest of the paper is organized as follows. In Section \ref{sec: setup}, we state the formal setup and define the estimator. In Section \ref{sec: limiting distributions}, we derive limiting distributions of the proposed estimator and develop inference methods for it. In Section \ref{sec: numerical results}, we conduct Monte Carlo simulations to assess the finite sample performance of the proposed estimator together with the analytical and subsampling confidence intervals. In addition,  we apply the proposed estimator to predicting the net hourly electrical energy output using Combined Cycle Power Plant Data.
Section \ref{sec: conclusion} concludes.
All the proofs are gathered in Appendix.

\section{Setup and estimator}
\label{sec: setup}

In this paper, we are interested in estimating the conditional mode of an outcome variable $Y \in \R$ given a vector of regressors $X = (X_{1},\dots,X_{d})^{T} \in \R^d$.
In what follows, we assume that there exists a conditional density $f(y \mid x)$ of $Y$ given $X$ that is (at least) continuous in $y$, and for each design point $x$ in the support of $X$, there exists a unique mode $m (x)$, i.e., there exists a unique maximizer of the  function  $y \mapsto f(y \mid x)$:
\[
 f(m(x) \mid x) = \max_{y \in \R} f(y \mid x).
\]
The function $m(x)$ is called the modal function.

We base our estimation strategy of the modal function $m (x)$ on inverting a quantile regression model.  Let $Q (\tau \mid X)$ denote the conditional $\tau$-quantile of $Y$ given $X$ for $\tau \in (0,1)$. For the notational convenience, we also write $Q_{x}(\tau) = Q(\tau \mid X=x)$. To see the link between the conditional quantile function and the modal function, we begin with observing that
\[
s_{x}(\tau) := Q_{x}'(\tau) = \frac{\partial Q_{x}(\tau)}{\partial \tau} = \frac{1}{f(Q_{x}(\tau) \mid x)}
\]
assuming some regularity conditions that will be clarified below. Hence, defining
\[
\tau_{x} = \argmin_{\tau \in (0,1)} s_{x}(\tau),
\]
which exists and is unique (by continuity and strict positivity of the function $y \mapsto f(y \mid x)$ around the mode $m(x)$),
we arrive at the key identity
\[
m(x) = Q_{x} (\tau_{x}).
\]
The function $\tau \mapsto s_{x}(\tau)$ (called the ``sparsity'' function) can be estimated by a numerical differentiation of an estimator of the conditional quantile function $\tau \mapsto Q_{x}(\tau)$, and so the problem boils down to estimating the conditional quantile function. To this end, we assume a linear quantile regression model:
\begin{equation}
Q (\tau \mid X) = X^{T} \beta(\tau), \ \tau \in (0,1),
\label{eq: QR model}
\end{equation}
where $\beta (\tau) \in \R^{d}$ is an unknown slope vector for each $\tau \in (0,1)$.

Pick any design point $x$ in the support of $X$, and consider to estimate $m(x)$.
Let $(Y_1,X_1),\dots,(Y_n,X_n)$ be i.i.d. observations of $(Y,X)$. We estimate the slope vector $\beta (\tau)$ by
\begin{equation}
\hat{\beta}(\tau) = \argmin_{\beta \in \R^d}  \sum_{i=1}^{n} \rho_{\tau}(Y_i-X_i^{T}\beta),
\label{eq: QR problem}
\end{equation}
where $\rho_{\tau} (u) = \{ \tau - I(u \le 0) \} u$ is the check function \cite{KoenkerBassett1978}. This leads to an estimator $\hat{Q}_{x}(\tau) = x^{T}\hat{\beta}(\tau)$ of $Q_{x}(\tau)$. To estimate $s_{x}(\tau) =Q_{x}'(\tau)$, let $h = h_{n} \to 0$ be a sequence of bandwidths such that $nh^{2} \to \infty$; then we estimate $s_{x}(\tau)$ by a numerical differentiation:
\[
\hat{s}_{x}(\tau) = \frac{\hat{Q}_{x}(\tau + h) - \hat{Q}_{x}(\tau-h)}{2h}.
\]
Finally, we estimate $m(x)$ by $\hat{m}(x) = \hat{Q}_{x}(\hat{\tau}_{x}) =x^{T}\hat{\beta}(\hat{\tau}_{x})$, where $\hat{\tau}_{x}$ is an approximate minimizer of $\hat{s}_{x}(\tau)$ on $[\varepsilon,1-\varepsilon]$ with sufficiently small parameter $\varepsilon \in (0,1/2)$ chosen by users, in the sense that
\[
\hat{s}_{x}(\hat{\tau}_{x}) \le \inf_{\tau \in [\varepsilon,1-\varepsilon]} \hat{s}_{x}(\tau) + o((nh^{2})^{-2/3}).
\]
The objective function $\hat{s}_{x}(\tau)$ may not admit strict minimizers, and so we allow $\hat{\tau}_{x}$ to be an approximate minimizer in the above sense, which always exists. In practice, our estimator requires to choose the bandwidth $h$, which will be discussed in Section \ref{subsec: bandwidth selection}.

Importantly,  our estimate $\hat{m}(x)$ is easy to compute even when the sample size $n$ and the dimension $d$ of $X$ are large. The quantile regression problem (\ref{eq: QR problem}) can be formulated as a linear programming problem and hence can be efficiently solved even when $n$ and $d$ are large (cf. Chapter 6 in \cite{Koenker2005}).
Furthermore, the entire path $\tau \mapsto \hat{\beta}(\tau)$ can be computed by a parametric linear programming or discretizing the interval $(0,1)$ into fine grids.
The minimization of $\hat{s}_{x}(\tau)$ is a one-dimensional optimization problem and can be solved by a grid search. On the other hand, the linear modal regression estimator  \cite{Lee1989, Lee1993, KempSantos2012,YaoLi2014} requires to solve a multi-dimensional  non-convex optimization problem.
For example, \cite{YaoLi2014} assume that the modal function is linear $m(x) = x^{T}\gamma$ for some $\gamma \in \R^{d}$ and propose the following estimator:
\begin{equation}
\hat{\gamma}_{YL} = \argmax_{\gamma} \sum_{i=1}^{n} \phi_{h}(Y_{i}-X_{i}^{T}\gamma),
\label{eq: YaoLi}
\end{equation}
where $\phi (y) = (2\pi)^{-1/2} e^{-y^{2}/2}$ is the density of the standard normal distribution and $\phi_{h}(y) = h^{-1} \phi(y/h)$. The optimization problem (\ref{eq: YaoLi}) is non-convex. \cite{YaoLi2014} propose an EM like algorithm for (\ref{eq: YaoLi}), but ``there is no guarantee that the algorithm will converge to the global optimal
solution'' (\cite{YaoLi2014}, p. 659).

\begin{remark}[Generality of linear quantile regression model]
\label{rem: linearity}
The linear quantile regression model (\ref{eq: QR model}) is flexible enough to cover many data generating processes. In general, if $\tau \mapsto \beta (\tau)$ is a function on $(0,1)$ such that the map $\tau \mapsto X^{T}\beta (\tau)$ is strictly increasing almost surely and $Y$ is generated as $Y = X^{T} \beta(U)$ for $U \sim U(0,1)$ independent of $X$, then the pair $(Y,X)$ satisfies the linear quantile regression model (\ref{eq: QR model}). In particular, it is worth pointing out that
the linear quantile regression model (\ref{eq: QR model}) does not imply linearity of the modal function $m(x)$.
For example, consider the simple case where $X=(1,X_2)^{T}$ with $X_2 \in (0,1)$ and $Y = U^{3}/3- X_2 (U-1)^{2}$ for $U \sim U(0,1)$ independent of $X$.
In this case, the pair $(Y,X)$ satisfies the model (\ref{eq: QR model}) with $\beta (\tau) = (\tau^{3}/3,-(\tau-1)^{2})^{T}$ and so $Q_{x}(\tau) = \tau^{3}/3 - (\tau-1)^{2} x_2$. Since $Q_{x}'(\tau) = \tau^{2} - 2(\tau-1)x_2$ is minimized at $\tau = x_2$,  the modal function $m(x) = Q_{x}(x_2) = -2x_{2}^{3}/3+2x_{2}^{2} - x_{2}$ is nonlinear.
\end{remark}

\begin{remark}[Case with no regressors]
\label{rem: Chernoff}
In the simple case where there are no regressors, i.e., $X=1$, our estimator of the mode reduces to $\hat{m} = \hat{Q}(\hat{\tau})$, where $\hat{Q}(\tau) = \hat{F}^{-1}(\tau) = \inf \{ y : \hat{F}(y) \ge \tau \}$ is the empirical quantile function (with $\hat{F}(y) = n^{-1} \sum_{i=1}^{n} I(Y_{i} \le y)$ being the empirical distribution function) and
\[
\hat{\tau} = \argmin_{\tau} \frac{\hat{Q}(\tau + h) - \hat{Q}(\tau - h)}{2h}.
\]
Our estimator $\hat{m}$ can also be  described by using order statistics $Y_{(1)} \le \cdots \le Y_{(n)}$.
Since in general $\hat{Q}(\tau) = Y_{(\lceil n\tau \rceil)}$ where $\lceil \cdot \rceil$ is the ceiling function, our estimator $\hat{m}$ coincides with the order statistic $Y_{(\lceil n\hat{\tau} \rceil)}$ where $\hat{\tau}$  minimizes the spacing $Y_{(\lceil n(\tau+h) \rceil)} - Y_{(\lceil n(\tau-h) \rceil)}$.

It is then clear that our estimator is (related to but) markedly different from Chernoff's \cite{Chernoff1964} estimator of  the unconditional mode of $Y$ that is defined by
\[
\hat{m}_{C} = \argmax_{y} \frac{\hat{F}(y + h) - \hat{F}(y-h)}{2h},
\]
namely, $\hat{m}_{C}$ is the point whose local neighborhood contains the most observations.
\end{remark}

\begin{remark}[Alternative objective function]
The estimator $\hat{s}_{x}(\tau)$ of $s_{x}(\tau)$ contains a deterministic bias of order $h^{2}$ under the conditions stated in the next section. Alternatively, we may estimate $s_{x}(\tau)$ by
\begin{equation}
\tilde{s}_{x}(\tau) = \frac{\frac{2}{3} \{ \hat{Q}_{x}(\tau+h) - \hat{Q}_{x}(\tau-h) \} - \frac{1}{12}\{\hat{Q}_{x}(\tau + 2h) - \hat{Q}_{x}(\tau-2h) \}}{h},
\label{eq: alternative objective function}
\end{equation}
which has a bias of order $h^{4}$ under additional smoothness conditions; cf. \cite{BCK2016}. In the present paper, however, we shall use a simpler objective function $\hat{s}_{x}(\tau)$.
\end{remark}

\begin{remark}[Implementation detail]
In the finite sample, $[\tau-h,\tau+h]$ may not be included in $(0,1)$ for some $\tau \in [\varepsilon,1-\varepsilon]$. To fix this, we suggest the following simple modification. Suppose that we compute $\hat{Q}_{x}(\tau)$ on  $[\tau_{\min},\tau_{\max}] \supset [\varepsilon,1-\varepsilon]$; then in practice we suggest to replace $\hat{s}_{x}(\tau)$ by
\[
\hat{s}_{x}(\tau) = \frac{\hat{Q}_{x}(\tau + \min \{ h, \tau_{\max}-\tau \}) - \hat{Q}_{x}(\tau - \min\{ h,\tau - \tau_{\min} \})}{ \min \{ h, \tau_{\max}-\tau \} + \min \{  h, \tau - \tau_{\min} \}},
\]
which asymptotically coincides with the original definition of $\hat{s}_{x}(\tau)$ uniformly in $\tau \in [\varepsilon,1-\varepsilon]$ (as long as $(\tau_{\min},\tau_{\max}) \supset [\varepsilon,1-\varepsilon]$).
\end{remark}

\begin{remark}[Alternative specifications to the conditional quantile function]
In the present paper, we assume that the conditional quantile function is linear in $X$.
The linear quantile regression model is the most fundamental modeling in conditional quantile estimation, and is computationally attractive since the computation of the Koenker-Bassett \cite{KoenkerBassett1978} estimate can be formulated as a linear programming problem. Indeed, the computational attractiveness is one of the main motivations to study the proposed estimator of the conditional mode.

Having said that, we could use alternative specifications to the conditional quantile function to estimate the conditional mode. One possible alternative is a nonlinear quantile regression model $Q_{x}(\tau) = g(x, \beta(\tau))$ where $g$ is some known smooth function (the dimensions of $x$ and $\beta(\tau)$ need not be matched); see e.g. Section 4.4 of \cite{Koenker2005}.
In this case, we can estimate $\beta(\tau)$ by
\[
\hat{\beta}(\tau) = \argmin_{\beta} \sum_{i=1}^{n} \rho_{\tau} (Y_{i} - g(X_{i},\beta)),
\]
and thus can estimate $s_{x}(\tau) = Q_{x}'(\tau)$ by $\hat{s}_{x}(\tau) = \{ \hat{Q}_{x}(\tau+h) - \hat{Q}_{x}(\tau-h) \}/(2h)$ with $\hat{Q}_{x}(\tau) = g(x,\hat{\beta}(\tau))$. Alternatively, we can use the expression
\[
s_{x}(\tau) = \Big [ \underbrace{\frac{\partial g (x,\beta)}{\partial \beta}|_{\beta = \beta(\tau)}}_{=: g_{\beta}(x,\beta(\tau))} \Big ]^{T} \frac{d \beta(\tau)}{d \tau},
\]
and estimate $s_{x}(\tau)$ by $\hat{s}_{x}(\tau) = g_{\beta}(x,\hat{\beta}(\tau))^{T} \{ \hat{\beta}(\tau+h) - \hat{\beta}(\tau-h)\}/(2h)$.
It is known that under regularity conditions, similar asymptotic properties to those of the linear quantile regression estimator hold for the nonlinear case (cf. Section 4.4 of \cite{Koenker2005}), and hence it is natural to expect that  asymptotic results analogous to those developed in the next section can be extended to the resulting conditional mode estimator under the nonlinear quantile regression model.

A yet alternative specification would be a semiparametric single index model $Q_{x}(\tau) = \psi (x^{T}\beta(\tau))$ where $\psi$ is some unknown function.  For given estimators $\hat{\psi}$ and $\hat{\beta}(\tau)$ of $\psi$ and  $\hat{\beta}(\tau)$, we can estimate $Q_{x}(\tau)$ and $s_{x}(\tau)$ by $\hat{Q}_{x}(\tau) = \hat{\psi}(x^{T}\hat{\beta}(\tau))$ and $\hat{s}_{x}(\tau) = \{ \hat{Q}_{x}(\tau + h) - \hat{Q}_{x}(\tau-h)\}/(2h)$, respectively. Alternatively, we can use the expression $s_{x}(\tau) = \psi'(x^{T}\beta(\tau)) d(x^{T}\beta(\tau))/d\tau$ and estimate $d(x^{T}\beta(\tau))/d\tau$ by a difference quotient. Methods to estimate the parametric and nonparametric components in the single index quantile regression model can be found in e.g. \cite{Chaudhuri1997, WuYuYu2010}, and \cite{MaHe2016}. In the single index case, the nonparametric estimation of the link function $\psi$ is involved, whose effect has to be taken into account  when considering asymptotic properties of the resulting conditional mode estimator, which would be a nontrivial challenge.
\end{remark}

\section{Limiting distributions}
\label{sec: limiting distributions}

\subsection{Limiting distributions}

In this section, we derive  limiting distributions of $\hat{\tau}_{x}$ and $\hat{m}(x)$. To this end, we make the following assumption. Let $\mathcal{X}$ denote the support of $X$.

\begin{assumption}
\label{assumption}
In addition to the baseline assumption stated in the previous section, we assume the following conditions.
\begin{enumerate}
\item[(i)]$\E[ X_{j}^{4}] < \infty$ for all $j=1,\dots,d$.
\item[(ii)] The matrix $\E[ XX^{T} ]$ is positive definite.
\item[(iii)] The conditional density $f(y \mid x)$ is three times continuously differentiable with respect to $y$ for each $x \in \mathcal{X}$. Let $f^{(j)}(y \mid x) = \partial^{j}f(y \mid x)/\partial y^{j}$ for $j=0,1,2,3$, where $f^{(0)}(y \mid x) =f(y \mid x)$. There exists a constant $C$ such that $|f^{(j)}(y \mid x)| \le C$ for all $(y,x) \in \R \times \mathcal{X}$ and $j=0,1,2,3$.
\item[(iv)] There exists a positive constant $c$ (that may depend on $\varepsilon$) such that $f(y \mid x) \ge c$ for all $y \in [Q_{x}(\varepsilon/2),Q_{x}(1-\varepsilon/2)]$ and $x \in \mathcal{X}$.
\item[(v)] As $n \to \infty$, $nh^{8} \to 0$ and $nh^{5} \to \infty$.
\end{enumerate}
\end{assumption}

Conditions (i)--(iv) are more or less standard in the quantile regression literature; cf. \cite{Koenker2005}.
In particular, they require no moment conditions on $Y$. For instance, they allow $\E[|Y|] = \infty$.
Conditions (iii) and (iv) allow $Q_{x}(\tau)$ to be four times continuously differentiable on $(\varepsilon/2,1-\varepsilon/2)$ with
\[
\begin{split}
&s_{x}(\tau) := Q_{x}'(\tau) =\frac{1}{f(Q_{x}(\tau) \mid x)}, \ s_{x}'(\tau) = \frac{-f^{(1)}(Q_{x}(\tau) \mid x)}{f(Q_{x}(\tau) \mid x)^{3}}, \\
&s_{x}''(\tau) =  \frac{3 f^{(1)}(Q_{x}(\tau) \mid x)^{2} -f(Q_{x}(\tau) \mid x) f^{(2)}(Q_{x}(\tau) \mid x)}{f(Q_{x}(\tau) \mid x)^{5}}.
\end{split}
\]
Condition (v) is concerned with the bandwidth. The condition $nh^{8} \to 0$ is an ``undersmoothing'' condition. The proof of Theorem \ref{thm: limiting distributions} shows that the estimator $\hat{m}(x)$ contains a deterministic bias of order $h^{2}$, while the stochastic error decreases at rate $(nh^{2})^{-1/3}$. To guarantee that $h^{2} = o((nh^{2})^{-1/3})$, we need $nh^{8} \to 0$.

Let $\{ B(t) : t \in \R \}$ be a two-sided standard Brownian motion, i.e., a centered Gaussian process with continuous sample paths and covariance function
\[
\E[B(t_1)B(t_2)] = \begin{cases}
t_{1} & \text{if} \ 0 \le t_{1} \le t_{2} \\
-t_{2} & \text{if} \ t_{1} \le t_{2} \le 0 \\
0 & \text{if} \ t_{1} \le 0 \le t_{2}
\end{cases}
.
\]
Such a two-sided standard Brownian motion can be constructed by generating independent standard Brownian motions $\{ W_{1}(t) : t \ge 0 \}$ and $\{ W_{2}(t) : t \ge 0 \}$, and then defining $B(t) = W_{1}(t)$ for $t \ge 0$ and $B(t) = W_{2}(-t)$ for $t < 0$. In addition, let
\[
Z = \argmax_{t \in \R} \{ B(t) - t^{2} \},
\]
which exists and is unique almost surely by Lemmas 2.5 and 2.6 in \cite{KimPollard1990}.
The distribution of $Z$ is called \textit{Chernoff's distribution} \cite{Chernoff1964}, and numerical values of quantiles of Chernoff's distribution can be found in \cite{GroeneboomWellner2001}.

Finally, define the matrix
\[
J(\tau) = \E[f (X^{T}\beta(\tau) \mid X) XX^{T}],
\]
which is positive definite for every $\tau \in [\varepsilon/2,1-\varepsilon/2]$ under our assumption.
We are now in position to state the main theorem of this paper.

\begin{theorem}[Limiting distributions]
\label{thm: limiting distributions}
Pick any $x \in \mathcal{X}$.
Suppose that Assumption \ref{assumption} holds, and in addition that $f^{(2)}(m(x) \mid x) < 0$ and $m(x) \in (Q_{x}(\varepsilon),Q_{x}(1-\varepsilon))$ (or equivalently $\tau_{x} \in (\varepsilon,1-\varepsilon)$).
Then we have
\begin{equation}
(nh^{2})^{1/3} (\hat{\tau}_{x} - \tau_{x}) \stackrel{d}{\to} (\sigma_{x}/v_{x})^{2/3} Z
\label{eq: first result}
\end{equation}
as $n \to \infty$, where $\sigma_{x}^{2} = \E[(x^{T}J(\tau_{x})^{-1} X)^{2}]/2, v_{x} = s_{x}''(\tau_{x})/2 = -f^{(2)}(m(x) \mid x)/\{ 2 f(m(x) \mid x)^{4} \} \ (> 0)$, and $\sigma_{x} = \sqrt{\sigma_{x}^{2}}$. In addition, we have
\begin{equation}
(nh^{2})^{1/3} (\hat{m}(x) - m(x)) \stackrel{d}{\to} s_{x}(\tau_{x})(\sigma_{x}/v_{x})^{2/3} Z.
\label{eq: second result}
\end{equation}
\end{theorem}

\begin{remark}[Rates of convergence]
The rate of convergence of $\hat{m}(x)$ toward $m(x)$ is $(nh^{2})^{-1/3}$ and can be arbitrarily close to $n^{-1/4}$ under Condition (v), which is independent of the dimension $d$ of the regressor vector.
The $n^{-1/4}$ rate is likely to be suboptimal from a minimax point of view since it is known that when $X=1$, the minimax rate of estimating the mode under three time differentiability of the underlying density is $n^{-2/7}$; see Theorem 3.1 in \cite{Romano1988}.
It is worth noting that if we use the alternative objective function $\tilde{s}_{x}(\tau)$ in (\ref{eq: alternative objective function}), the bias of the resulting estimator $\hat{m}(x)$ is reduced to $O(h^{4})$ (this however requires additional smoothness conditions on the conditional density), and therefore the rate of convergence can be arbitrarily close to $n^{-2/7}$.
\end{remark}

\begin{remark}[Case with no regressors]
In the simple case where there are no regressors, i.e., $X=1$, the limiting distribution of our estimator $\hat{m}$ is as follows. Let $f$ denote the density of $Y$ with mode $m$; then
$(nh^{2})^{1/3} (\hat{m} - m) \stackrel{d}{\to} \{ 2f(m)^{3}/f''(m)^{2} \}^{1/3} Z$.
In contrast, the limiting distribution of Chernoff's mode estimator (see Remark \ref{rem: Chernoff}) is $(nh^{2})^{1/3} (\hat{m}_{C} - m) \stackrel{d}{\to} \{ 2f(m)/f''(m)^{2} \}^{1/3} Z$, which is slightly different from our limiting distribution. It is worth mentioning that Chernoff's derivation of the preceding limiting distribution in \cite{Chernoff1964} is only heuristic, but can be made rigorous (under regularity conditions) by mimicking the proof of Theorem \ref{thm: limiting distributions}.
\end{remark}

Interestingly, despite the presence of regressors, the limiting distribution of our estimator $\hat{m}(x)$ is a scale transformation of Chernoff's distribution, which is in contrast to e.g. Manski's  maximum score \cite{Manski1975} whose limiting distribution is given by a maximizer of a Gaussian process  whose covariance function  depends on the distribution of regressors; see Example 6.4 in \cite{KimPollard1990}.
The fact that the limiting distribution is a scale transformation of Chernoff's distribution makes inference for our estimator $\hat{m}(x)$  relatively simple.
Namely, an asymptotic confidence interval can be constructed by consistently estimating the constant $s_{x}(\tau_{x})(\sigma_{x}/v_{x})^{2/3}$, which will be discussed in the next section.

The main part of the proof is the proof of the first result (\ref{eq: first result}).
The second result (\ref{eq: second result}) follows from the $\sqrt{n}$-uniform consistency of the quantile regression estimator and the delta method.
To prove the first result (\ref{eq: first result}), we begin with expanding the objective function $\hat{s}_{x}(\tau)$ and showing that $\hat{\tau}_{x}$ is an approximate minimizer of the sample average of kernel functions with a uniform kernel; see (\ref{eq: optimality condition}) in the proof.
Since those kernel functions depend on the sample size $n$ via the bandwidth $h = h_n$,
the result (\ref{eq: first result}) does not follow from the general theorem, Theorem 1.1, in  \cite{KimPollard1990}, which is a pioneering work on cube root asymptotic theory. Theorem 1.1 in \cite{KimPollard1990} covers the case where the objective function is the sample average of functions that do not depend on $n$ and the estimator is $n^{1/3}$-consistent, but its proof does not carry over to our case (cf. the second paragraph in page 192 of \cite{KimPollard1990}). The recent work of \cite{SeoOtsu2018} extends the results of \cite{KimPollard1990} to allow the objective function to depend on the bandwidth (and the data to be dependent), but some of their assumptions are severely restrictive or difficult to verify in our problem. Specifically, Assumption M (i) in \cite{SeoOtsu2018} requires $h_n f_{n,\theta}$ (in their notation) to be uniformly bounded, which in our problem requires the regressor vector $X$ to be bounded (recall that we only assume each coordinate of $X$ to have finite fourth moment); and we (the authors) found that Assumption M (ii) is difficult to verify in our problem.
Hence, instead of checking the assumptions of \cite{SeoOtsu2018}, we provide a separate and self-contained proof of the result (\ref{eq: first result}), which requires a substantial work.
Specifically, we show that the ``rescaled'' objective function for which the rescaled estimator $\hat{t} = (nh^{2})^{1/3}(\hat{\tau}_{x}-\tau_{x})$ is an approximate maximizer converges weakly to the process $\{ \sigma_{x} B(t) - v_{x}t^{2} : t \in \R \}$ in the space of locally bounded functions on $\R$, and apply Theorem 2.7 in \cite{KimPollard1990} to conclude that the approximate maximizer $\hat{t} = (nh^{2})^{1/3}(\hat{\tau}_{x}-\tau_{x})$ converges weakly to $\argmax_{t \in \R} \{ \sigma_{x} B(t) - v_{x}t^{2} \}$, which is shown to be equal in distribution to $(\sigma_{x}/v_{x})^{2/3} Z$; see Step 5 of the proof.

Next, we consider a multivariate limit theorem for the proposed estimator. Let $x^{1},\dots,x^{L} \in \mathcal{X}$ be a finite number of design points with $L$ independent of $n$, and let
\[
\tau_{(1)} > \tau_{(2)} > \cdots > \tau_{(M)}
\]
denote the distinct values of  $\tau_{x^{1}},\dots,\tau_{x^{L}}$. Set $S_{k} = \{ j \in \{ 1,\dots, L \} : \tau_{x^{j}} = \tau_{(k)} \} $ with $s_{k} = \Card (S_{k})$ for $k=1,\dots,M$. For each $k=1,\dots,M$, let $\{ \mathbb{B}_{k}((t_{j})_{j \in S_{k}}) : (t_{j})_{j \in S_{k}} \in \R^{s_{k}} \}$ denote a centered Gaussian process with covariance function
\[
\Cov \left ( \mathbb{B}_{k}((t_{i})_{i \in S_{k}}),\mathbb{B}_{k}((t_{j}')_{j \in S_{k}}) \right) = \frac{1}{2} \sum_{i,j \in S_{k}} (x^{i})^{T} J(\tau_{(k)})^{-1} \E[XX^{T}] J(\tau_{(k)})^{-1} x^{j} \E[B(t_{i})B(t_{j}')].
\]
We note that the construction of the  $\mathbb{B}_{k}$-process depends on the design points $x^{1}.\dots,x^{L}$.
Recall that a version of a stochastic process is another process with the same finite dimensional distributions.

\begin{corollary}
\label{cor: multivariate}
Suppose that Assumption \ref{assumption} holds, and in addition that $f^{(2)}(m(x) \mid x) < 0$ and $m(x) \in (Q_{x}(\varepsilon),Q_{x}(1-\varepsilon))$ for all $x \in \{ x^{1},\dots,x^{L} \}$. Then, for each $k=1,\dots,M$, there exists a version of the $\mathbb{B}_{k}$-process with continuous paths, and denoting the continuous version by the same symbol $\mathbb{B}_{k}$, we have
\[
(nh^{2})^{1/3} ( \hat{\tau}_{x^{1}},\dots,\hat{\tau}_{x^{L}})^{T} \stackrel{d}{\to} (W_{1},\dots,W_{L})^{T}
\]
as $n \to \infty$, where $(W_{j})_{j \in S_{k}}, k=1,\dots,M$ are independent, and for each $k=1,\dots,M$,
\[
(W_{j})_{j \in S_{k}} \stackrel{d}{=} \argmax_{(t_{j})_{j \in S_{k}} \in \R^{s_{k}}} \left \{ \mathbb{B}_{k}((t_{j})_{j \in S_{k}})-\sum_{j \in S_{k}} v_{x^{j}} t_{j}^{2} \right \}.
\]
In addition, we have
\[
(nh^{2})^{1/3} \left ( \hat{m}(x^{1}) - m(x^{1}),\dots,\hat{m}(x^{L}) - m(x^{L}) \right )^{T}  \stackrel{d}{\to} \left(s_{x^{1}}(\tau_{x^{1}})W_{1},\dots,s_{x^{L}}(\tau_{x^{L}}) W_{L} \right)^{T}.
\]
\end{corollary}

In the special case when $\tau_{x^{1}},\dots,\tau_{x^{L}}$ are all distinct, we have
\[
\begin{split}
&(nh^{2})^{1/3} \left ( \hat{m}(x^{1}) - m(x^{1}),\dots,\hat{m}(x^{L}) - m(x^{L}) \right )^{T} \\
&\quad \stackrel{d}{\to} \left(s_{x^{1}}(\tau_{x^{1}})(\sigma_{x^{1}}/v_{x^{1}})^{2/3} Z_{1},\dots,s_{x^{L}}(\tau_{x^{L}})(\sigma_{x^{L}}/v_{x^{L}})^{2/3} Z_{L} \right)^{T},
\end{split}
\]
where $Z_{1},\dots,Z_{L}$ are independent Chernoff random variables.

Corollary \ref{cor: multivariate} implies that
\[
(nh^{2})^{1/3} \max_{1 \le j \le L} | \hat{m}(x^{j}) - m(x^{j}) | \stackrel{d}{\to} \max_{1 \le j \le L} | s_{x^{j}}(\tau_{x^{j}}) W_{j}|,
\]
which can be used to construct simultaneous confidence intervals for $m(x)$ over the design points $x^{1},\dots,x^{L}$; see Remark \ref{rem: simultaneous interval} ahead.

\begin{remark}[Uniform rate over expanding sets of design points]
\label{rem: uniform rate}
It is of interest to study the rate of convergence and limiting distribution of the $L^{\infty}$-distance between the proposed estimator and the true modal function on a continuum set of design points or expanding sets of design points, since e.g. such limiting distribution enables us to construct simultaneous confidence bands. To the best of our knowledge, however, much less is known about the rate of convergence and (especially) limiting distribution for the $L^{\infty}$-distance in nonstandard nonparametric estimation problems than standard nonparametric estimation problems with Gaussian limits, and we believe that the problem is challenging. One exception is the work of \cite{Durot2012}, which derives the uniform rate of convergence and the limiting distribution of  the $L^{\infty}$-distance for the Grenander \cite{Grenander1956} estimator (precisely speaking \cite{Durot2012} cover more general Grenander-\textit{type} estimators); see also the recent review article by \cite{Durot2018}. Their argument depends substantially on the specific construction of the Grenander estimator and can not be directly extended to our estimator. It is thus beyond the scope of this paper to formally study the uniform rate and the limiting distribution of the $L^{\infty}$-distance to our estimator, but we will give some heuristic discussion on this question, which we believe is of some interest to the reader.

To simplify the question, we confine ourselves to the maximum distance on expanding sets of design points $x^{1},\dots,x^{L}$ with $L=L_{n} \to \infty$. Suppose in addition that $\tau_{x^{1}},\dots,\tau_{x^{L_{n}}}$ are all distinct.
Then by Corollary \ref{cor: multivariate} it is expected that  $\max_{1 \le j \le L_{n}} (nh^{2})^{1/3}|\hat{m}(x^{j}) - m(x^{j})|/\{ s_{x^{j}}(\tau_{x^{j}})(\sigma_{x^{j}}/v_{x^{j}})^{2/3}\}$ could be approximated by $\max_{1 \le j \le L_{n}} | Z_{j} | =: |Z|_{(L_{n})}$ as long as $L_{n} \to \infty$ sufficiently slowly. In Appendix \ref{sec: Gumbel}, we will show that, for the norming constants
\[
\begin{split}
a_{L_{n}} &=  3 \left ( \frac{2}{3} \right )^{1/3} (\log L_{n})^{2/3}, \\
b_{L_{n}}' &=  \left(\frac{3}{2}\log L_{n} \right) ^{1/3} - \frac{1}{a_{L_{n}}} \left [\kappa \left ( \frac{3}{2} \log L_{n} \right)^{1/3} + \frac{1}{3} \log \log L_{n} + \frac{1}{3} \log \frac{3}{2} - \log (2\lambda)  \right ],
\end{split}
\]
where $\lambda$ and $\kappa$ are positive constants (see Appendix \ref{sec: Gumbel}), we have
\[
a_{L_{n}} (|Z|_{(L_{n})} - b_{L_{n}}') \stackrel{d}{\to} \underbrace{\Lambda}_{\text{Gumbel distribution}}.
\]
In particular, $|Z|_{(L_{n})} = b_{L_{n}}'/a_{L_{n}} + O_{\Prob}(1/a_{L_{n}}) = O_{\Prob} ((\log L_{n})^{1/3})$, and as long as $L_{n}$ grows at most polynomially fast in $n$, $|Z|_{(L_{n})} = O_{\Prob} ((\log n)^{1/3})$. This suggests that the uniform rate of the proposed estimator would be $O_{\Prob} ((nh^{2}/\log L_{n})^{-1/3})$ and the maximum distance would converge in distribution to the Gumbel distribution after normalization. The preceding argument is heuristic since Corollary \ref{cor: multivariate} only holds with fixed $L$ (and extending the corollary to the case where $L = L_{n} \to \infty$ is a substantial technical challenge), and the rigorous result is left to future research.
\end{remark}

\subsection{Inference}
\label{subsec: inference}

\subsubsection{Analytical confidence intervals}
Theorem \ref{thm: limiting distributions} allows us to construct pointwise confidence intervals for $m(x)$ by consistently estimating the nuisance parameters $\sigma_{x}^{2},v_{x}$, and $s_{x}(\tau_{x})$.

The parameter $s_{x}(\tau_{x})$ can be estimated by $\hat{s}_{x}(\hat{\tau}_{x})$.
Next, consider to  estimate $\sigma_{x}^{2}$. For the notational convenience, let $\Sigma = \E[XX^{T}]$ and so  $\sigma_{x}^{2} =x^{T} J(\tau_{x})^{-1} \Sigma J(\tau_{x})^{-1} x/2$.
The matrices  $\Sigma$ and $J(\tau)$ can be estimated by
\[
\hat{\Sigma} = \frac{1}{n}\sum_{i=1}^{n}X_{i}X_{i}^{T} \quad \text{and} \quad
\hat{J}(\tau) = \frac{1}{2nh} \sum_{i=1}^{n} I(|Y_{i} - X_{i}^{T}\hat{\beta}(\tau)| \le h) X_{i}X_{i}^{T},
\]
respectively, so that we can estimate $\sigma_{x}^{2}$ by
\[
\hat{\sigma}_{x}^{2} = \frac{1}{2}x^{T}\hat{J}(\hat{\tau}_{x})^{-1}\hat{\Sigma}\hat{J}(\hat{\tau}_{x})^{-1}x,
\]
where $\hat{J}(\tau)$ is Powell's kernel estimator \cite{Powell1986}.
Finally, consider to estimate $v_{x} = s_{x}''(\tau_{x})/2$. To this end, we estimate $s_{x}''(\tau) = Q_{x}'''(\tau)$ by a numerical differentiation of $\hat{Q}_{x}(\tau)$.
Namely, define the operator $\Delta_{h}$ by $\Delta_{h} g(\tau) = (g(\tau+h)-g(\tau-h))/(2h)$, and $\Delta_{h}^{j} g = \Delta_{h} (\Delta_{h}^{j-1} g)$ recursively for $j=2,3,\dots$.
Then we can estimate $v_{x}$ by
\begin{equation}
\hat{v}_{x} = \frac{1}{2}\Delta_{h}^{3} \hat{Q}_{x}(\hat{\tau}_{x}).
\label{eq: third derivative}
\end{equation}
The bandwidths used in $\hat{J}(\tau)$ and $\hat{v}_{x}$ can be different from that for $\hat{\tau}_{x}$.
See Remark \ref{rem: estimation of v} ahead for alternative estimators for $v_{x}$.
The following proposition shows that these estimators are indeed consistent under almost the  same conditions as in Theorem \ref{thm: limiting distributions}.

\begin{proposition}[Consistency of estimators for nuisance parameters]
\label{prop: inference}
Suppose that the conditions of Theorem \ref{thm: limiting distributions} hold and  in addition that $nh^{5}/\log n \to \infty$. Then we have
$\hat{\sigma}_{x}^{2} \stackrel{\Prob}{\to} \sigma_{x}^{2}, \ \hat{v}_{x} \stackrel{\Prob}{\to} v_{x}$, and $\hat{s}_{x}(\hat{\tau}_{x}) \stackrel{\Prob}{\to} s_{x}(\tau_{x})$
as $n \to \infty$.
\end{proposition}

Now, since Chernoff's distribution is symmetric about the origin, an asymptotic $(1-\alpha)$-confidence interval for $m(x)$ is given by
\[
\left [ \hat{m}(x) \pm \frac{\hat{s}_{x}(\hat{\tau}_{x}) (\hat{\sigma}_{x}/\hat{v}_{x})^{2/3}}{(nh^{2})^{1/3}} q_{1-\alpha/2} \right],
\]
where $q_{1-\alpha/2}$ is the $(1-\alpha/2)$-quantile of Chernoff's distribution.
For example, Table 2 in \cite{GroeneboomWellner2001} yields that $q_{0.975} \approx 0.998181$.

\begin{remark}[Alternative estimators for $v_{x}$]
\label{rem: estimation of v}
Alternatively to the estimator $\hat{v}_{x}$, we may use
\[
\tilde{v}_{x} = \frac{1}{2h^{3}} \left (  \hat{Q}_{x}(\hat{\tau}_{x}+2h) - \hat{Q}_{x}(\hat{\tau}_{x} -2h) - 2\{\hat{Q}_{x}(\hat{\tau}_{x}+h) - \hat{Q}_{x}(\hat{\tau}_{x} -h) \} \right ),
\]
which is consistent under additional smoothness conditions on the conditional density.

Still, higher order numerical differentials tend to be unstable in the finite sample.
Instead, we may use the expression $v_{x} = -f^{(2)}(m(x) \mid x)s(\tau_{x})^{4}/2$, and estimate $f^{(2)}(m(x) \mid x)$ by a kernel method.
Suppose that $X$ is decomposed as $X= (X^{C},X^{D})$ where $X^{C} \in \R^{d_{C}}$ is continuous and $X^{D} \in \R^{d-d_{C}}$ is discrete.  Let $\mathbb{K}_1: \R \to \R$ and $\mathbb{K}_2: \R^{d_{C}} \to \R$ be kernel functions (i.e., functions that integrate to $1$) where $\mathbb{K}_1$ is twice differentiable.
For  given bandwidths $b_{X} =b_{X,n} \to 0$ and $b_{Y} = b_{Y,n} \to 0$, we may estimate $f^{(2)}(m(x) \mid x)$ with $x = (x^{C},x^{D})$ by
\[
\hat{f}^{(2)} (\hat{m}(x) \mid x) = \frac{(nb_{Y}^{3}b_{X}^{d_{C}})^{-1}\sum_{i=1}^{n} \mathbb{K}_1''((\hat{m}(x)-Y_{i})/b_{Y}) \mathbb{K}_2((x^{C}-X_{i}^{C})/b_{X})I(X_{i}^{D}=x^{D})}{(nb_{X}^{d_{C}})^{-1} \sum_{i=1}^{n} \mathbb{K}_2((x^{C}-X_{i}^{C})/b_{X})I(X_{i}^{D} = x^{D})},
\]
which is consistent under appropriate conditions. This leads to an alternative estimator for $v_{x}$:
\begin{equation}
\check{v}_{x} = -\hat{f}^{(2)}(\hat{m}(x) \mid x) \hat{s}(\hat{\tau}_{x})^{4}/2.
\label{eq: kernel estimation}
\end{equation}
In the simulation study, we use the kernel-based estimator $\check{v}_{x}$ for $v_{x}$.
\end{remark}

\subsubsection{Subsampling}

It is known that the nonparametric bootstrap in general fails to be consistent for $n^{1/3}$-consistent estimators (cf. \cite{AbrevayaHuang2005, LegerMacGibbon2006, Kosorok2008, Sen2010}) and so it is unlikely that the bootstrap would be consistent for our estimator $\hat{m}(x)$. Instead, since the limiting distribution is a scale transformation of Chernoff's distribution that is absolutely continuous, the subsampling
provides a valid inference method for our estimator $\hat{m}(x)$; see \cite{PolitisRomano1994,Politis1999}.
Let $\hat{m} (x) = \hat{m}_{n}(x) = \hat{m}_{n}(x;(Y_{1},X_{1}),\dots,(Y_{n},X_{n}))$ and $h=h_n$, and let $W_{1},\dots,W_{N}$ be the $N=\binom{n}{\ell}$ subsets of $\{ (Y_{1},X_{1}),\dots,(Y_{n},X_{n}) \}$ of size $\ell \ (<n)$. Consider the subsampling distribution
\begin{equation}
U_{n,\ell}(x;t) = \frac{1}{N} \sum_{i=1}^{N} I\left \{ (\ell h_{\ell}^{2})^{1/3}(\hat{m}_{\ell}(x;W_{i}) - \hat{m}_{n}(x)) \le t \right \}.
\label{eq: subsampling distribution}
\end{equation}
Then, under the same conditions as in Theorem \ref{thm: limiting distributions}, we have
\[
\sup_{t \in \R} \left | U_{n,\ell} (x;t) - \Prob \left ( s_{x}(\tau_{x})(\sigma_{x}/v_{x})^{2/3}Z \le t \right ) \right| \stackrel{\Prob}{\to} 0,
\]
provided that $\ell=\ell_n \to \infty$ and $\ell = o(n)$. Hence, denoting by $\hat{q}_{n,\ell}(x;1-\alpha)$ the $(1-\alpha)$-quantile of $U_{n,\ell}(x;\cdot)$, i.e.,
\[
\hat{q}_{n,\ell}(x;1-\alpha) = \inf \{ t : U_{n,\ell}(x;t) \ge 1-\alpha \},
\]
an asymptotic $(1-\alpha)$-confidence interval for $m(x)$ is given by
\[
\left [ \hat{m}_{n}(x) - \frac{\hat{q}_{n,\ell}(x;1-\alpha/2)}{(nh_{n}^{2})^{1/3}}, \hat{m}_{n}(x) - \frac{\hat{q}_{n,\ell}(x;\alpha/2)}{(nh_{n}^{2})^{1/3}}\right].
\]
Some comments on the subsampling confidence interval are in order.
\begin{remark}[Comments on subsampling confidence interval]
(i) In practice, $N = \binom{n}{\ell}$ is too large and so the computation of the complete average over $i=1,\dots,N$ in (\ref{eq: subsampling distribution}) is too demanding. Instead, we can take the average of a randomly selected subset of $\{ 1,\dots, N \}$; see Corollary 2.4.1 in \cite{Politis1999}.

(ii) The bandwidth $h_{\ell}$ used in each subsample may be taken as $h_{n}$ as long as $\ell_{n}h_n^{8} \to 0$ and $\ell_{n}h_n^{5} \to \infty$.
\end{remark}

\begin{remark}[Simultaneous confidence intervals over finite design points]
\label{rem: simultaneous interval}
Consider the setting of Corollary \ref{cor: multivariate}, and let $\nu_{1-\alpha}$ denote the $(1-\alpha)$ quantile of $\max_{1 \le j \le L} |s_{x^{j}}(\tau^{j}) W_{j}|$. Then a simultaneous confidence interval for $m(x)$ over the design points $x^{1},\dots,x^{L}$ is given by
\[
\left [ \hat{m}(x^{j}) \pm \frac{\nu_{1-\alpha}}{(nh^{2})^{1/3}} \right ], \ j=1,\dots,L.
\]
In general the distribution of $(W_{1},\dots,W_{L})^{T}$ is complicated as it depends on whether there are ties in $\tau_{x^{1}},\dots,\tau_{x^{L}}$, so analytical estimation of $\nu_{1-\alpha}$ is difficult.
Instead, we can use the subsampling to estimate $\nu_{1-\alpha}$. The procedure is analogous to the pointwise case and hence omitted.
\end{remark}

\section{Numerical results}
\label{sec: numerical results}

\subsection{Bandwidth selection}
\label{subsec: bandwidth selection}
The proposed estimator requires to choose the bandwidth $h$.
We suggest here a simple method to choose the bandwidth, which is based on a modification to the bandwidth selection rule suggested in \cite{KoenkerMachado1999}.
The baseline idea of our approach is to select the bandwidth in such a way that the sparsity function $s_{x}(\tau)$ is well estimated. A similar approach is used in \cite{EinbecTutz2006} who adapt the smoothing bandwidth to kernel estimation of multi-modal regression by optimizing the conditional density estimation rate. The performance of the sparsity function estimate $\hat{s}_{x}(\tau)$ depends on the quantile $\tau$ of interest, and so the constant involved in the bandwidth should adapt to $\tau$. Since we are interested in $s_{x}(\tau)$ around $\tau = \tau_{x}$, we aim at choosing $h$ in such a way that $\hat{s}_{x}(\tau)$ around $\tau = \tau_{x}$ tends to be accurate but modify the rate of $h$ so that it satisfies Condition (v) in Assumption \ref{assumption}.

For estimation of $s_{x}(\tau)$ based on quantile regression, \cite{KoenkerMachado1999} suggest to use the $\tau$-dependent bandwidth
\[
h^{KM}(\tau)= n^{-1/3} z_\alpha^{2/3} \left\{1.5 \frac{\phi(\Phi^{-1}(\tau))}{2\Phi^{-1}(\tau)^2 +1} \right\}^{1/3},
\]
where $\phi$ and $\Phi$ are the density and distribution functions of $N(0,1)$, and $z_{\alpha} = \Phi^{-1}(1-\alpha/2)$.
We set $\alpha = 0.05$. The bandwidth $h^{KM}(\tau)$ does not satisfy Condition (v) in Assumption \ref{assumption} and is $\tau$-dependent, and so we shall modify $h^{KM}(\tau)$ as follows: (i)  pick any design point $x$ in the support of $X$; (ii) use the pilot bandwidth $h^{\mathrm{pilot}} = n^{1/6} h^{KM}(0.5) \propto n^{-1/6}$ to construct a preliminary estimator $\hat{\tau}_{x}^{\mathrm{prelim}}$ of $\tau_{x}$; (iii) and  use $h_n = h_{n,x} = n^{1/6} h^{KM} (\hat{\tau}_x^{\mathrm{prelim}})$ to construct a final estimator $\hat{m}(x)$. The simulation results suggest that, although it would not be optimal, this bandwidth selection rule works reasonably well.

\subsection{Simulation results}
\subsubsection{Comparison of RMSEs}
We compare the performance of our estimator with that of the linear modal regression estimator of \cite{KempSantos2012,YaoLi2014} via the root mean square error (RMSE) $\sqrt{\E_{X^{*}}[\{ \hat{m}(X^{*}) - m(X^{*}) \}^{2}]}$ where $X^{*} \stackrel{d}{=} X$ is independent of the data and $\E_{X^{*}}$ is the expectation with respect to $X^{*}$.
We consider two settings: the first one is the case where the modal function is linear while the second one is the case where the modal function is nonlinear.

Case (i). Consider a linear location-scale model
\[
Y = 1 + X_2 -3X_3 + X_4 + X_2 \nu,
\]
where $X=(1,X_2,X_3,X_4)^{T}, \ X_2, X_3 \sim U(0,1), \ X_4 \sim N(0,1)$, and $\nu \sim Ga(3,0.5)$ (the Gamma distribution with shape parameter $3$ and scale parameter $0.5$).
In this case, both the conditional quantile and modal functions are linear in $X$. In fact, $Q_{\tau}(X) = 1 + (1+F^{-1}(\tau))X_2 - 3X_3 + X_4$, where $F$ denotes the distribution function of $\nu$.
In addition, since the mode of $Ga(3,0.5)$ is $1$, the modal function is $m(X) = 1+2X_2-3X_3+X_4$.

Case (ii). Consider the following data generating process
\[
Y = U^{3}/3- X_2 (U-1)^{2},
\]
where $X = (1,X_2)^{T}$, $X_2 \sim U(0,1)$, and $U \sim U(0,1)$ independent of $X$. In this case, the conditional quantile function is linear, $Q_{\tau}(X) = \tau^{3}/3 - X_2 (\tau-1)^{2}$, but the modal function is nonlinear, $m(X) = -2X_2^{3}/3+2X_2^{2}-X_2$; see Remark \ref{rem: linearity}.

In this simulation study, we choose $\varepsilon=0.1$ and compute $\hat{Q}_{x}(\tau)$ for 100 equally spaced grids on $[\tau_{\min},\tau_{\max}] = [0.05,0.95]$. To implement the linear modal regression estimator, we follow the EM algorithm and the bandwidth selection rule suggested in \cite{YaoLi2014}.
The number of Monte Carlo repetitions is $1000$ for each case.

Figures \ref{fig: RMSE Case1} and \ref{fig: RMSE Case2} present the box plots of RMSEs of the linear modal regression and proposed estimators for Cases (i) and (ii), respectively, with $n=500, 1000$, and $2000$.
These figures lead to the following observations. First, in both cases, the RMSE of the proposed estimator overall decreases as the sample size increases. Second, the proposed estimator tends to be more variable than the linear modal regression estimator, so that the interquartile range of the RMSE is wider for the proposed estimator than the linear modal regression estimator.
Third, in both cases, the proposed estimator outperforms the linear modal regression estimator. The superior performance of the proposed estimator in Case (ii) is not surprising since the true modal function is nonlinear in that case and so the linear modal regression estimator is not consistent.
Interestingly,  even when the true modal function is linear (Case (i)), the proposed estimator performs substantially better than the linear modal regression estimator. This may be partly because the EM algorithm used to compute linear modal regression estimates failed to find global optimal solutions.
Overall, the figures confirm that the proposed estimator works  well in practice.

\begin{figure}[t!]
   \centering
       \includegraphics[scale=0.33]{n500_lin_modified_zero.png}
       \includegraphics[scale=0.33]{n1000_lin_modified_zero.png}
       \includegraphics[scale=0.33]{n2000_lin_modified_zero.png}
    \caption{Box plots of RMSEs of the linear modal regression and proposed estimators for Case (i) with $n=500$ (left), $n=1000$ (middle), and $n=2000$ (right). }
    \label{fig: RMSE Case1}
\end{figure}

\begin{figure}[t!]
   \centering
       \includegraphics[scale=0.75]{n500_nonlin_v3.png}
       \includegraphics[scale=0.75]{n1000_nonlin_v3.png}
       \includegraphics[scale=0.75]{n2000_nonlin_v3.png}
    \caption{Box plots of RMSEs of the linear modal regression and proposed estimators for Case (ii) with $n=500$ (left), $n=1000$ (middle), and $n=2000$ (right). }
    \label{fig: RMSE Case2}
\end{figure}

\subsubsection{Coverage probabilities of confidence intervals}

Next, we assess the performance of analytical and subsampling confidence intervals considered in Section \ref{subsec: inference}.
We follow the data generating process of Case (ii) and evaluate Monte Carlo average and median lengths,  and coverage probabilities of confidence intervals at three design points $x_2 = 0.25, 0.5$, and $0.75$. We consider two nominal coverage probabilities of 99\% and 95\%. To implement the analytical confidence interval, we use the kernel-based estimator $\check{v}_{x}$  given in (\ref{eq: kernel estimation}) for $v_{x}$. To construct $\check{v}_{x}$, we use the Gaussian kernel for $\mathbb{K}_1$ and the Epanechnikov kernel for $\mathbb{K}_2$ together with bandwidths $b_{Y} = n^{-1/9}\hat{\sigma}_{Y}$ and $b_{X} = n^{-1/5} \hat{\sigma}_{X}$ where $\hat{\sigma}_{Y}$ and $\hat{\sigma}_{X}$ are the sample standard deviations of  $Y$ and $X$, respectively.
To implement the subsampling confidence interval, we examine two subsample sizes: $\ell = 0.1 n$ and $0.2 n$.
In this simulation study, instead of taking the average of whole subsamples in (\ref{eq: subsampling distribution}), we take the average of 250 randomly chosen subsamples.
When applying the bandwidth selection rule to the subsample, we use the pilot bandwidth computed using the full sample.

Tables \ref{table: analytic 99}--\ref{table: subsampling 95} present the simulation results on the confidence intervals. The tables show that both confidence intervals work reasonable well, given that the convergence rate of the estimator is relatively slow.
It is worth noting that the estimators for the nuisance parameters $s_{x}(\tau_{x})$ and $v_{x}$ tend to be unstable, which results in the discrepancy between the average and median lengths of the analytical confidence interval. The subsample confidence interval is able to avoid estimation of those nuisance parameters, and so the length of the subsampling confidence interval tends to be shorter than that of the analytical confidence interval. In terms of the coverage probability, the subsampling confidence interval with subsample size $0.2n$ works the best.

{\small

\begin{table}[h]

	\begin{tabular}{|c|c|c|c|c|} \hline
		Design point & Sample size & Ave. length &Med. length& Cov. probability \\ \hline \hline
		$x_2=0.25$ & $n=500$ & 0.494 & 0.419&0.981
		\\
		& $n=1000$&0.359 &0.315 &	0.986
		\\
		& $n=2000$ &0.247  &0.220 &0.985 \\
		\hline

		$x_2=0.50$ & $n=500$ &0.715 &0.599 &1.000\\
		& $n=1000$&0.506 &0.475  &0.997	\\
		& $n=2000$ &0.392  &0.380 & 0.992\\
		\hline

		$x_2=0.75$ & $n=500$ &1.045 &0.878 &0.978\\
		& $n=1000$&0.724 & 0.653&0.977	\\
		& $n=2000$ &0.524  &0.488 &0.956 \\
		\hline
	\end{tabular}
	\medskip
	\caption{Monte Carlo average and median lengths, and coverage probabilities of the $99\%$ analytical confidence interval. }
	\label{table: analytic 99}
\end{table}

\begin{table}[h]

	\begin{tabular}{|c|c|c|c|c|} \hline
	Design point & Sample size & Ave. length &Med. length& Cov. probability \\ \hline \hline
	$x_2=0.25$ & $n=500$ &0.309  &0.242 &0.948\\
	& $n=1000$&0.207 &0.175 &0.941	\\
	& $n=2000$ & 0.139 &0.128 &0.952 \\
	\hline

	$x_2=0.50$ & $n=500$ &0.459 &0.343 &0.987\\
	& $n=1000$& 0.302 & 0.269&0.933	\\
	& $n=2000$ &0.226  &0.221&0.894 \\
	\hline

	$x_2=0.75$ & $n=500$ &0.660  &0.534 &0.873\\
	& $n=1000$&0.429 &0.371 &0.869	\\
	& $n=2000$ &0.302  &0.278 &0.845 \\
	\hline
\end{tabular}
	\medskip
	\caption{Monte Carlo average and median lengths, and coverage probabilities  of the $95\%$ analytical confidence interval.}
\label{table: analytic 95}

\end{table}

\begin{table}[h]

	\begin{tabular}{|c|c|c|c|c|c|} \hline
		Design point & Sample size & Subsample size & Ave. length  &Med. length& Cov. probability \\ \hline \hline
		$x_2=0.25$ & $n=500$ & $0.1n$&0.232 &0.234 &0.959\\
		& & $0.2n$&0.250 &0.262 &0.991\\\hline

		& $n=1000$  &$0.1n$ &0.208 &0.214 &0.966 \\
		&   &$0.2n$ &0.191&0.184&0.997\\ \hline

		& $n=2000$ &$0.1n$ &0.148&0.146&1.000\\
		&  &$0.2n$ &0.146&0.143&1.000\\\hline

		$x_2=0.50$ & $n=500$ & $0.1n$&0.336 &0.337&0.946
		\\
		& & $0.2n$&0.405&0.407 &0.999\\\hline

		& $n=1000$  &$0.1n$ &0.326&0.327 &0.973\\
		&   &$0.2n$ &0.391& 0.395&0.998\\ \hline

		& $n=2000$ &$0.1n$ &0.371&0.382&1.000\\
		&  &$0.2n$ &0.371&0.382&0.999\\\hline

		$x_2=0.75$ & $n=500$ & $0.1n$&0.447 &0.450&0.822
		\\
		& & $0.2n$&0.529 &0.538 &0.917 \\\hline

		& $n=1000$  &$0.1n$ &0.430&0.433 &0.847\\
		&   &$0.2n$ &0.488&0.508&0.961\\ \hline

		& $n=2000$ &$0.1n$ &0.416&0.415&0.971\\
		&  &$0.2n$ &0.423&0.416	&0.971\\\hline

	\end{tabular}
	\medskip

	\caption{Monte Carlo average and median lengths, and coverage probabilities of the $99\%$ subsampling confidence interval.}

\label{table: subsampling 99}
\end{table}

\begin{table}[h]

	\begin{tabular}{|c|c|c|c|c|c|} \hline
		Design point & Sample size & Subsample size & Ave. length  &Med. length& Cov. probability \\ \hline \hline
		$x_2=0.25$ & $n=500$ & $0.1n$&0.203 & 0.208&0.926\\
		& & $0.2n$&0.198 & 0.195&0.982\\\hline

		& $n=1000$  &$0.1n$ &0.166 & 0.166&0.947 \\
		&   &$0.2n$ &0.148&0.145&0.993\\ \hline

		& $n=2000$ &$0.1n$ &0.120&0.119&0.997\\
		&  &$0.2n$ &0.118&0.116&0.998\\\hline

		$x_2=0.50$ & $n=500$ & $0.1n$&0.313 &0.314&0.899
		\\
		& & $0.2n$&0.374&0.380 &0.989\\\hline

		& $n=1000$  &$0.1n$ &0.304&0.306 &0.968\\
		&   &$0.2n$ &0.353&0.366 &0.997\\ \hline

		& $n=2000$ &$0.1n$ &0.316&0.326&0.994\\
		&  &$0.2n$ &0.318&0.328&0.996\\\hline

		$x_2=0.75$ & $n=500$ & $0.1n$&0.413 &0.416&0.779
		\\
		& & $0.2n$&0.473 &0.490 &0.887 \\\hline

		& $n=1000$  &$0.1n$ &0.388&0.396 &0.808\\
		&   &$0.2n$ &0.412&0.415&0.937\\ \hline

		& $n=2000$ &$0.1n$ &0.335&0.328&0.958\\
		&  &$0.2n$ &0.342&	0.336&0.959\\\hline

	\end{tabular}
	\medskip

	\caption{Monte Carlo average and median lengths, and coverage probabilities of the $95\%$ subsampling confidence interval.}

\label{table: subsampling 95}
\end{table}
}

\subsection{Combined Cycle Power Plant Data}

The electric energy output provided by a power plant fluctuates through the year because of several environmental conditions, and prediction of the electricity output given such environmental conditions is of interest.
We apply the proposed estimator to predicting the net hourly electrical energy output using Combined Cycle Power Plant Data \cite{Kaya2012,Tufekci2014}.
The data set is taken from \texttt{https://archive.ics.uci.edu/ml/datasets/
Combined+Cycle+Power+Plant} and consists  of 9568 data points collected from a Combined Cycle Power Plant over 6 years (2006-2011). It contains hourly average ambient variables Temperature, Ambient Pressure, Relative Humidity, Exhaust Vacuum, and the net hourly electrical energy output, where the first four variables are regressors and the last variable is a response.
For this data, the conditional distribution tends to be skewed, and therefore it would be natural to estimate the conditional mode.
Figure \ref{fig:power plant} shows the estimate of the conditional density given one of the regressors (Exhaust Vacuum).  It is seen that the conditional density estimate is highly skewed and the pattern of the skewness depends on the value of the regressor.

\begin{figure}[t!]
   \centering
       \includegraphics[scale=1]{cd4.png}
\label{fig:power plant}
\caption{The conditional density estimate of the electronic energy output given Exhaust Vacuum.}
\end{figure}

To construct prediction intervals, we combine the proposed estimator with the  split conformal prediction of \cite{Lei2018}. Specifically:
\begin{enumerate}
\item[1.] Randomly split the index set $\{1,\dots,9568 \}$ into three parts $\mathcal{I}_{1},\mathcal{I}_{2}$, and $\mathcal{I}_{3}$.
\item[2.] Use the data $\{ (Y_{i},X_{i}) : i \in \mathcal{I}_{1} \}$ to construct the estimator $\hat{m}(\cdot)$ for the modal function $m(\cdot)$.
\item[3.] Compute the $\alpha/2$- and $(1-\alpha/2)$-quantiles of $\{ Y_{i} - \hat{m}(X_{i}) : i \in \mathcal{I}_{2} \}$ and they are denoted by $\hat{\xi}_{\alpha/2}$ and $\hat{\xi}_{1-\alpha/2}$, respectively. In this experiment, $\alpha=0.05$ is used.
\item[4.] Construct $C_{\text{split}}(x) = [ \hat{m}(x) + \hat{\xi}_{\alpha/2}, \hat{m}(x) + \hat{\xi}_{1-\alpha/2}]$.
\item[5.] Compute the empirical coverage probability:
\[
\frac{1}{|\mathcal{I}_{3}|} \sum_{i \in \mathcal{I}_{3}} I\{ Y_{i} \in C_{\text{split}}(X_{i}) \}.
\]
\end{enumerate}
In this experiment, we take $\mathcal{I}_{1},\mathcal{I}_{2}$, and $\mathcal{I}_{3}$ in such a way that $| \mathcal{I}_{1} \cup \mathcal{I}_{2} | : | \mathcal{I}_{3} | \approx 0.95 : 0.05$ and $| \mathcal{I}_{1} | : | \mathcal{I}_{2} | \approx 8:2$.
We repeated this procedure 250 times and report the average of the empirical coverage probabilities together with the average and median lengths. In addition, we compare the proposed estimator with the linear modal regression estimator.
Table \ref{table:conformal} shows the results.
For both methods, the empirical coverage probabilities are surprisingly close to the nominal coverage probability of $95 \%$, which is consistent with the theory developed in \cite{Lei2018}. On the other hand, the average and median lengths of the conformal prediction band with the proposed estimator are substantially smaller than those with the linear modal regression estimator, which is an encouraging sign for the proposed estimator.

{\small
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|} \hline
Method &  Average length &Median length& Coverage probability \\ \hline \hline
Proposed method &19.01 &19.02 & 0.950 \\
Modal linear regression &23.71 &23.32 &0.950 \\
\hline
\end{tabular}
\medskip
\caption{Monte Carlo average and median lengths, and empirical coverage probabilities of the $95\%$ conformal prediction intervals.}
\label{table:conformal}
\end{table}
}

\section{Discussion}
\label{sec: conclusion}

In the present paper we have proposed a new estimator for the conditional mode based on quantile regression.
The proposed estimate is computationally scalable since the quantile regression problem can be formulated as a linear programming problem. We have developed asymptotic distributional theory for the proposed estimator, which turns out to be nonstandard. Specifically, we have shown that the rate of convergence of the proposed estimator is $(nh^{2})^{1/3}$ where $h = h_n \to 0$ is a sequence of bandwidths, and that the limiting distribution is a scale transformation of Chernoff's distribution. For inference, we have discussed analytical and subsampling confidence intervals.
Finally we have verified the practical usefulness of the proposed method through numerical experiments.

In the present paper, we use the naive quantile regression estimator $\hat{\beta}(\tau)$ that is not smooth in $\tau$ to estimate the conditional quantile function, while the true slope vector $\beta (\tau)$ is smooth in $\tau$ under our assumption. An interesting alternative approach is to impose smoothness to $\hat{\beta}(\tau)$ so that the estimated conditional quantile function is differentiable in $\tau$. We expect that the resulting conditional mode estimator would have a Gaussian limit (under regularity conditions), which is a reminiscent of the smoothed maximum score estimator of \cite{Horowitz1992}. Developing this alternative approach requires a whole new theory and is left as future research.

\section*{Acknowledgments}

The authors would like thank the Editor Domenico Marinucci, an AE, and an anonymous referee for their careful review and  constructive comments that helped improve on the quality of the paper.

\clearpage

redpajama_set_name: RedPajamaArXiv


"When I offer distance energy healing sessions and Oneness Blessings, I sense them as either condensed forms of energy, perceived as a ball or bubble of energy, or a steadily flowing stream of energy. Both of which are divinely guided. As I channel in and open up to the energy of the divine universal consciousness, I form a "bubble" or stream and send it to the client.
To receive a session, schedule here.
At the time of your appointment, relax and receive the energy. The energy in distance sessions comes in very concentrated, so for me as it flows to the client, it may take longer than the scheduled time to fully integrare, so be sure relax until the energy subsides. Often the receiver will continue integrating the energy for the next day or so.
In the distance sessions the scheduled time is spent tuning into the client's energy body, channeling healing energies, and making notes. For the scheduled "first-time client" sessions, additional time is provided for the assessment period. In distance sessions I am involved in the healing work by moving and directing energies, clearing low vibrations, balancing chakras, etc.. and using other techniques that I would use in an in-person session. As I tune in with the client, I am able to sense through clairvoyance (sight) and Clairsentience (feeling) what is happening, and often receive information for the client. At the close of this session, I send session notes via text or email, which lets the client know all that I experienced and what I perceived to be happening. This can be very validating, as the client often will recognize aspects within the notes that match their experience. The energy comes in concentrated from time to time and may continue adjusting and working with the client for hours after I have finished working.
The format of a remote session includes a brief connection with the client to establish that we are both ready to begin, and a lengthier conversation for First-Time clients. We then hang up the phone, unless guided or requested otherwise, and the client relaxes for the next 45-60 min. Some clients opt to not connect by phone and relax at the scheduled time.
During the session, I tune into the client and work with the assistance of healing energies. When the energy has completed it's channel through me, I make notes and text the client to let them know of the session's closure on my end. Often the energies will continue working through the remainder of the day/evening and sometimes into the next day and week as the client assimilates the session's adjustments.
I offer Distance sessions for Oneness Blessings, Reconnective Healings, Reiki, and Earth-Based energy healing. If you prefer one frequency over the other, please let me know by email. To book a session, please visit the scheduling link here.
to request the assistance of your Spiritual Healing Team, Archangels, and Spiritual Figures. This will set up the space for your session to begin.
If at any time a sensation is too strong, just breathe through it. Stay with the sensation as it clears and releases the low vibe blocks and energy from your field.
From time to time, I am asked to explain the experience and science behind distance / remote healing sessions. While I'm not a scientist, I will do my best to provide resources. I can only speak from my personal experiences and the feedback of others who have received distance bubbles, sessions or blessings.
When I first heard of and learned this concept, I too questioned, "how does this work". However, after I experienced a remote Reiki offering, I didn't feel I needed to know "how" in order to receive the benefits. I had received my own personal experience in recognizing and sensing the energy as it greeted my field and worked with me. The strongest sensation for me was a very distinct, high-pitched humming tone that hovered over my ears. I then felt my ears receiving warmth from the energy. After the session, the Reiki practitioner explained she started working with my head and ears. This experience happened back in 2001 and is still very significant for me. This simple phenomenon and the undeniable way in which the energy offered its presence to me was enough to get me started in further working with distance sessions.
There is a scientific, quantum physics explanation of this process that I will leave up to the experts in the field of science. I attended a lecture presented in 2013 by Dr. William Tiller, of the William Tiller Foundation. It all made sense as he was presenting the information, but I would have a hard time explaining it myself. He does a beautiful job in explaining healing through time and space with quantum physics in this article. Dr. Tiller has published many papers and books and appeared in the movie What the Bleep do we Know.


The Algona Police Department is an agency comprised of eighteen full and part-time employees dedicated to the simple principle of making Algona a better place to live and work. This department not only provides 24 hour patrol coverage and dispatch services but also offers many other services. Those programs that we are honored to offer the public include DARE, canine, house checks, ALICE, police reserves, and the providing of guest speakers. We thank you for visiting our website but also encourage everyone to Like our Facebook page.
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