# Generalized extremiles and risk measures of distorted random variables

Dieter Debrauwer111Department of Mathematics, KU Leuven, Leuven, Belgium ,
Irène Gijbels222Corresponding Author ††footnotemark:  , Klaus
Herrmann333Département de mathématiques, Université de Sherbrooke, Sherbrooke,
Canada

###### Abstract

Quantiles, expectiles and extremiles can be seen as concepts defined via an
optimization problem, where this optimization problem is driven by two
important ingredients: the loss function as well as a distributional weight
function. This leads to the formulation of a general class of functionals that
contains next to the above concepts many interesting quantities, including
also a subclass of distortion risks. The focus of the paper is on developing
estimators for such functionals and to establish asymptotic consistency and
asymptotic normality of these estimators. The advantage of the general
framework is that it allows application to a very broad range of concepts,
providing as such estimation tools and tools for statistical inference (for
example for construction of confidence intervals) for all involved concepts.
After developing the theory for the general functional we apply it to various
settings, illustrating the broad applicability. In a real data example the
developed tools are used in an analysis of natural disasters.

Keywords: Distortion risk measure, expected loss minimization, extremile,
nonparametric estimation, risk measure, statistical functional.

##  1 Introduction

There is a vast literature on quantiles and expectiles, and their applications
in a variety of settings, in particular as risk measures. See Artzner et al.
(1999), Taylor (2008), Bellini and Di Bernardina (2017), among others. For a
random variable $X$, its $\delta$th quantile (with $\delta\in(0,1)$) can be
defined as

$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\mathbb{E}[\ell_{\delta}(X,c)]\qquad\mbox{with}\quad\ell_{\delta}(x,c)=\left|\delta-1_{\\{x\leq
c\\}}\right||x-c|,$ (1)

see for example Ferguson (1967) and Koenker and Bassett (1978). Herein $1_{A}$
denotes the indicator function on a set $A$, i.e., $1_{A}=1$, if $A$ holds,
and zero otherwise. Expectiles, introduced and first studied by Aigner et al.
(1967) and Newey and Powell (1987), are defined as follows: the $\delta$th
expectile (with $\delta\in(0,1)$) of $X$ is given by

$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\mathbb{E}[\ell_{\delta}(X,c)]\qquad\mbox{with}\quad\ell_{\delta}(x,c)=\left|\delta-1_{\\{x\leq
c\\}}\right||x-c|^{2}.$ (2)

Quantiles can also be viewed via an alternative optimization problem, namely
the $\tau$th quantile (with $\tau\in(0,1)$) of $X$ (with cumulative
distribution function $F_{X}$) is obtained via

$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\mathbb{E}[J_{\tau}(F_{X}(X))\ell_{\delta}(X,c)]\qquad\mbox{with}\quad\ell_{\delta}(x,c)=|x-c|,$
(3)

and

$J_{\tau}(u)=\begin{cases}s(\tau)(1-u)^{s(\tau)-1}&\text{ if
}0<\tau<\frac{1}{2}\\\ r(\tau)u^{r(\tau)-1}&\text{ if
}\frac{1}{2}\leq\tau<1,\end{cases}$ (4)

with $r(\tau)=s(1-\tau)=\ln(1/2)/\ln(\tau)$. This alternative formulation of
quantiles can be found in Newey and Powell (1987). Consequently (1) and (3)
(taking in the latter $\tau=\delta$) provides two approaches to obtain the
$\delta$th quantile of a random variable $X$. Recently Daouia et al. (2019)
studied the class of extremiles, where the $\tau$th extremile of $X$ is
obtained via

$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\mathbb{E}[J_{\tau}(F_{X}(X))\ell_{\delta}(X,c)]\qquad\mbox{with}\quad\ell_{\delta}(x,c)=(x-c)^{2}\quad\mbox{and}\quad
J_{\tau}\,\mbox{as in \eqref{densitytauExtremiles}}.$ (5)

Of interest is further to note that the function $J_{\tau}$ is in fact the
derivative of a function $K_{\tau}$, which is a specific cumulative
distribution function defined on the interval $[0,1]$ (see expression (9) and
Figure 1).

The above observations inspire to look at a more general form of an
optimization problem, and to study functionals defined via

$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\mathbb{E}[d_{\tau}(F_{X}(X))\,\ell_{\delta}(X,c)],$

where $d_{\tau}$ is the density associated to a cumulative distribution
function $D_{\tau}$, such that $D_{\tau}(0)=0$ and $D_{\tau}(1)=1$. As shown
in Section 2, this minimization problem can also be looked upon as
$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\mathbb{E}[\ell_{\delta}(X_{{\scriptsize
D}_{\tau}},c)]$, where $X_{{\scriptsize D}_{\tau}}$ denotes the random
variable which has as cumulative distribution function the transformed
function $D_{\tau}\circ F_{X}$, and the expectation is taken with respect to
this distributional distorted (shortly distorted) random variable
$X_{{\scriptsize D}_{\tau}}$. Moreover, with $\ell_{\delta}(x,c)=(x-c)^{2}$,
i.e. square loss, this leads to
$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\mathbb{E}[\ell_{\delta}(X_{{\scriptsize
D}_{\tau}},c)]=\mathbb{E}\left[X_{{\scriptsize D}_{\tau}}\right]$ and
subsequently $\min_{c\in\mathbb{R}}\mathbb{E}[\ell_{\delta}(X_{{\scriptsize
D}_{\tau}},c)]=\mbox{Var}(X_{{\scriptsize D}_{\tau}})$.

In the special case of square loss, the functional thus relates to the mean of
the transformed random variable $X_{{\scriptsize D}_{\tau}}$. This is linked
to the probability distortions studied by Liu et al. (2021), as a particular
setting of distributional transforms. The paper investigates probabilistic
properties of these distributional transforms, and discusses (new) risk
measures generated from such distributional transforms. In Section 5 we rely
on some results of Liu et al. (2021) to study risk measures of the distorted
random variable $X_{{\scriptsize D}_{\tau}}$. Distortion risk measures have
found widespread application in the literature, for example, in finance,
economics, insurance; see Wang (1996), Wang et al. (1997), Föllmer and Schied
(2002), and Cherny and Madan (2017), among others. Our general functional
involves a loss function $\ell_{\delta}(x,c)$. There are numerous examples of
loss functions appearing in the literature. Tables 1 and 2, as well as Section
2.4 review some of these.

As our main contribution we introduce and study a general class of
functionals, which cover and generalize these in (1), (2), (3), and (5), as
special cases. This generalization allows an interesting link to distortion
risks as well as to risks of non-distortion type. We call this general class
of functionals _generalized extremiles_. The main contribution of this paper
consists of developing estimators for generalized extremiles, and establish
consistency and asymptotic normality of these estimators, with explicit
expression for the asymptotic variance. This general theory can then be
applied to a variety of population target quantities. The asymptotic normality
result allows to construct asymptotic confidence intervals of estimators in
the general framework.

Specific optimization problems capture a number of situations studied in
actuarial science, financial mathematics or statistics which are a special
case of our framework. For example, our framework applies if a functional
$\rho$ can be defined as the expected loss minimizer

$\displaystyle\rho(Y)=\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\,\mathbb{E}[\ell(Y,c)]$
(6)

for a suitable loss function
$\ell\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (where we surpressed the
index $\delta$), and a random variable $Y$. In this case $\rho(X_{{\scriptsize
D}_{\tau}})$ can be the mean, quantile or expectile of the distorted random
variable $X_{{\scriptsize D}_{\tau}}$. Equally, one could for example consider
functionals defined as the expected loss minimum

$\displaystyle\rho(Y)=\min_{c\in\mathbb{R}}\mathbb{E}[\ell(Y,c)],$ (7)

in which case $\rho(X_{{\scriptsize D}_{\tau}})$ can for example be the
variance or expected shortfall of the distorted random variable
$X_{{\scriptsize D}_{\tau}}$. See Embrechts et al. (2021) for details and a
discussion connecting risk measures of the form in (6) and (7).

While we present some general theory for risk functionals applied to distorted
random variables $X_{{\scriptsize D}_{\tau}}$ in Section 5, our study mainly
focusses on the specific case of expected loss minimizers as in (6), due to
their ubiquitous nature in the literature.

The paper is organized as follows. In Section 2 we introduce the generalized
extremile functionals, establish some basic properties, and explain the link
with distortion risks. Section 3 deals with estimation of the broad class of
generalized extremiles, and proves consistency and asymptotic normality
results. These are then applied to various settings, revealing the breath of
the studied functionals. Numerical studies and real data analysis are given in
Sections 4 and 6. The proofs of the main theoretical results are provided in
the Appendices. Some further discussions are given in a final section. Some
additional information and explanations are in the Supplementary Material.

##  2 Generalized extremiles and their properties

In this section we introduce the concept of generalized extremile, study its
basic probabilistic properties, highlight some issues toward solving the
associated optimization problem, and review and discuss some loss functions.
In a final subsection we discuss how generalized extremiles link up to
distortion risk measures. In Section 3 we then turn to statistical estimation
of generalized extremiles.

###  2.1 Generalized extremiles and distorted random variables

The class of functionals that we study is based on a pair of key ingredients,
denoted by $(D_{\boldsymbol{\tau}},\ell_{\delta})$, consisting of a
distribution function $D_{\boldsymbol{\tau}}$ and a loss function
$\ell_{\delta}$. Their formal definitions are provided below.

######  Definition 2.1.

Consider a set $\Delta\subset\mathbb{R}$. For every $\delta\in\Delta$ we
denote by $\ell_{\delta}$ the loss function

$\ell_{\delta}(x,c):\mathbb{R}\times\mathbb{R}\to\mathbb{R}:(x,c)\mapsto\ell_{\delta}(x,c).$

Where appropriate we use
$\ell_{\delta}^{\prime}(x,c):=\frac{\mathrm{d}}{\mathrm{d}c}\ell_{\delta}(x,c)$
to denote the partial derivative of $\ell_{\delta}$ with respect to $c$.

######  Definition 2.2.

Let $\mathrm{T}$ be some subset of $\mathbb{R}^{k}$ where $k$ is some positive
integer. For each $\boldsymbol{\tau}\in\mathrm{T}$ denote by
$D_{\boldsymbol{\tau}}$ an absolutely continuous cumulative distribution
function supported on $[0,1]$, with density $d_{\boldsymbol{\tau}}$. We denote
by $\mathbb{D}$ for the set of all such $D_{\boldsymbol{\tau}}$.

######  Definition 2.3 (Generalized extremile).

Consider a cumulative distribution function
$D_{\boldsymbol{\tau}}\in\mathbb{D}$, and a loss function $\ell_{\delta}$. For
a random variable $X\sim F_{X}$ we define the generalized extremile of index
$(\boldsymbol{\tau},\delta)$, denoted
$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})$, as

$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})\in\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\mathbb{E}[d_{\boldsymbol{\tau}}(F_{X}(X))(\ell_{\delta}(X,c)-\ell_{\delta}(X,0))].$
(8)

Taking $D_{\tau}$, with $\boldsymbol{\tau}=\tau\in(0,1)$, equal to

$K_{\tau}(u)=\begin{cases}1-(1-u)^{s(\tau)}&\text{ if }0<\tau<\frac{1}{2}\\\
u^{r(\tau)}&\text{ if }\frac{1}{2}\leq\tau<1,\end{cases}$ (9)

with $r(\tau)=s(1-\tau)=\ln(1/2)/\ln(\tau)$, in minimization problem (8),
leads on the one hand to the concept of $\tau$th quantiles when taking as loss
function $\ell_{\delta}(x,c)=|x-c|$ (see (3)), and on the other hand to the
concept of $\tau$th extremile with squared loss $\ell_{\delta}(x,c)=(x-c)^{2}$
(see (5)). We refer to the class of functionals in Definition 2.3 as
generalized extremiles, since it steps away from the specific choice of the
pair $(D_{\boldsymbol{\tau}},\ell_{\delta})$ in quantiles and extremiles, but
having its inspiration strongly influenced by these two concepts.

Note that the choice of the distribution function $D_{\tau}$ in (9) already
reveals why we allow this distribution function to depend on a parameter
(vector) $\boldsymbol{\tau}$. From the viewpoint on quantiles expressed in (1)
it is also clear why we allow the loss function to depend on a parameter
$\delta$. Figure 1 depicts the function $K_{\tau}$ and its derivative
$d_{\tau}$ for various values of $\tau$.

Figure 1: The functions $K_{\tau}$ in (9) and its derivative $d_{\tau}$ for various values of $\tau$. Table 1: Commonly used nonnegative loss functions and their basic properties. | loss function | sign symmetric | shift invariant | positive homogeneous
---|---|---|---|---
|  |  |  | of degree $k$
| $\ell_{\delta}(x,c)$ | $\ell_{\delta}(-x,c)=\ell_{\delta}(x,-c)$ | $\ell_{\delta}(x+b,c+b)$ | $\ell_{\delta}(ax,ac)=a^{k}\ell_{\delta}(x,c)$
|  |  | $\quad=\ell_{\delta}(x,c)$ |
1 | $|x-c|$ |  |  |
| absolute value loss | yes | yes | yes, of degree 1
2 |  $|x-c|^{p}$ with $p>0$ |  |  |
| power $p$ absolute loss | yes | yes | yes, of degree $p$
3 | $\left|\delta-1_{\\{x\leq c\\}}\right||x-c|$ |  |  |
| quantile loss | no | yes | yes, of degree 1
4 | $\left|\delta-1_{\\{x\leq c\\}}\right||x-c|^{2}$ |  |  |
| expectile loss | no | yes | yes, of degree 2
5 | $\ell_{\delta}(x,c)$ | yes | yes | no
| $=\left\\{\begin{array}[]{lll}0.5(x-c)^{2}&\mbox{if}&|x-c|\leq\delta\\\ \delta(|x-c|-0.5\delta)&\mbox{if}&|x-c|>\delta\end{array}\right.$ |  |  |
| Huber loss function |  |  |
6 | $\ell_{\delta}(x,c)=(c-x)^{2}e^{\delta x}$ | no | no | no
| Esscher loss (see Van Heerwaarden et al. (1989)) |  |  |
Table 2: Commonly used real-valued loss functions and their basic properties. | loss function | sign symmetric | shift invariant | positive homogeneous
---|---|---|---|---
|  |  |  | of degree $k$
| $\ell_{\delta}(x,c)$ | $\ell_{\delta}(-x,c)=\ell_{\delta}(x,-c)$ | $\ell_{\delta}(x+b,c+b)$ | $\ell_{\delta}(ax,ac)=a^{k}\ell_{\delta}(x,c)$
|  |  | $\quad=\ell_{\delta}(x,c)$ |
G1 | $\ell_{\delta}(x,c)=-c(x^{2}-x)+c^{2}/2$ | no | no | no
G2 |  $\ell_{\delta}(x,c)=-c\lvert x-b\rvert^{\delta}+c^{2}/2$, |  |  |
| (see e.g. Hürlimann (2006)) |  |  |
|  with $b\in\mathbb{R}$ | no | no | no
G3 | $\ell_{\delta}(x,c)=1_{\\{c<x\\}}(x-c)+cx$ | no | no | no
G4 | $\ell_{\delta}(x,c)=\frac{c^{2}}{2}-(1+\delta)cx$ | yes | no | yes, of degree 2
| (see Dickson (2005)) |  |  |

Note that in Definition 2.1 we allow a loss function to take values in
$\mathbb{R}$. Several commonly used loss functions are restricted to take
values in $[0,+\infty)$. Table 1 lists some of these. Table 2 provides some
examples of real-valued loss functions. Throughout the paper we allow for a
general loss function, and state explicitly when we restrict it to be
nonnegative.

When defining generalized extremiles in Definition 2.3 we require the
distribution function $D_{\boldsymbol{\tau}}$ to be absolutely continuous
admitting a density function $d_{\boldsymbol{\tau}}$. The latter can be
relaxed, as we illustrate next.

Recall that $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$ denotes the random
variable with cumulative distribution function $D_{\boldsymbol{\tau}}\circ
F_{X}$, i.e., $X_{{\scriptsize D}_{\boldsymbol{\tau}}}\sim
D_{\boldsymbol{\tau}}\circ F_{X}$.

######  Proposition 2.1.

Consider $X_{{\scriptsize D}_{\boldsymbol{\tau}}}\sim
D_{\boldsymbol{\tau}}\circ F_{X}$. We then have

$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\mathbb{E}[d_{\boldsymbol{\tau}}(F_{X}(X))(\ell_{\delta}(X,c)-\ell_{\delta}(X,0))]=\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\
\mathbb{E}[\ell_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}},c)-\ell_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}},0)],$

where the expectation on the right-hand side is taken with respect to the
random variable $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$.

###### Proof.

The proof is straightforward by noting that

$\displaystyle\mathbb{E}[d_{\boldsymbol{\tau}}(F_{X}(X))(\ell_{\delta}(X,c)-\ell_{\delta}(X,0)]$
$\displaystyle=\int_{\mathbb{R}}d_{\boldsymbol{\tau}}(F_{X}(x))(\ell_{\delta}(x,c)-\ell_{\delta}(x,0))\mathrm{d}F_{X}(x)$
$\displaystyle=\int_{\mathbb{R}}(\ell_{\delta}(x,c)-\ell_{\delta}(x,0))\mathrm{d}D_{\boldsymbol{\tau}}(F_{X}(x))$
$\displaystyle=\mathbb{E}[\ell_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}},c)-\ell_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}},0)].$

∎

An alternative would be to define generalized extremiles as stipulated in
Proposition 2.1, via

$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\
\mathbb{E}[\ell_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}},c)-\ell_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}},0)],$

as such not requiring the existing of the density $d_{\boldsymbol{\tau}}$.
Although this would be a slightly broader definition, it would make it more
tedious to establish (asymptotic) properties when dealing with statistical
inference for the studied functionals. The latter is the main goal of the
paper.

In Tables 4 and 5 (see later) we indicate for each of the loss functions in
Tables 1 and 2 this alternative view, with focus on the loss function applied
to the random variable $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$. Table 3
lists a selection of distribution (distortion) functions (with their
associated densities).

Table 3: Commonly used distribution functions/distortion functions and their basic properties. | distrib. function or distortion function | density | bounded density
---|---|---|---
|  $D_{\boldsymbol{\tau}}(u)\quad$ or $\quad g_{\boldsymbol{\tau}}(u)=1-D_{\boldsymbol{\tau}}(1-u)$ | $d_{\boldsymbol{\tau}}(u)$ |
|  and $D_{\boldsymbol{\tau}}(u)=1-g_{\tau}(1-u)$ |  |
1 | $D_{\boldsymbol{\tau}}(u)=u$ | $d_{\tau}(u)=1$ | yes
| uniform distr. |  |
2 |  $D_{\boldsymbol{\tau}}(u)=K_{\tau}(u)$ as in (9) |  $d_{\tau}(u)=J_{\tau}(u)$ as in (4) | yes
3 | $D_{\boldsymbol{\tau}}(u)=[B(a,b)]^{-1}\int_{0}^{u}t^{a-1}(1-t)^{b-1}\mathrm{d}t$ | $d_{\tau}(u)=[B(a,b)]^{-1}u^{a-1}(1-u)^{b-1}$ | possibly unbounded
|  $a,b>0$ |  |  in 0 and/or 1
| Beta distr. |  | bounded if $a,b>1$
4 | $D_{\boldsymbol{\tau}}(u)=1-(1-u^{a})^{b}\quad a,b>0$ | $d_{\boldsymbol{\tau}}(u)=abu^{a-1}(1-u^{a})^{b-1}$ | possibly unbounded
| Kumaraswamy distr. |  |  in 0 and/or 1
|  |  | bounded if $a,b>1$
5 | $g_{\tau}(u)=u(1-\tau)^{-1}1_{\\{u\leq 1-\tau\\}}+1_{\\{u>1-\tau\\}}\quad\tau\in[0,1]$ | $d_{\tau}(u)=(1-\tau)^{-1}1_{\\{u>\tau\\}}$ | yes, for $\tau\in[0,1)$
| or $D_{\tau}(u)=(1-\tau)^{-1}(u-\tau)1{\\{u\geq\tau\\}}$ |  |
|  or shortly $D_{\tau}\sim\mbox{U}[\tau,1]$ |  |
| expected shortfall |  |
6 | $g_{\boldsymbol{\tau}}(u)=\Phi\left(\Phi^{-1}(u)+\tau\right)\quad\tau\in\mathbb{R}$ | $d_{\boldsymbol{\tau}}(u)=\displaystyle\frac{\phi\left(\Phi^{-1}(1-u)+\tau\right)}{\phi\left(\Phi^{-1}(1-u)\right)}$ | for $\tau=0$ on $[0,1]$
|  |  | unbounded in 1 for $\tau>0$
| Wang Transform risk measure |  | unbounded in 0 for $\tau<0$
7 | $g_{\boldsymbol{\tau}}(u)=u^{1/\tau}\quad\tau\geq 1$ | $d_{\tau}(u)=\tau^{-1}(1-u)^{\\{\tau^{-1}-1\\}}$ |
| Proportional Hazard transform |  | unbounded in 1
8 |  $D_{\tau}(u)=u^{\tau+1}$, with $\tau\in\mathbb{R}_{+}$ | $d_{\boldsymbol{\tau}}(u)=(\tau+1)u^{\tau}$ | bounded
|  minvar (see Cherny and Madan (2017)) |  |
9 |  $D_{\tau}(u)=1-(1-u)^{1/(\tau+1)}$, with $\tau\in\mathbb{R}_{+}$ | $d_{\boldsymbol{\tau}}(u)=(\tau+1)^{-1}(1-u)^{-\tau/(\tau+1)}$ | unbounded in 1,
|  maxvar (see Cherny and Madan (2017)) |  | unless $\tau=0$
|  |  |  (but then $\mbox{U}[0,1]$)
10 |  $D_{\tau}(u)=\left(1-(1-u)^{1/(\tau+1)}\right)^{\tau+1}$, with $\tau\in\mathbb{R}_{+}$ | $d_{\boldsymbol{\tau}}(u)$ | unbounded in 1
|  minmaxvar (see Cherny and Madan (2017)) |  $=\left(1-(1-u)^{1/(\tau+1)}\right)^{\tau}(1-u)^{-\tau/(\tau+1)}$ |
11 |  $D_{\tau}(u)=1-\left(1-u^{\tau+1}\right)^{1/(\tau+1)}$, with $\tau\in\mathbb{R}_{+}$ | $d_{\tau}(u)=\left(1-u^{\tau+1}\right)^{-\tau/(\tau+1)}u^{\tau}$ | unbounded in 1
|  maxminvar (see Cherny and Madan (2017)) |  |
12 | $g_{\boldsymbol{\tau}}(u)=G\left(G^{-1}(u)+\tau\right)\quad\tau\geq 0$ | $d_{\tau}(u)=\displaystyle\frac{g\left(G^{-1}(1-u)+\tau\right)}{g\left(G^{-1}(1-u)\right)}$ | yes, on $(0,1)$
|  $G$ cumul. distr. function, with $G^{\prime}=g$ log-concave |  | possibly unbounded
| Junike’s distortion function (see Junike (2019)) |  | in 0 or 1, depending on $g$
13 |  $g_{\tau}(u)=\tau^{-1}C(u,\tau)$ $\quad\tau\in(0,1]$ | $d_{\tau}(u)=\tau^{-1}\frac{\partial C}{\partial u}(1-u;\tau)$ | yes
|  $C(\cdot,\cdot)$ bivariate copula |  |
| (see Yin and Zhu (2018)) |  |

###  2.2 Generalized extremiles: properties

Before stating some probabilistic properties of generalized extremiles, we
first define some properties of loss functions, as well as the notion of a
dual of a cumulative distribution function $D_{\boldsymbol{\tau}}$.

######  Definition 2.4.

Consider a loss function
$\ell_{\delta}:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$. It is called

  1. 1.

sign symmetric if $\ell_{\delta}(-x,c)=\ell_{\delta}(x,-c)\text{ for all
}x,c\in\mathbb{R}$;

  2. 2.

shift invariant if $\ell_{\delta}(x+b,c+b)=\ell_{\delta}(x,c)$ for all
$x,c,b\in\mathbb{R}$;

  3. 3.

positive homogeneous of degree $k>0$ if
$\ell_{\delta}(ax,ac)=a^{k}\ell_{\delta}(x,c)$ for all $a>0$ and
$x,c\in\mathbb{R}$.

In Tables 1 and 2 we indicate for each of the listed loss functions whether
they possess the property of sign symmetry, shift invariance and positive
homogeneity.

For a given cumulative distribution function $D_{\boldsymbol{\tau}}$ in the
set $\mathbb{D}$ it is of interest to consider its dual.

######  Definition 2.5.

Given $D_{\boldsymbol{\tau}}\in\mathbb{D}$ we define its dual
$\widetilde{D}_{\boldsymbol{\tau}}$ as
$\widetilde{D}_{\boldsymbol{\tau}}(t)=1-D_{\boldsymbol{\tau}}(1-t)$.

It is clear that the dual function is again part of the set $\mathbb{D}$.
Furthermore, the dual of $\widetilde{D}_{\boldsymbol{\tau}}$ is equal to
$D_{\boldsymbol{\tau}}$.

The proofs of all propositions in this section can be found in Appendix A.
Proposition 2.2 provides sufficient conditions under which the expectation in
Definition 2.3 is finite. The proof is straightforward and omitted.

######  Proposition 2.2.

If the cumulative distribution function $D_{\boldsymbol{\tau}}$ is
differentiable with bounded density $d_{\boldsymbol{\tau}}$ and
$\mathbb{E}[\lvert\ell_{\delta}(X,c)-\ell_{\delta}(X,0))\rvert]<\infty$ for
all $c\in\mathbb{R}$ then

$\lvert\mathbb{E}[d_{\boldsymbol{\tau}}(F_{X}(X))(\ell_{\delta}(X,c)-\ell_{\delta}(X,0))]\rvert<\infty$

for all $c\in\mathbb{R}$.

######  Remark 2.1.

The condition on the loss function (and $X$) in Proposition 2.2 is very mild,
and fulfilled for several commonly used loss functions. In case of the
absolute value loss $\ell(x,c)=\lvert x-c\rvert$ the reverse triangle
inequality yields $\lvert\lvert x-c\rvert-\lvert x\lvert\lvert\leq\lvert
c\rvert$ and hence $\mathbb{E}[\lvert\ell(X,c)-\ell(X,0)\rvert]<\infty$ for
all random variables $X$. In case of the square loss $\ell(x,c)=(x-c)^{2}$ we
have $\lvert(x-c)^{2}-x^{2}\rvert=\lvert c^{2}-2cx\rvert\leq c^{2}+2\lvert
c\rvert\cdot\lvert x\rvert$ and hence
$\mathbb{E}[\lvert\ell(X,c)-\ell(X,0)\rvert]<\infty$ for all random variables
$X$ with $\mathbb{E}[\lvert X\rvert]<\infty$.

From Remark 2.1 it also becomes clear why the term $\ell_{\delta}(X,0)$ is
included in (8), whereas it does not depend on the argument $c$. Due to the
inclusion of that term we have, for example, that quantiles obtained by taking
$D_{\tau}=K_{\tau}$ and $\ell_{\delta}(x,c)=|x-c|$, exist for all random
variables. Similarly, expectiles of $X$ only require the finiteness of the
first absolute moment of $X$.

Propositions 2.3, 2.4 and 2.5 establish the behaviour of generalized
extremiles when the underlying random variable is transformated either by
taking the negative of it (Proposition 2.3), or by an affine transformation
(Proposition 2.4), or by applying a strictly monotone function to it
(Proposition 2.5). From these propositions we can conclude that generalized
extremiles have similar properties as quantiles and expectiles, among others.

######  Proposition 2.3 (Sign symmetry).

Denote by $\ell_{\delta}$ a sign symmetric loss function. If the distribution
function $F_{X}$ of $X$ is continuous then we have

$e_{\boldsymbol{\tau},\delta}\left(X;D_{\boldsymbol{\tau}},\ell_{\delta}\right)=-e_{\boldsymbol{\tau},\delta}(-X;\widetilde{D}_{\boldsymbol{\tau}},\ell_{\delta})\qquad\text{
and hence also }\qquad
e_{\boldsymbol{\tau},\delta}(X;\widetilde{D}_{\boldsymbol{\tau}},\ell_{\delta})=-e_{\boldsymbol{\tau},\delta}(-X;D_{\boldsymbol{\tau}},\ell_{\delta}).$

######  Proposition 2.4.

Consider scalars $a>0$ and $b\in\mathbb{R}$, and a random variable $X\sim
F_{X}$ such that $\lvert
e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})\rvert<\infty$.
If the loss function $\ell_{\delta}$ is shift invariant, positive homogeneous
of degree $k>0$ and if

$\lvert\mathbb{E}[d_{\boldsymbol{\tau}}(F_{X}(X))(\ell_{\delta}(aX,0)-\ell_{\delta}(aX+b,0))]\rvert<\infty$
(10)

then

$e_{\boldsymbol{\tau},\delta}(aX+b;D_{\boldsymbol{\tau}},\ell_{\delta})=ae_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})+b.$

If in addition the loss function $\ell_{\delta}$ is sign symmetric, $F_{X}$ is
continuous and (10) holds for $a<0$ and $b\in\mathbb{R}$ then

$e_{\boldsymbol{\tau},\delta}(aX+b;D_{\boldsymbol{\tau}},\ell_{\delta})=ae_{\boldsymbol{\tau},\delta}(X;\widetilde{D}_{\boldsymbol{\tau}},\ell_{\delta})+b\qquad\text{
and hence also }\qquad
e_{\boldsymbol{\tau},\delta}(aX+b;\widetilde{D}_{\boldsymbol{\tau}},\ell_{\delta})=ae_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})+b.$

######  Proposition 2.5 (Monotone transformations).

Suppose $\alpha:\mathbb{R}\to\mathbb{R}$ is a strictly increasing function and
hence measurable function, then

$e_{\boldsymbol{\tau},\delta}(\alpha(X);D_{\boldsymbol{\tau}},\ell_{\delta})=e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\widetilde{\ell}_{\delta}),$

where $\widetilde{\ell}_{\delta}(x,c)=\ell_{\delta}(\alpha(x),c)$. If
$\alpha:\mathbb{R}\to\mathbb{R}$ is strictly decreasing and $X$ is a random
variable with continuous $F_{X}$ then

$e_{\boldsymbol{\tau},\delta}(\alpha(X);D_{\boldsymbol{\tau}},\ell_{\delta})=e_{\boldsymbol{\tau},\delta}(X;\widetilde{D}_{\boldsymbol{\tau}},\widetilde{\ell}_{\delta}).$

When a random variable $X$ is symmetric and has finite expectation it is
possible to express this mean in terms of differences or sums of generalized
extremiles, as stated in Proposition 2.6.

######  Proposition 2.6 (Mean symmetry).

Denote by $\ell_{\delta}$ a shift invariant loss function. If $\mathbb{E}[X]$
is finite and $X$ is symmetric, i.e., $X\stackrel{{\scriptstyle d}}{{=}}-X$,
then

$\frac{1}{2}(e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})-e_{\boldsymbol{\tau},\delta}(-X;D_{\boldsymbol{\tau}},\ell_{\delta}))=\mathbb{E}[X]$

If in addition $\ell_{\delta}$ is a sign symmetric loss function and the
distribution function $F_{X}$ of $X$ is continuous then

$\frac{1}{2}\left(e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})+e_{\boldsymbol{\tau},\delta}(X;\widetilde{D}_{\boldsymbol{\tau}},\ell_{\delta})\right)=\mathbb{E}[X].$

For two random variables $X$ and $Y$, it is interesting to know how the
generalized extremile of their sum, i.e.,
$e_{\boldsymbol{\tau},\delta}(X+Y;D_{\boldsymbol{\tau}},\ell_{\delta})$,
relates to the sum of the generalized extremiles of each of the random
variables. Proposition 2.7 below provides some answer to this question,
establishing the property of comonotonic additivity for the absolute value and
square loss functions.

######  Proposition 2.7 (Comonotonic additivity).

Denote by $X\sim F_{X}$ and $Y\sim F_{Y}$ two random variables that have a
comonotonic dependence structure, i.e.,
$F_{X,Y}(x,y)=\min\\{F_{X}(x),F_{Y}(y)\\}$. If $\ell_{\delta}$ is either the
absolute value loss or the square loss we have (assuming the considered
generalized extremiles are finite) that

$e_{\boldsymbol{\tau},\delta}(X+Y;D_{\boldsymbol{\tau}},\ell_{\delta})=e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})+e_{\boldsymbol{\tau},\delta}(Y;D_{\boldsymbol{\tau}},\ell_{\delta}).$

A final result states the equivalence of an injective transformation of a
generalized extremile and a generalized extremile of the same random variable
with a transformed loss function. It is similar to results in (Gneiting, 2011,
see Theorem 4) and in (Osband, 1985, see p. 9) known as revelation principle.

######  Proposition 2.8 (Revelation principle).

Let $\beta:\mathbb{R}\to\mathbb{R}$ be an injective function. Define
$\widetilde{\ell}_{\delta}(x,c)=\ell_{\delta}(x,\beta^{-1}(c))$, then

$\beta(e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta}))=e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\widetilde{\ell}_{\delta}).$

The result in Proposition 2.8 has some interesting consequences. Consider the
square loss function $\ell_{\delta}(x,c)=(x-c)^{2}$. Then we know that

$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},(x-c)^{2})=\mathbb{E}[X_{{\scriptsize
D}_{\boldsymbol{\tau}}}].$

Applying Proposition 2.8 with injective function
$\beta:\mathbb{R}\to\mathbb{R}:x\mapsto-x$, gives that

$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},(x+c)^{2})=-\mathbb{E}[X_{{\scriptsize
D}_{\boldsymbol{\tau}}}].$

###  2.3 Towards solving the minimization problem

When it comes to finding a generalized extremile, defined in (8), Proposition
2.1 is a good starting point, showing that we need to find a minimizer of
$\mathbb{E}[\ell_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}},c)-\ell_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}},0)],$ with respect to $c\in\mathbb{R}$. Proposition
2.9 below provides a way to find such a minimizer, and states sufficient
conditions on the pair $(D_{\boldsymbol{\tau}},\ell_{\delta})$.

######  Proposition 2.9.

Let $\ell_{\delta}$ be a function such that $c\mapsto\ell_{\delta}(x,c)$ is
(strictly) convex for every $x$ and $c$. Additionally, assume that both
$D_{\boldsymbol{\tau}}$ and $F_{X}$ are differentiable functions. Under these
conditions, the function
$c\mapsto\mathbb{E}\left[\ell_{\delta}\left(X_{D_{\boldsymbol{\tau}}},c\right)-\ell_{\delta}\left(X_{D_{\boldsymbol{\tau}}},0\right)\right]$
is (strictly) convex, and the generalized extremile is equal to any value $c$
for which

$\frac{\mathrm{d}}{\mathrm{d}c}\left(\int_{-\infty}^{+\infty}\ell_{\delta}(x,c)\,\mathrm{d}D_{\boldsymbol{\tau}}\left(F_{X}(x)\right)\right)=0,$
(11)

provided the derivative exists. When the mapping $c\mapsto\ell_{\delta}(x,c)$
is strictly convex, there exists at most one $c$ value fulfilling (11). The
existence of such a $c$ value is ensured when $c\mapsto\ell_{\delta}(x,c)$ is
convex and
$c\mapsto\mathbb{E}\left[\ell_{\delta}\left(X_{D_{\boldsymbol{\tau}}},c\right)-\ell_{\delta}\left(X_{D_{\boldsymbol{\tau}}},0\right)\right]$
is coercive.

###### Proof.

To solve the minimization problem we want to find the value $c$ that minimizes

$I(c)=\mathbb{E}\left[\ell_{\delta}\left(X_{D_{\tau}},c\right)\right]=\int_{-\infty}^{+\infty}\ell_{\delta}(x,c)d_{\boldsymbol{\tau}}\left(F_{X}(x)\right)f_{X}(x)\mathrm{d}x.$

By Boyd and Vandenberghe (2004), p. 79, we know that $I(c)$ is convex in $c$
if $c\mapsto\ell_{\delta}(x,c)$ is convex for every $x$. It is clear that
strict convexity of $c\mapsto\ell_{\delta}(x,c)$ implies that $I(c)$ is also
strict convex and hence the minimum is unique (see Theorem 3.4.4 on p. 114 in
Niculescu and Persson (2006)).
The existence of a value $c$ follows by Theorem 4.4 on p. 638 in Herrmann et
al. (2018). ∎

In Proposition 2.9 uniqueness of $c$ solving (11) is guaranteed when
$c\mapsto\mathbb{E}\left[\ell_{\delta}\left(X_{D_{\boldsymbol{\tau}}},c\right)-\ell_{\delta}\left(X_{D_{\boldsymbol{\tau}}},0\right)\right]$
is a coercive function. Some sufficient conditions for this to hold are
discussed in the following remark.

######  Remark 2.2.

Consider a random variable $Y$ (with associated probability measure
$\mathbb{P}_{Y}$) such that $E_{0}=\mathbb{E}[\ell_{\delta}(Y,0)]$ is finite.
Additional to convexity, assume that $c\mapsto\ell_{\delta}(y,c)$ is (i)
coercive for $\mathbb{P}_{Y}$ almost all $y\in\mathbb{R}$, i.e.,
$\lim_{c_{n}\to\pm\infty}\ell_{\delta}(y,c_{n})=\infty$, and (ii) bounded from
below, i.e., $\exists\,C>0$, such that $\ell_{\delta}(y,c)\geq C$
$\mathbb{P}_{Y}$-a.s. Then also
$c\mapsto\mathbb{E}[\ell_{\delta}(Y,c)-\ell_{\delta}(Y,0)]$ is coercive. To
see this consider a sequence $(c_{n})$ such that $\lvert c_{n}\rvert\to\infty$
and define $Y_{n}:=\ell_{\delta}(Y,c_{n})-C$ to have $Y_{n}\geq 0$
$\mathbb{P}_{Y}$-a.s. and $Y_{n}\to\infty$ due to the coercivity of
$\ell_{\delta}$. A slight modification of the monotone convergence theorem,
see Feinstein (2007), then implies
$\lim_{n\to\infty}\mathbb{E}[\ell_{\delta}(Y,c_{n})-\ell_{\delta}(Y,0)]=\lim_{n\to\infty}\mathbb{E}[\ell_{\delta}(Y,c_{n})-C]+C-E_{0}=\mathbb{E}[\lim_{n\to\infty}\ell_{\delta}(Y,c_{n})-C]+C-E_{0}=\infty$,
showing that $c\mapsto\mathbb{E}[\ell_{\delta}(Y,c)-\ell_{\delta}(Y,0)]$ is
coercive. The result is now applicable in the context of Proposition 2.9 if
the necessary conditions apply for the choice of $Y=X_{D_{\tau}}$.

As is clear from the above result, convexity of the function
$c\mapsto\ell_{\delta}(x,c)$ will be an important requirement when it comes to
calculating generalized extremiles. In addition it also makes it transparant
that interchanging integral and differentiation operators will be helpful. For
any random variable $X$ (not necessarily absolutely continuous) and any
distribution function $D_{\boldsymbol{\tau}}$ (not necessarily admitting a
density) the next theorem (stating a type of Leibniz rule) is helpful.

######  Theorem 2.1 (see Klenke (2013), Theorem 6.28, p. 142).

Denote $\mu$ the measure induced by $D_{\boldsymbol{\tau}}\circ F_{X}$.
Suppose that a loss function $\ell_{\delta}(x,c)$ satisfies the following
properties.

  * •

For every $c$, the map $x\mapsto\ell_{\delta}(x,c)$ is integrable (with
respect to the measure $\mu$);

  * •

For $\mu$-almost all $x$, the map $c\mapsto\ell_{\delta}(x,c)$ is
differentiable with derivative $\ell_{\delta}^{\prime}(x,c)$;

  * •

There is an integrable function $h:\mathbb{R}\to(0,\infty)$ (with respect to
the measure $\mu$) such that $\lvert\ell_{\delta}^{\prime}(x,c)\rvert\leq
h(x)$ for all $c$ and $\mu$-almost all $x$.

Then for all $c\in\mathbb{R}$ we have

$\frac{\mathrm{d}}{\mathrm{d}c}\Big{(}\int_{-\infty}^{+\infty}\ell_{\delta}(x,c)\mathrm{d}D_{\boldsymbol{\tau}}(F_{X}(x))\Big{)}=\int_{-\infty}^{+\infty}\ell_{\delta}^{\prime}(x,c)\mathrm{d}D_{\boldsymbol{\tau}}(F_{X}(x)).$

For absolutely continuous $X$ with density function $f_{X}$, and
$D_{\boldsymbol{\tau}}$ admitting a density $d_{\boldsymbol{\tau}}$, the
adaptation of this theorem is as follows.

######  Theorem 2.2.

Consider an absolutely continuous random variable $X$, and a cumulative
distribution function $D_{\boldsymbol{\tau}}\in\mathbb{D}$ with density
$d_{\boldsymbol{\tau}}$. Suppose that a loss function $\ell_{\delta}(x,c)$
satisfies the following properties.

  * •

For every $c$, the map
$x\mapsto\ell_{\delta}(x,c)d_{\boldsymbol{\tau}}(F_{X}(x))f_{X}(x)$ is
integrable;

  * •

For almost all $x$, the map $c\mapsto\ell_{\delta}(x,c)$ is differentiable
with derivative $\ell_{\delta}^{\prime}(x,c)$;

  * •

There is an integrable function $h:\mathbb{R}\to(0,\infty)$ (with respect to
the measure $\mu$) such that
$\lvert\ell_{\delta}^{\prime}(x,c)d_{\boldsymbol{\tau}}(F_{X}(x))f_{X}(x)\rvert\leq
h(x)$ for all $c$ and almost all $x$.

Then for all $c\in\mathbb{R}$ we have

$\frac{\mathrm{d}}{\mathrm{d}c}\Big{(}\int_{-\infty}^{+\infty}\ell_{\delta}(x,c)d_{\boldsymbol{\tau}}(F_{X}(x))f_{X}(x)\mathrm{d}x\Big{)}=\int_{-\infty}^{+\infty}\ell_{\delta}^{\prime}(x,c)d_{\boldsymbol{\tau}}(F_{X}(x))f_{X}(x)\mathrm{d}x.$

In Tables 4 and 5 we return to the loss functions listed in respectively
Tables 1 and 2, and now focus on the properties of convexity and
differentiability of these loss functions.

Table 4: Commonly used nonnegative loss functions and basic properties
(convexity, differentiability, … ).

| loss function | alternative view | convex | differentiability | derivative
---|---|---|---|---|---
|  | from $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$ | in $c$ | in $c$ | with respect to $c$
1 | $|x-c|$ | $\mbox{Median}(X_{{\scriptsize D}_{\boldsymbol{\tau}}})$ | yes | yes, for $c\neq x$ | $\ell^{\prime}_{\delta}(x,c)=1_{\\{x\leq c\\}}-1_{\\{x>c\\}}$
| absolute value loss |  |  |  |
2 |  $|x-c|^{p}$ with $p>0$ |  $\mathbb{E}[X_{{\scriptsize D}_{\boldsymbol{\tau}}}]$ for $p=2$ | yes, | depends on the |
| power $p$ absolute loss |  | if $p\geq 1$ | value of $p$ |
3 | $\left|\delta-1_{\\{x\leq c\\}}\right||x-c|$ |  $\delta$th quantile of $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$ | yes | yes, for $c\neq x$ | $\ell^{\prime}_{\delta}(x,c)=1_{\\{x\leq c\\}}-\delta$
| quantile loss |  |  |  |
4 | $\left|\delta-1_{\\{x\leq c\\}}\right||x-c|^{2}$ |  $\delta$th expectile of $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$ | yes | yes, $\forall c$ | $\ell^{\prime}_{\delta}(x,c)$
| expectile loss |  |  |  | $\quad=-2\delta\lvert x-c\rvert-2(x-c)1_{\\{x\leq c\\}}$
5 | $\ell_{\delta}(x,c)$ | (⋆1) | yes | yes, for $c$ | $\ell^{\prime}_{\delta}(x,c)=-(x-c)1_{\\{|x-c|\leq\delta\\}}$
| $=\left\\{\begin{array}[]{ll}0.5(x-c)^{2}&\mbox{if }|x-c|\leq\delta\\\ \delta(|x-c|-0.5\delta)&\mbox{if }|x-c|>\delta\end{array}\right.$ |  |  | such that | $\hskip 45.52458pt-\mbox{sign}(x-c)\delta\,1_{\\{|x-c|>\delta\\}}$
| Huber loss function |  |  | $|x-c|\neq\delta$ |
6 | $\ell_{\delta}(x,c)=(c-x)^{2}e^{\delta x}$ | $\displaystyle\frac{\mathbb{E}[X_{{\scriptsize D}_{\boldsymbol{\tau}}}e^{\delta X_{{\scriptsize D}_{\boldsymbol{\tau}}}}]}{\mathbb{E}[e^{\delta X_{{\scriptsize D}_{\boldsymbol{\tau}}}}]}$ | yes | yes, $\forall c$ | $\ell_{\delta}^{\prime}(x,c)=2(c-x)e^{\delta x}$
| Esscher loss |  |  |  |

(⋆1): Value c for which $c=\frac{1}{\mathbb{E}[1_{\\{\lvert X_{{\scriptsize
D}_{\boldsymbol{\tau}}}-c\rvert\leq\delta\\}}]}\Big{(}\mathbb{E}[X_{{\scriptsize
D}_{\boldsymbol{\tau}}}1_{\\{\lvert X_{{\scriptsize
D}_{\boldsymbol{\tau}}}-c\rvert\leq\delta\\}}]-\delta\mathbb{E}[1_{\\{X_{{\scriptsize
D}_{\boldsymbol{\tau}}}-c\leq-\delta\\}}]+\delta\mathbb{E}[1_{\\{X_{{\scriptsize
D}_{\boldsymbol{\tau}}}-c\geq\delta\\}}]\Big{)}$).

Table 5: Commonly used real-valued loss functions and basic properties (convexity, differentiability, ….). | loss function | alternative view | convex | differentiability | derivative
---|---|---|---|---|---
|  | from $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$ | in $c$ | in $c$ | with respect to $c$
G1 | $\ell_{\delta}(x,c)=-c(x^{2}-x)+c^{2}/2$ | $\mathbb{E}[(X_{{\scriptsize D}_{\boldsymbol{\tau}}})^{2}-X_{{\scriptsize D}_{\boldsymbol{\tau}}}]$ | yes | yes, for all $c\neq x$ | $\ell^{\prime}_{\delta}(x,c)=-(x^{2}-x)+c$
G2 |  $\ell_{\delta}(x,c)=-c\lvert x-b\rvert^{\delta}+c^{2}/2$, | $\mathbb{E}[\lvert X_{{\scriptsize D}_{\boldsymbol{\tau}}}-b\rvert^{\delta}]$ | yes | yes, $\forall c$ | $\ell_{\delta}^{\prime}(x,c)=-\lvert x-b\rvert^{\delta}+c$
|  with $b\in\mathbb{R}$ |  |  |  |
G3 | $\ell_{\delta}(x,c)=1_{\\{c<x\\}}(x-c)+cx$ | value $c$ for which | yes | yes, for all $c\neq x$ | $\ell^{\prime}_{\delta}(x,c)=-1_{\\{c<x\\}}+x$
|  | $P(X_{{\scriptsize D}_{\boldsymbol{\tau}}}\geq c)=\mathbb{E}[X_{{\scriptsize D}_{\boldsymbol{\tau}}}]$ |  |  |
G4 | $\ell_{\delta}(x,c)=\frac{c^{2}}{2}-(1+\delta)cx$ | $(1+\delta)\mathbb{E}[X_{{\scriptsize D}_{\boldsymbol{\tau}}}]$ | yes | yes, $\forall c$ | $\ell_{\delta}^{\prime}(x,c)=c-(1+\delta)x$
|  | expected value |  |  |
|  | premium principle |  |  |

###  2.4 Examples

Complementary to the loss functions listed in Tables 1 and 2 we review here
some loss functions that have been proposed in recent literature, illustrating
as such the broad framework we consider. Some more loss functions are given in
Section S1 in the Supplementary Material (non exhaustive list).

####  2.4.1 Nonnegative valued loss functions

Adaptive quantile loss function for censoring
In the context of quantile estimation under right random censoring, De Backer
et al. (2019) established an approach based on considering a loss function
that adapts to the cumulative distribution function $F_{C}$ of the censoring
variable $C$:

$\ell_{\delta}(x,c)=\left|\delta-1_{\\{x\leq
c\\}}\right||x-c|-(1-\delta)\displaystyle\int_{0}^{c}F_{C}(s)\mathrm{d}s.$
(12)

Note that the mapping $c\mapsto\ell_{\delta}(x,c)$ is in general not convex,
nor is it monotone. In case of no censoring, i.e. when $P\\{C<\infty\\}=0$,
the adapted loss function reduces to the usual quantile loss function.

M-quantile loss
For a function $\psi:\mathbb{R}\to[0,+\infty)$, consider the loss function

$\ell_{\delta}(x,c)=\delta\psi(x-c)1_{\\{x>c\\}}+(1-\delta)\psi(x-c)1_{\\{x\leq
c\\}}.$

This is the M-quantile loss function. Taking $\psi(u)=|u|^{p}$ leads to the
power $p$ absolute loss function, also called $L_{p}-$quantiles as defined in
Chen (1996). See also Jiang et al. (2021). Obviously, the qualitative
properties of $\ell_{\delta}(x,c)$ depend on the qualitative properties of the
function $\psi$.

Generalized quantile loss
An extension of the M-quantiles are obtained by involving two different
functions $\psi_{1}$ and $\psi_{2}$. Namely consider
$\psi_{1},\psi_{2}:[0,+\infty)\to[0,+\infty)$, convex and strictly increasing
functions satisfying $\psi_{j}(0)=0\text{ and }\psi_{j}(1)=1\text{ for
}j=1,2.$ The generalized quantile loss function is then

$\ell_{\delta}(x,c)=\delta\psi_{1}(x-c)1_{\\{x>c\\}}+(1-\delta)\psi_{2}(x-c)1_{\\{x\leq
c\\}},$

leading to the $\delta$th generalized quantile of a random variable. See
Bellini et al. (2014). The qualitative properties of the generalized quantile
loss depend on these of the functions $\psi_{1}$ and $\psi_{2}$.

In Mao and Cai (2018), the authors further generalized this class by using
rank-dependent expected utility (RDEU) theory. This corresponding class of
generalized quantiles based on RDEU theory is given by

$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\Big{\\{}\delta\int_{c}^{+\infty}\phi_{1}(x-c)dh_{1}(F_{X}(x))+(1-\delta)\int_{-\infty}^{c}\phi_{2}(c-x)dh_{2}(F_{X}(x))\Big{\\}}.$
(13)

The functions $h_{1},h_{2}$ are two distortion functions on $[0,1]$, which
means they are right-continuous and increasing with $h_{j}(0)=0$ and
$h_{j}(1)=1$ for $j=1,2$, with no jumps at $0$ and $1$. Furthermore,
$\phi_{1}$ and $\phi_{2}$ are nondegenerate increasing convex functions on
$[0,\infty)$. When $h_{1}=h_{2}:=D_{\boldsymbol{\tau}}$ we can describe (part
of) this class using the generalized extremile framework. Indeed, to do so, we
have to take

$D_{\boldsymbol{\tau}}=h_{1}=h_{2}\text{ and
}\ell_{\delta}(x,c)=\delta\phi_{1}((x-c)_{+})+(1-\delta)\phi_{2}((c-x)_{-}).$

Here we used the notations: $x_{+}=\max\\{x,0\\}$ and $x_{-}=-\min\\{x,0\\}$.

Generalized shortfall loss
In Mao and Cai (2018) and Mao et al. (2023), the authors study generalized
shortfall risk measures. These are obtained by replacing $\phi_{1},\phi_{2}$
with their derivatives $\phi_{1}^{\prime},\phi_{2}^{\prime}$ in (13). Just as
before, part of these generalized shortfall risk measures can be described in
our generalized extremile framework when $h_{1}=h_{2}:=D_{\boldsymbol{\tau}}$.

Loss function for trimmed mean
The Hampel family of functions (see Hampel et al. (1986)) is defined as

$\ell(x,c)=\begin{cases}\frac{1}{2}(x-c)^{2}&\text{ if }\lvert x-c\rvert<a\\\
a(x-c-a/2)&\text{ if }a\leq\lvert x-c\rvert<b\\\
\displaystyle\frac{a(x-c-d)^{2}}{2(b-d)}+a\frac{(b+d-a)}{2}&\text{ if
}b\leq\lvert x-c\rvert<d\\\ \frac{1}{2}a(b+d-a)&\text{ if }\lvert
x-c\rvert\geq d,\\\ \end{cases}$

where the function $\ell$ depends on parameters $a,b$ and $d$, with $0<a<b<d$.
This is a non-convex function.

A limiting case of this family is obtained by letting $a,b$ and $d$ taking the
limiting value $\delta>0$, leading to the loss function (see p. 79 in Huber
(1964))

$\ell_{\delta}(x,c)=\begin{cases}(x-c)^{2}&\text{ if }\lvert
x-c\rvert<\delta\\\ \delta^{2}&\text{ if }\lvert
x-c\rvert\geq\delta,\end{cases}$

appearing in the context of trimmed means. Note that the function
$c\mapsto\ell_{\delta}(x,c)$ is quadratic in $c$ around $x$, but attributes
always the same constant value ($\delta^{2}$) when $c$ is such that $\lvert
x-c\rvert\geq\delta$. Consequently the function $c\mapsto\ell_{\delta}(x,c)$
is not convex.

####  2.4.2 Real-valued loss functions

Various functionals involving moments
Let $I\subset\mathbb{R}$ be an interval. Consider $r:I\to\mathbb{R}$ and
$s:I\to(0,+\infty)$ measurable functions. Let $\phi$ be a strictly convex
function with subgradient $\phi^{\prime}$. Let $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$ be a random variable supported on $I$ such that
$\mathbb{E}[r(X_{{\scriptsize D}_{\boldsymbol{\tau}}})]$,
$\mathbb{E}[s(X_{{\scriptsize D}_{\boldsymbol{\tau}}})]$,
$\mathbb{E}[r(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})\phi^{\prime}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})]$, $\mathbb{E}[s(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})\phi(X_{{\scriptsize D}_{\boldsymbol{\tau}}})]$ and
$\mathbb{E}[X_{{\scriptsize D}_{\boldsymbol{\tau}}}s(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})\phi^{\prime}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})]$ exist and are finite. Consider the loss function

$\ell(x,c)=s(x)(\phi(x)-\phi(c))-\phi^{\prime}(c)(r(x)-cs(x))+\phi^{\prime}(x)(r(x)-xs(x)).$

The corresponding generalized extremile equals

$\frac{\mathbb{E}[r(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})]}{\mathbb{E}[s(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})]}.$

We refer to Gneiting (see e.g. Theorem 8 on p. 754 in Gneiting (2011)) for an
in-depth discussion.

Taking $r(x)=x^{2}$, $s(x)=x$ and $\phi(x)=x^{2}$ leads to the special case of

$\frac{\mathbb{E}[\left(X_{{\scriptsize
D}_{\boldsymbol{\tau}}}\right)^{2}]}{\mathbb{E}[X_{{\scriptsize
D}_{\boldsymbol{\tau}}}]}=\frac{\text{var}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})}{\mathbb{E}[X_{{\scriptsize
D}_{\boldsymbol{\tau}}}]}+\mathbb{E}[X_{{\scriptsize
D}_{\boldsymbol{\tau}}}].$

In this special case the loss function is $\ell(x,c)=x(c-x)^{2}$ (see also p.
83 in Heilmann (1989)). This is a convex function in $c$ for $x>0$, and the
derivative of $c\mapsto\ell(x,c)$ exists for every $x$ and is given by
$\ell_{\delta}^{\prime}(x,c)=-2x(x-c)$.

Consider $w:[0,+\infty)\to[0,+\infty)$; and take $r(x)=xw(x)$ and $s(x)=w(x)$.
The interpretation of the resulting generalized extremile

$\frac{\mathbb{E}[X_{{\scriptsize D}_{\boldsymbol{\tau}}}w(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})]}{\mathbb{E}[w(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})]},$

for different weighting functions $w$ is discussed in Furman and Zitikis
(2009).

###  2.5 Square loss and distortion risk measures

A special subclass of functionals of generalized extremiles is obtained when
focussing on the square loss $\ell_{\delta}(x,c)=(x-c)^{2}$. Indeed, as
already pointed out in Section 1, we know that
$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})=\mathbb{E}[X_{{\scriptsize
D}_{\boldsymbol{\tau}}}]$ where $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$ is
distributed with cumulative distribution function $D_{\boldsymbol{\tau}}\circ
F_{X}$. This implies that

$e_{\boldsymbol{\tau}}(X;D_{\boldsymbol{\tau}}):=e_{\boldsymbol{\tau}}(X,D_{\boldsymbol{\tau}},(x-c)^{2})=\int_{-\infty}^{0}[g(1-F_{X}(y))-1]\mathrm{d}y+\int_{0}^{+\infty}g(1-F_{X}(y))\mathrm{d}y,$
(14)

where
$g(y)=1-D_{\boldsymbol{\tau}}(1-y)=\widetilde{D}_{\boldsymbol{\tau}}(y)$. An
expression as in (14), with $g:[0,1]\to[0,1]$, a non-decreasing function and
$g(0)=0$ and $g(1)=1$, is termed a distortion risk of $X$. For this distortion
risk to be a coherent risk measure (see for example Wang (1996) and Artzner et
al. (1999)) one needs to require in addition that $g$ is a concave function.
See, for example, Wang et al. (1997) and CaiWangMao2017. Note that generalized
extremiles, in case of square loss function, can be seen as a distortion risk
measure with distortion function $1-D_{\boldsymbol{\tau}}(1-y)$. This implies
that

$D_{\boldsymbol{\tau}}(y)=1-g(1-y).$

Note that concavity of $g$ is equivalent with convexity of
$D_{\boldsymbol{\tau}}$. Therefore not every coherent distortion risk measure
is necessarily part of our class. If the function $g$ is differentiable then
$d_{\tau}(u)=g^{\prime}(1-u)$. In summary, the class of generalized extremiles
contains part of the distortion risk measures, but it also contains non-
distortion risk measures, for example, expectiles. Table 3 lists several
distortion risk measures that can be seen as part of our general class of
functionals.

##  3 Estimation

Based on an i.i.d. sample $X_{1},\dots,X_{n}$ from $X$, with unknown
distribution function $F_{X}$, the aim is to estimate the generalized
extremile
$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})$ defined
in Definition 2.3, and this for a general pair
$(D_{\boldsymbol{\tau}},\ell_{\delta})$. We firstly focus on the setting of a
square loss function, but with general $D_{\boldsymbol{\tau}}$; and secondly
treat the setting of a general pair $(D_{\boldsymbol{\tau}},\ell_{\delta})$.
The reason is that in the former setting estimation of the generalized
extremile is simpler, and subsequently establishing (asymptotic) properties
for it is somewhat less involved.

###  3.1 Square loss and general distribution function $D_{\boldsymbol{\tau}}$

As discussed in Section 2.5 in case of square loss the generalized extremile
reduces to

$e_{\boldsymbol{\tau}}(X;D_{\boldsymbol{\tau}})=\mathbb{E}[X_{{{\scriptsize
D}}_{\boldsymbol{\tau}}}]=\int_{0}^{1}F_{X}^{-1}(u)\mathrm{d}D_{\boldsymbol{\tau}}(u).$
(15)

An estimator of $e_{\boldsymbol{\tau}}(X;D_{\boldsymbol{\tau}})$ (and hence of
the corresponding distortion risk) can be easily derived from (15). Denote by
$X_{k:n}$ the $k$th order statistic in the random sample $X_{1},\dots,X_{n}$
of size $n$. Let $F_{n}(x)=(n+1)^{-1}\sum_{i=1}^{n}1_{\\{X_{i}\leq x\\}}$ be
the empirical cumulative distribution function (with factor $1/(n+1)$ instead
of $1/n$), and denote by $F_{n}^{-1}(p)$ the corresponding empirical quantile
function, i.e. for $p\in(0,1)$, $F_{n}^{-1}(p)=X_{k:n}$, where $k$ is chosen
such that $(k-1)/(n+1)<p\leq k/(n+1)$. A plug-in type of estimator for
$e_{\boldsymbol{\tau}}(X;D_{\boldsymbol{\tau}})$ is then

$\widehat{e}_{\boldsymbol{\tau},n}^{\mbox{ {\scriptsize
L}}}=\sum_{k=1}^{n}\left(D_{\boldsymbol{\tau}}(k/(n+1))-D_{\boldsymbol{\tau}}((k-1)/(n+1))\right)X_{k:n}:=\frac{1}{n+1}\sum_{k=1}^{n}w_{k,n}X_{k:n},$

where it is thus readily seen that $\widehat{e}_{\boldsymbol{\tau},n}^{\mbox{
{\scriptsize L}}}$ is proportional (with factor $n/(n+1)$) to an
$\mathrm{L}$-statistic with weights
$w_{k,n}=(n+1)\left(D_{\boldsymbol{\tau}}(k/(n+1))-D_{\boldsymbol{\tau}}((k-1)/(n+1))\right)$.

A second estimator, denoted $\widehat{e}_{\boldsymbol{\tau},n}^{\mbox{
{\scriptsize LM}}}$, relies on the fact that in case of square loss we have

$e_{\boldsymbol{\tau}}(X;D_{\boldsymbol{\tau}})=\mathbb{E}[Xd_{\boldsymbol{\tau}}(F_{X}(X))],$

leading to the estimator

$\widehat{e}^{\mbox{ {\scriptsize
LM}}}_{\boldsymbol{\tau},n}=\frac{1}{n}\sum_{i=1}^{n}d_{\boldsymbol{\tau}}\left(\frac{i}{n+1}\right)X_{i:n}.$

A third estimator is obtained by solving the empirical minimization problem.
That is by solving

$\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\frac{1}{n}\sum_{i=1}^{n}d_{\boldsymbol{\tau}}\left(\frac{i}{n+1}\right)\Big{(}(X_{i:n}-c)^{2}-X_{i:n}^{2}\Big{)}=\operatorname*{arg\,min\,}_{c\in\mathbb{R}}\frac{1}{n}\sum_{i=1}^{n}d_{\boldsymbol{\tau}}\left(\frac{i}{n+1}\right)\Big{(}-2cX_{i:n}+c^{2}\Big{)}.$

Differentiating the objective function with respect to $c$ and equating this
to zero gives

$\frac{1}{n}\sum_{i=1}^{n}d_{\boldsymbol{\tau}}\left(\frac{i}{n+1}\right)(-2X_{i:n}+2c)=0\quad\Longleftrightarrow\quad
c=\displaystyle\frac{\displaystyle\sum_{i=1}^{n}d_{\boldsymbol{\tau}}\left(\frac{i}{n+1}\right)X_{i:n}}{\displaystyle\sum_{i=1}^{n}d_{\boldsymbol{\tau}}\left(\frac{i}{n+1}\right)}.$

We thus obtain a new M-type estimator

$\widehat{e}^{\mbox{ {\scriptsize
M}}}_{\boldsymbol{\tau},n}=\frac{\displaystyle\sum_{i=1}^{n}d_{\boldsymbol{\tau}}\left(\frac{i}{n+1}\right)X_{i:n}}{\displaystyle\sum_{i=1}^{n}d_{\boldsymbol{\tau}}\left(\displaystyle\frac{i}{n+1}\right)}=\frac{\widehat{e}^{\mbox{
{\scriptsize
LM}}}_{\boldsymbol{\tau},n}}{\displaystyle\frac{1}{n}\displaystyle\sum_{i=1}^{n}d_{\boldsymbol{\tau}}\left(\frac{i}{n+1}\right)}.$

Note that, since we work with the empirical cumulative distribution function
rescaled with the factor $1/(n+1)$, which takes values between $1/(n+1)$ and
$n/(n+1)$, when evaluated in the observations $X_{1},\dots,X_{n}$, we avoid
evaluation of the density $d_{\boldsymbol{\tau}}$ in the endpoints 0 and 1.

All the above three estimators are consistent and have the same (first order)
asymptotic distribution. The proofs of the results stated in this section are
given in Appendix B.1.

######  Theorem 3.1.

Given is a fixed index $\boldsymbol{\tau}$. Assume that
$D_{\boldsymbol{\tau}}$ is differentiable with density
$d_{\boldsymbol{\tau}}$. Let $\widehat{e}_{\boldsymbol{\tau},n}$ be any of the
two estimators $\widehat{e}_{\boldsymbol{\tau},n}^{\mbox{ {\scriptsize LM}}}$
or $\widehat{e}_{\boldsymbol{\tau},n}^{\mbox{ {\scriptsize M}}}$.

  * (i)

[Consistency] Assume $\mathbb{E}[\lvert X\rvert^{\kappa}]<\infty$ for some
$\kappa>1$. Let $d_{\boldsymbol{\tau}}$ be continuous almost everywhere. If
for some $0<M<\infty$

$\lvert d_{\boldsymbol{\tau}}(t)\rvert\leq
M[t(1-t)]^{-1+1/\kappa+\gamma}\quad\text{ for }t\in(0,1)$

for some $\gamma>0$, then
$\widehat{e}_{\boldsymbol{\tau},n}\xrightarrow{\text{a.s.}}e_{\boldsymbol{\tau}}(X;D_{\boldsymbol{\tau}})$
as $n\to\infty$.
If in addition $d_{\boldsymbol{\tau}}$ is Lipschitz continuous or bounded
(uniformly) on $(0,1)$ then $\widehat{e}_{\boldsymbol{\tau},n}^{\mbox{
{\scriptsize
L}}}\xrightarrow{\text{a.s.}}e_{\boldsymbol{\tau}}(X;D_{\boldsymbol{\tau}})$
as $n\to\infty$

  * (ii)

[Asymptotic normality result] Assume $\mathbb{E}[\lvert
X\rvert^{\kappa}]<\infty$ for some $\kappa>2$. Suppose
$d_{\boldsymbol{\tau}}^{\prime}$ exists and is continuous on $(0,1)$.
Furthermore, suppose there exists some $\gamma>0$ such that

$\lvert d_{\boldsymbol{\tau}}(t)\rvert\leq
M[t(1-t)]^{-1/2+1/\kappa+\gamma}\quad\text{ for }t\in(0,1)$

and

$\lvert d_{\boldsymbol{\tau}}^{\prime}(t)\rvert\leq
M[t(1-t)]^{-3/2+1/\kappa+\gamma}\quad\text{ for }t\in(0,1).$

Then
$\sqrt{n}(\widehat{e}_{\boldsymbol{\tau},n}-e_{\boldsymbol{\tau}}(X;D_{\boldsymbol{\tau}}))\xrightarrow{d}\mathcal{N}(0;\sigma_{\boldsymbol{\tau}}^{2})$
as $n\to\infty$ where

$\sigma_{\boldsymbol{\tau}}^{2}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(F_{X}(\min(s,t))-F_{X}(s)F_{X}(t))d_{\boldsymbol{\tau}}(F_{X}(s))d_{\boldsymbol{\tau}}(F_{X}(t))\mathrm{d}s\mathrm{d}t.$
(16)

If in addition $d_{\boldsymbol{\tau}}$ is Lipschitz continuous or bounded
(uniformly) then $\sqrt{n}(\widehat{e}_{\boldsymbol{\tau},n}^{\mbox{
{\scriptsize
L}}}-e_{\boldsymbol{\tau}}(X;D_{\boldsymbol{\tau}}))\xrightarrow{d}\mathcal{N}(0;\sigma_{\boldsymbol{\tau}}^{2})$
as $n\to\infty$.

When $d_{\boldsymbol{\tau}}$ is bounded, we obtain the almost sure convergence
for any random variable $X$ with a finite absolute $\kappa$th moment (for some
$\kappa>1$). In particular, when
$D_{\boldsymbol{\tau}}(u)=K_{\boldsymbol{\tau}}(u)$, that is the case of
extremiles, we obtain the same result as in Theorem 1(i) on p. 1370 of Daouia
et al. (2019). In the case of extremiles, also the function $\lvert
d_{\boldsymbol{\tau}}^{\prime}(t)\rvert t(1-t)$ is bounded. Hence, also the
asymptotic normality result in Theorem 1(ii) on p. 1370 of Daouia et al.
(2019) is included as a special case of Theorem 3.1 in our more general
framework.

###  3.2 General loss functions and distribution functions

We now turn to the general setting of any pair
$(D_{\boldsymbol{\tau}},\ell_{\delta})$. From now on we assume that $X$ is an
absolutely continuous random variable, i.e., $F_{X}$ has an associated density
function $f_{X}$. We only consider loss functions $\ell_{\delta}(x,c)$ for
which $c\mapsto\ell_{\delta}(x,c)\,\mbox{ is convex for every $x$}.$ By
Proposition 2.9 we know that solving the minimization problem is equivalent to
finding a value $c$ for which

$\frac{\mathrm{d}}{\mathrm{d}c}\Big{(}\int_{-\infty}^{+\infty}\ell_{\delta}(x,c\,)\mathrm{d}D_{\boldsymbol{\tau}}(F_{X}(x))\Big{)}=0.$

Since $X$ is an absolutely continuous random variable, Theorem 2.2 then
ensures the right-hand side equals

$\frac{\mathrm{d}}{\mathrm{d}c}\Big{(}\int_{-\infty}^{+\infty}\ell_{\delta}(x,c)d_{\boldsymbol{\tau}}(F_{X}(x))f_{X}(x)\mathrm{d}x\Big{)}=\int_{-\infty}^{+\infty}\ell_{\delta}^{\prime}(x,c)d_{\boldsymbol{\tau}}(F_{X}(x))f_{X}(x)\mathrm{d}x.$

We are now in the setting of an $M$-functional problem: finding a value $c$
such that

$\lambda_{F}(c):=\int_{-\infty}^{+\infty}d_{\boldsymbol{\tau}}(F_{X}(x))\ell_{\delta}^{\prime}(x,c)f_{X}(x)\mathrm{d}x=0.$

Given a random sample $X_{1},\dots,X_{n}$ from $X$, the $M$-estimator is
defined as the solution $T_{n}$ to

$\lambda_{F,n}(c):=\frac{1}{n}\sum_{i=1}^{n}d_{\boldsymbol{\tau}}(F_{X}(X_{i}))\ell_{\delta}^{\prime}(X_{i},c)=0.$

However, the appearance of $F_{X}$ leads to an extra complication. Since
$F_{X}$ is unknown we replace it by a consistent estimator $\widehat{F}_{n}$
for it, for example, by its empirical version
$F_{n}(x)=(n+1)^{-1}\sum_{i=1}^{n}1_{\\{X_{i}\leq x\\}}$. The estimator
$T_{n}^{*}$ is then defined as the value $c$ for which

$\lambda_{\widehat{F}_{n}}^{*}(c):=\frac{1}{n}\sum_{i=1}^{n}d_{\boldsymbol{\tau}}(\widehat{F}_{n}(X_{i}))\ell_{\delta}^{\prime}(X_{i},c)=0.$

If there is no such value $c$, then we define $T_{n}^{*}$ as the value
minimizing $\lvert\lambda_{\widehat{F}_{n}}^{*}(c)\rvert$. If, in practice,
there are multiple values of $c$ for which
$\lambda_{\widehat{F}_{n}}^{*}(c)=0$, we take the smallest of them.

In what follows we establish the consistency and the asymptotic normality of
the estimator $T_{n}^{*}$. Before stating the results, we introduce some
assumptions, needed in various parts of the theoretical results.

  * (A1)

    1. 1.

$X$ is an absolutely continuous random variable;

    2. 2.

$\lambda_{F}(c)$ is finite for every $c$ and has a unique root $t_{0}$;

    3. 3.

$c\mapsto\ell_{\delta}(x,c)$ is convex for every $x$;

    4. 4.

$x\mapsto\ell_{\delta}^{\prime}(x,c)$ is measurable for every $c$.

  * (A2)

$\mathbb{E}[\lvert\ell_{\delta}^{\prime}(X,c)\rvert]<\infty$.

  * (A3)

There exist functions $h_{1},h_{2}$ with $x\mapsto h_{j}(x,c)$ non-decreasing
and left continuous for $j=1,2$ such that
$\ell_{\delta}^{\prime}(x,c)=h_{1}(x,c)-h_{2}(x,c)$ for every $c$ and $x$.

  * (A4)

There exists an $r>0$, such that $\mathbb{E}[\lvert
h_{j}(X,c)\rvert^{r}]<\infty$ for $j=1,2$ with $h_{j}$ from (A3).

  * (A5)

There exist constants $M,\gamma>0$ such that $\lvert
d_{\boldsymbol{\tau}}(t)\rvert\leq M[t(1-t)]^{-1+1/r+\gamma}$ with $r$ from
(A4).

  * (A6)

$d_{\boldsymbol{\tau}}$ has a continuous derivative on $(0,1)$. Furthermore,
with $r$ from (A4) there exist $M,\gamma>0$ such that $\lvert
d_{\boldsymbol{\tau}}(t)\rvert\leq M[t(1-t)]^{-1/2+1/r+\gamma}$ and $\lvert
d_{\boldsymbol{\tau}}^{\prime}(t)\rvert\leq M[t(1-t)]^{-3/2+1/r+\gamma}$.

  * (A7)

$d_{\boldsymbol{\tau}}$ is continuous except at possibly a finite number of
points.

  * (A8)

    1. 1.

$d_{\boldsymbol{\tau}}$ is bounded on $[0,1]$;

    2. 2.

$d_{\boldsymbol{\tau}}$ is locally uniformly continuous except at a finite
number of points.

  * (A9)

$\lambda_{F}(c)$ is differentiable at $t_{0}$ with derivative
$\lambda_{F}^{\prime}(t_{0})\neq 0$.

  * (A10)

For any sequence $\omega_{n}\rightarrow 0$ as $n\to\infty$

$\sup_{\lvert
c-t_{0}\rvert\leq\omega_{n}}\frac{\sqrt{n}\lvert\lambda_{\widehat{F}_{n}}^{*}(c)-\lambda_{\widehat{F}_{n}}^{*}(t_{0})-\lambda_{F}(c)\rvert}{1+\sqrt{n}\lvert
c-t_{0}\rvert}\xrightarrow{\text{P}}0.$

Throughout this section we assume the existence (i.e. finiteness), as well as
uniqueness, of the target quantity of interest
$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})$ (also
shortly denoted as $t_{0}$).

Appendix B.2 contains the proofs of the results stated in the various lemmas
in this section. Lemma 3.1 expresses the consistency of $T_{n}^{*}$ when using
a general estimator $\widehat{F}_{n}$ of $F_{X}$, satisfying some condition.
Note that this lemma requires the Lipschitz continuity of
$d_{\boldsymbol{\tau}}$ which might be a quite restrictive condition. For this
reason, we also prove the consistency of $T_{n}^{*}$, when using as an
estimator for $F_{X}$ the empirical distribution function. This then allows to
show consistency under weaker conditions on $d_{\boldsymbol{\tau}}$. See Lemma
3.2.

######  Lemma 3.1.

Suppose Assumption (A1) and Assumption (A2) hold. Furthermore, assume that
$d_{\boldsymbol{\tau}}$ is Lipschitz continuous. Then for any estimator
$\widehat{F}_{n}$ of $F_{X}$ for which
$\sup_{x\in\mathbb{R}}\lvert\widehat{F}_{n}(x)-F_{X}(x)\rvert\xrightarrow{\text{a.s.}}0$
as $n\to\infty$, we have that
$T_{n}^{*}\xrightarrow{\text{a.s.}}e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})$
as $n\to\infty$.

######  Remark 3.1.

(i) Suppose in addition to the assumptions of Lemma 3.1 that
$\ell_{\delta}^{\prime}(x,c)$ is continuous in $c$ in a neighborhood of
$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})$, for
every $x$. Then $\lambda_{\widehat{F}_{n}}^{*}$ has a root and this root is
equal to $T_{n}^{*}$.
(ii) The requirement on the nonparametric estimator $\widehat{F}_{n}$ for
$F_{X}$ is satisfied by the usual empirical distribution function (or its
rescaled version $F_{n}$). Indeed, this is expressed by the Glivenko-Cantelli
theorem (see p. 62 in Serfling (1980)).

In Lemma 3.2, and in all subsequent results, we work with the estimator
$\widehat{F}_{n}(x):=F_{n}(x)=(n+1)^{-1}\sum_{i=1}^{n}1_{\\{X_{i}\leq x\\}}$.
This allows us to weaken the assumption on the density
$d_{\boldsymbol{\tau}}$.

######  Lemma 3.2.

Suppose Assumptions (A1), (A3), (A4), (A5) and (A7) hold, then
$T_{n}^{*}\xrightarrow{\text{a.s.}}e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})$
as $n\to\infty$.

We next turn to establishing an asymptotic normality result for the estimator
$T_{n}^{*}$. We therefore first prove the asymptotic normality of
$\lambda_{F_{n}}^{*}(t_{0})$. In Lemma 3.3 this is done under, among others,
the assumption that the density $d_{\boldsymbol{\tau}}$ has a continuous
derivative. In Lemma 3.4 this assumption is relaxed. To ease calculations in
practice, we provide two equivalent expressions for the asymptotic variance in
Lemma 3.3 (and subsequently Lemma 3.4).

######  Lemma 3.3.

Suppose Assumptions (A1), (A3), (A4) and (A6) are satisfied. Then
$\sqrt{n}\lambda_{F_{n}}^{*}(t_{0})\xrightarrow{\text{d}}\mathcal{N}(0;\sigma^{2}_{t_{0}})$
as $n\to\infty$, where

$\displaystyle\sigma^{2}_{t_{0}}$
$\displaystyle=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}[F_{X}(\min\\{x,y\\})-F_{X}(x)F_{X}(y)]\cdot
d_{\boldsymbol{\tau}}(F_{X}(x))d_{\boldsymbol{\tau}}(F_{X}(y))\mathrm{d}\ell_{\delta}^{\prime}(x,t_{0})\mathrm{d}\ell_{\delta}^{\prime}(y,t_{0})$
(17)
$\displaystyle=\sum_{i,j\in\\{1,2\\}}(-1)^{i+j}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}[\min\\{F_{h_{i}(X,t_{0})}(x),F_{h_{j}(X,t_{0})}(y)\\}-F_{h_{i}(X,t_{0})}(x)F_{h_{j}(X,t_{0})}(y)]$
$\displaystyle\hskip 142.26378pt\cdot
d_{\boldsymbol{\tau}}(F_{h_{i}(X,t_{0})}(x))d_{\boldsymbol{\tau}}(F_{h_{j}(X,t_{0})}(y))\mathrm{d}x\mathrm{d}y.$
(18)

As Lemma 3.3 does not allow for a discontinuous density
$d_{\boldsymbol{\tau}}$, it does not cover the case for
$D_{\boldsymbol{\tau}}$ corresponding to expected shortfall (entry number 5 in
Table 3). The following result covers this specific case.

######  Lemma 3.4.

Suppose Assumptions (A1), (A3), (A4) with $r>2$ and (A8) are satisfied. Then
$\sqrt{n}\lambda_{F_{n}}^{*}(t_{0})\xrightarrow{\text{d}}\mathcal{N}(0;\sigma^{2}_{t_{0}})$
as $n\to\infty$.

We can now state (and prove) the asymptotic normality result for $T_{n}^{*}$.

######  Theorem 3.2.

Suppose that in addition to Assumptions (A1), (A3), (A9) and (A10) one of the
following sets of assumptions is satisfied

  * •

(A4) and (A6)

  * or

  * •

(A4) with $r>2$ and (A8).

Then
$\sqrt{n}(T_{n}^{*}-t_{0})\xrightarrow{\text{d}}\mathcal{N}\left(0;\frac{\sigma^{2}_{t_{0}}}{[\lambda_{F}^{\prime}(t_{0})]^{2}}\right)$
as $n\to\infty$, with $\sigma^{2}_{t_{0}}$ as in (17) (or (18)) in Lemma 3.3.

###### Proof.

This follows by Theorem 7.2 on p. 2186 in Newey and McFadden (1986). See also
Theorem 3.3 on p. 1040 in Pakes and Pollard (1989). Both references state the
assumption (using our notations)

$(\lambda_{F_{n}}^{*}(T_{n}^{*}))^{2}\leq\inf_{c\in\mathbb{R}}\Big{(}\lambda_{F_{n}}^{*}(c)\Big{)}^{2}+o_{P}(n^{-1}).$

By definition of $T_{n}^{*}$ this is always satisfied. ∎

######  Remark 3.2.

(i) Assumption (A10) is a kind of equicontinuity assumption. In Section S2 of
the Supplementary Material it is shown that this assumption is satisfied for
several loss functions, among which the square loss, the Esscher loss (entry
number 6 in Table 1), and the loss function with entry G2 in Table 2.
(ii) In Section 3.1 we established an asymptotic normality result for the
considered estimators, in the specific setting of square loss
($\ell_{\delta}(x,c)=(x-c)^{2}$). See Theorem 3.1. Theorem 3.2 on the other
hand is for general loss functions. It is easily seen that in case of square
loss the expression for the asymptotic variance in (18) reduces to that in
(16). Indeed, note that for the square loss function:
$\ell_{\delta}^{\prime}(x,c)=-2(x-c)$ and $\lambda_{F}^{\prime}(c)=-2$. A
simple substitution into (18) leads to (16). A setting with square loss was
also considered in Jones and Zitikis (2003). Although the authors consider
another estimator, the asymptotic variances coincide. Since Jones and Zitikis
(2003) (see pp. 49–50) also discuss a consistent estimator for the asymptotic
variance, this estimator can also be used in our context.

###  3.3 Applications of the asymptotic normality result

The asymptotic normality results established in Theorem 3.1(ii) and Theorem
3.2 allow, for example, to create asymptotic (pointwise) confidence intervals
for the generalized extremile
$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})$.

In this section, we apply Theorem 3.2 for various loss functions, establishing
as such results on the corresponding estimator $T_{n}^{*}$ for the generalized
extremile
$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})$ under
study. In each of the illustrative examples, we focus on the asymptotic
normality result and in particular the expression for the asymptotic variance.

We start by looking into the two loss functions encountered when estimating
quantiles. Recall from Section 1 that quantiles of $X$ can be obtained using
at least two different pairs $(D_{\boldsymbol{\tau}},\ell_{\delta})$.

$\displaystyle\mbox{approach 1}:\qquad$ $\displaystyle
D_{\boldsymbol{\tau}}(u)=u\qquad\mbox{and}\qquad\ell_{\delta}(x,c)=\left|\delta-1_{\\{x\leq
c\\}}\right||x-c|$ (19) $\displaystyle\mbox{approach 2}:\qquad$ $\displaystyle
D_{\boldsymbol{\tau}}(u)=K_{\tau}(u)\quad\mbox{in
\eqref{DtauExtremiles}}\qquad\mbox{and}\qquad\ell_{\delta}(x,c)=|x-c|,$ (20)

where in approach 2, one takes $\tau=\delta$ to obtain the $\delta$th quantile
of $X$.

In Corollary 3.1 we consider an absolute value loss function, whereas in
Corollary 3.2 we apply Theorem 3.2 with the loss function
$\ell_{\delta}(x,c)=\left|\delta-1_{\\{x\leq c\\}}\right||x-c|$. Before
stating the first corollary we make an important remark.

######  Remark 3.3.

Consider the absolute value loss $\ell_{\delta}(x,c)=\lvert x-c\rvert.$ The
derivative with respect to $c$ of this function is not defined in the point
$c=x$. In order for $x\mapsto\ell_{\delta}^{\prime}(x,c)$ to be left
continuous (see Assumption (A3)), we will choose $\ell_{\delta}^{\prime}$ to
be the right-hand side derivative. In the specific case of the absolute value
loss, this means $\ell_{\delta}^{\prime}(x,c)=1_{\\{x\leq
c\\}}-1_{\\{x>c\\}}$. This choice for the right-hand side derivative is
justified since we are integrating with respect to $x$. Meaning that the
integrand is uniquely defined almost everywhere.

######  Corollary 3.1.

Consider the absolute value loss function $\ell_{\delta}(x,c)=\lvert
x-c\rvert.$ Consider a random variable $X$ and a cumulative distribution
function $D_{\boldsymbol{\tau}}$ such that the assumptions of Theorem 3.2 are
satisfied. Then the asymptotic variance reduces to

$\frac{\text{Median}(D_{\boldsymbol{\tau}})(1-\text{Median}(D_{\boldsymbol{\tau}}))}{\left[f_{X}\left(\mbox{Median}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}})\right)\right]^{2}},$ (21)

where $\mbox{Median}(D_{\tau})$ denotes the median of a random variable with
cumulative distribution function $D_{\boldsymbol{\tau}}$.

###### Proof.

We have $\ell^{\prime}_{\delta}(x,c)=1_{\\{x\leq c\\}}-1_{\\{x>c\\}}$, with
thus $x\mapsto\ell^{\prime}_{\delta}(x,c)$ non-increasing. This means we can
take $h_{1}(x,c)=0$ and $h_{2}(x,c)=-\ell^{\prime}_{\delta}(x,c)$ for
Assumption (A3). When integrating a function, say $k(s)$, with respect to this
measure, we get

$\int_{\mathbb{R}}\,k(s)\mathrm{d}\ell^{\prime}_{\delta}(s,t_{0})=-2k(t_{0}).$

This implies
$\sigma_{t_{0}}^{2}=4[F_{X}(t_{0})-F_{X}^{2}(t_{0})][d_{\boldsymbol{\tau}}(F_{X}(t_{0}))]^{2}$.
Further, we have that

$\lambda_{F}(c)=\int_{-\infty}^{c}d_{\boldsymbol{\tau}}(F_{X}(x))f_{X}(x)\mathrm{d}x-\int_{c}^{+\infty}d_{\boldsymbol{\tau}}(F_{X}(x))f_{X}(x)\mathrm{d}x=P\left\\{X_{{\scriptsize
D}_{\boldsymbol{\tau}}}\leq c\right\\}-P\left\\{X_{{\scriptsize
D}_{\boldsymbol{\tau}}}>c\right\\}=2D_{\boldsymbol{\tau}}\left(F_{X}(c)\right)-1,$

from which it follows that $t_{0}=\mbox{Median}\left(X_{{\scriptsize
D}_{\boldsymbol{\tau}}}\right)$, and also
$F_{X}(t_{0})=\mbox{Median}(D_{\boldsymbol{\tau}})$.

Furthermore
$\lambda_{F}^{\prime}(c)=2f_{X}(c)d_{\boldsymbol{\tau}}(F_{X}(c))$. In
conclusion, $\sigma_{t_{0}}^{2}/\left[\lambda_{F}^{\prime}(t_{0})\right]^{2}$
leads to expression (21). ∎

######  Remark 3.4.

When taking $D_{\boldsymbol{\tau}}(u)=K_{\tau}(u)$, expression (21) leads to
the asymptotic variance for the quantile estimator, obtained via approach 2
(see (20)). It is easily seen from (9) that $\mbox{Median}(K_{\tau})=\tau$ and
$\mbox{Median}\left(X_{{\scriptsize D}_{\tau}}\right)=q_{\tau}(X)$ the
$\tau$th order quantile of $X$. Hence for the setting of approach 2 in (20),
expression (21) reduces to

$\frac{\tau(1-\tau)}{\left[f_{X}\left(q_{\tau}(X)\right)\right]^{2}},$ (22)

which is the well-known asymptotic variance for the empirical quantile
estimator (see p. 77 in Serfling (1980)).

We next apply Theorem 3.2 for the loss function
$\ell_{\delta}(x,c)=\left|\delta-1_{\\{x\leq c\\}}\right||x-c|$, leading to
the statement in Corollary 3.2.

######  Corollary 3.2.

Consider the quantile loss function
$\ell_{\delta}(x,c)=\left|\delta-1_{\\{x\leq c\\}}\right||x-c|$ (with
$\delta\in(0,1)$). Consider a random variable $X$ and a cumulative
distribution function $D_{\boldsymbol{\tau}}$ such that the assumptions of
Theorem 3.2 are satisfied. Then the asymptotic variance reduces to

$\frac{F_{X}(q_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}}))\left[1-F_{X}(q_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}}))\right]}{\left[f_{X}(q_{\delta}(X_{{\scriptsize
D}_{\boldsymbol{\tau}}}))\right]^{2}}.$ (23)

###### Proof.

We have $\ell^{\prime}_{\delta}(x,c)=1_{\\{x\leq c\\}}-\delta$ (see also Table
4), with $x\mapsto\ell^{\prime}_{\delta}(x,c)$ non-increasing. This means we
can take $h_{1}(x,c)=0$ and $h_{2}(x,c)=-\ell^{\prime}_{\delta}(x,c)$ for
Assumption (A3). If we integrate with respect to this measure then

$\int k(s)\mathrm{d}\ell^{\prime}_{\delta}(s,t_{0})=-k(t_{0}).$

For this reason
$\sigma_{t_{0}}^{2}=[F_{X}(t_{0})-F^{2}_{X}(t_{0})][d_{\boldsymbol{\tau}}(F_{X}(t_{0}))]^{2}$.
Furthermore,

$\lambda_{F}(c)=\int_{-\infty}^{c}d_{\boldsymbol{\tau}}(F_{X}(x))f_{X}(x)\mathrm{d}x-\delta=P\left\\{X_{{\scriptsize
D}_{\boldsymbol{\tau}}}\leq c\right\\}-\delta,$

leading to $t_{0}=q_{\delta}(X_{{\scriptsize D}_{\boldsymbol{\tau}}})$, the
$\delta$th quantile of the random variable $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$. Moreover,
$\lambda_{F}^{\prime}(t_{0})=d_{\boldsymbol{\tau}}(F_{X}(t_{0}))f_{X}(t_{0})$.
This concludes the proof. ∎

######  Remark 3.5.

When estimating a quantile, via approach 1 (see (19)), we take
$D_{\boldsymbol{\tau}}(u)=u$, and hence $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}=X$, and expression (23) reduces to

$\displaystyle\frac{\delta(1-\delta)}{\left[f_{X}(q_{\delta}(X))\right]^{2}},$

corresponding to (22) (where herein $\tau=\delta$).

The asymptotic variance stated in Corollary 3.2 can be estimated by

$\frac{[F_{n}(T_{n}^{*})-(F_{n}(T_{n}^{*}))^{2}]}{\left[\widehat{f}_{X}(T_{n}^{*})\right]^{2}},$
(24)

where $F_{n}$ is the empirical cumulative distribution and $\widehat{f}_{X}$ a
kernel density estimator.

Corollary 3.3 results from applying Theorem 3.2 with the loss function
$\ell_{\delta}(x,c)=\left|\delta-1_{\\{x\leq c\\}}\right||x-c|^{2}$, i.e. the
expectile loss function (entry number 4 in Table 1).

######  Corollary 3.3.

Consider the loss function $\ell_{\delta}(x,c)=\left|\delta-1_{\\{x\leq
c\\}}\right||x-c|^{2}$ (with $\delta\in(0,1)$), corresponding to expectiles.
Consider a random variable $X$ and a cumulative distribution function
$D_{\boldsymbol{\tau}}$ such that the assumptions of Theorem 3.2 are
satisfied. Then the asymptotic variance reduces to $\sigma_{\mbox{{\scriptsize
E}}}^{2}=\mbox{N}_{\mbox{{\scriptsize E}}}/\mbox{D}_{\mbox{{\scriptsize E}}}$,
where

$\displaystyle\mbox{N}_{\mbox{{\scriptsize E}}}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}(-2\delta
1_{\\{y>t_{0}\\}}+(2\delta-2)1_{\\{y<t_{0}\\}})\int_{-\infty}^{+\infty}(-2\delta
1_{\\{x>t_{0}\\}}+(2\delta-2)1_{\\{x<t_{0}\\}})$ $\displaystyle\hskip
28.45274pt\times[F_{X}(\min\\{x,y\\})-F_{X}(x)F_{X}(y)]d_{\boldsymbol{\tau}}(F_{X}(x))d_{\boldsymbol{\tau}}(F_{X}(y))\mathrm{d}x\mathrm{d}y$
$\displaystyle=$ $\displaystyle
4\delta^{2}\int_{t_{0}}^{+\infty}\int_{t_{0}}^{+\infty}[F_{X}(\min\\{x,y\\})-F_{X}(x)F_{X}(y)]d_{\boldsymbol{\tau}}(F_{X}(x))d_{\boldsymbol{\tau}}(F_{X}(y))\mathrm{d}x\mathrm{d}y$
$\displaystyle\quad-8\delta(\delta-1)\int_{t_{0}}^{+\infty}\int_{-\infty}^{t_{0}}[F_{X}(x)-F_{X}(x)F_{X}(y)]d_{\boldsymbol{\tau}}(F_{X}(x))d_{\boldsymbol{\tau}}(F_{X}(y))\mathrm{d}x\mathrm{d}y$
$\displaystyle\quad+4(\delta-1)^{2}\int_{-\infty}^{t_{0}}\int_{-\infty}^{t_{0}}[F_{X}(\min\\{x,y\\})-F_{X}(x)F_{X}(y)]d_{\boldsymbol{\tau}}(F_{X}(x))d_{\boldsymbol{\tau}}(F_{X}(y))\mathrm{d}x\mathrm{d}y,$
$\displaystyle\mbox{D}_{\mbox{{\scriptsize E}}}$ $\displaystyle=$
$\displaystyle(-4\delta
D_{\boldsymbol{\tau}}(F_{X}(t_{0}))+2\delta+2D_{\boldsymbol{\tau}}(F_{X}(t_{0})))^{2}.$

###### Proof.

We have $\ell^{\prime}_{\delta}(x,c)=-2\delta\lvert
x-c\rvert-2(x-c)1_{\\{x\leq c\\}}$, for which
$x\mapsto\ell^{\prime}_{\delta}(x,c)$ is non-increasing. This means we can
take $h_{1}(x,c)=0$ and $h_{2}(x,c)=-\ell^{\prime}_{\delta}(x,c)$ for
Assumption (A3). If we integrate with respect to this measure, then we obtain

$\int_{a}^{b}g(x)\mathrm{d}\ell^{\prime}_{\delta}(x,c)=\int_{a}^{b}(-2\delta
1_{\\{x>c\\}}+(2\delta-2)1_{\\{x<c\\}})g(x)\mathrm{d}x.$

This explains the factors in the numerator. By expanding the factors we obtain
the second expression for the numerator $\mbox{N}_{\mbox{{\scriptsize E}}}$.
Here we take into account that by changing the order of integration

$\displaystyle
4\delta(\delta-1)\int_{t_{0}}^{+\infty}\int_{-\infty}^{t_{0}}[F_{X}(\min\\{x,y\\})-F_{X}(x)F_{X}(y)]d_{\boldsymbol{\tau}}(F_{X}(x))d_{\boldsymbol{\tau}}(F_{X}(y))\mathrm{d}x\mathrm{d}y$
$\displaystyle=4\delta(\delta-1)\int_{-\infty}^{t_{0}}\int_{t_{0}}^{+\infty}[F_{X}(\min\\{x,y\\})-F_{X}(x)F_{X}(y)]d_{\boldsymbol{\tau}}(F_{X}(x))d_{\boldsymbol{\tau}}(F_{X}(y))\mathrm{d}y\mathrm{d}x.$

This explains the term with factor $8\delta(\delta-1)$. ∎

######  Remark 3.6.

When using the expectile loss function, the corresponding generalized
extremile equals the $\delta$th expectile of $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$. Hence, if we consider $D_{\boldsymbol{\tau}}$ the
uniform cumulative distribution function, we can use our estimator to estimate
the $\delta$th expectile of $X$. Doing so, our estimator coincides with the
estimator studied in Holzmann and Klar (2016). The latter authors also
established the asymptotic normality of their estimator (see p. 2359 in
Holzmann and Klar (2016)), with asymptotic variance given by

$\frac{\mathbb{E}\left[\left(I_{\delta}(t_{0},X)\right)^{2}\right]}{\left[\delta(1-F_{X}(t_{0}))+(1-\delta)(F_{X}(t_{0}))\right]^{2}},$
(25)

where

$I_{\delta}(t_{0},X)=\delta(X-t_{0})1_{\\{X\geq
t_{0}\\}}-(1-\delta)(t_{0}-X)1_{\\{X<t_{0}\\}}.$

In Section S3 of the Supplementary Material we prove that the asymptotic
variance as stated in Corollary 3.3, equals that in (25), for the specific
case that $D_{\boldsymbol{\tau}}(u)=u$. Hence the asymptotic normality result
for expectiles is obtained as special case of our Theorem 3.2 and Corollary
3.3.

Applying Theorem 3.2 to the loss function $\ell_{\delta}(x,c)=-c\lvert
x-b\rvert^{\delta}+c^{2}/2$ establishes the asymptotic normality results for
the proposed estimator for $\mathbb{E}[\lvert X_{{\scriptsize
D}_{\boldsymbol{\tau}}}-b\rvert^{\delta}]$.

######  Corollary 3.4.

Consider the loss function $\ell_{\delta}(x,c)=-c\lvert
x-b\rvert^{\delta}+c^{2}/2$, for $\delta>0$. This corresponds to
$t_{0}=\mathbb{E}[\lvert X_{{\scriptsize
D}_{\boldsymbol{\tau}}}-b\rvert^{\delta}]$. Consider a random variable $X$ and
a cumulative distribution function $D_{\boldsymbol{\tau}}$ such that the
assumptions of Theorem 3.2 are satisfied. Then the asymptotic variance equals

$\displaystyle\int_{-\infty}^{t_{0}}\int_{-\infty}^{t_{0}}[F_{h_{1}(X,t_{0})}(\min\\{x,y\\})-F_{h_{1}(X,t_{0})}(x)F_{h_{1}(X,t_{0})}(y)]d_{\boldsymbol{\tau}}(F_{h_{1}(X,t_{0})}(x))d_{\boldsymbol{\tau}}(F_{h_{1}(X,t_{0})}(y))\mathrm{d}x\mathrm{d}y$
$\displaystyle+\int_{0}^{+\infty}\int_{0}^{+\infty}[F_{h_{2}(X,t_{0})}(\min\\{x,y\\})-F_{h_{2}(X,t_{0})}(x)F_{h_{2}(X,t_{0})}(y)]d_{\boldsymbol{\tau}}(F_{h_{2}(X,t_{0})}(x))d_{\boldsymbol{\tau}}(F_{h_{2}(X,t_{0})}(y))\mathrm{d}x\mathrm{d}y$
$\displaystyle-2\int_{0}^{+\infty}\int_{-\infty}^{t_{0}}[\min\\{F_{h_{1}(X,t_{0})}(x),F_{h_{2}(X,t_{0})}(y)\\}-F_{h_{1}(X,t_{0})}(x)F_{h_{2}(X,t_{0})}(y)]d_{\boldsymbol{\tau}}(F_{h_{1}(X,t_{0})}(x))d_{\boldsymbol{\tau}}(F_{h_{2}(X,t_{0})}(y))\mathrm{d}x\mathrm{d}y,$

where

$F_{h_{1}(X,t_{0})}(z)=\begin{cases}1&\text{ if }z\geq t_{0}\\\
F_{X}(b-(t_{0}-z)^{1/\delta})&\text{ if }z<t_{0}\end{cases}$

and

$F_{h_{2}(X,t_{0})}(z)=\begin{cases}0&\text{ if }z<0\\\
F_{X}(z^{1/\delta}+b)&\text{ if }z\geq 0.\end{cases}$

###### Proof.

We calculate $\sigma_{t_{0}}^{2}$ using (18), where we use
$\ell_{\delta}^{\prime}(x,c)=h_{1}(x,c)-h_{2}(x,c)$ with

$h_{1}(x,c)=-(-x+b)^{\delta}1_{\\{x\leq b\\}}+c\hskip 14.22636pt\text{ and
}\hskip 14.22636pth_{2}(x,c)=(x-b)^{\delta}1_{\\{x>b\\}},$

non-decreasing and left-continuous in $x$. First, we find an expression for
$F_{h_{1}(X,t_{0})}(z)$. This is done by observing that $h_{1}(x,t_{0})\leq
t_{0}$, consequently $F_{h_{1}(X,t_{0})}(z)=1$ for $z\geq t_{0}$. For
$z<t_{0}$ the result follows by noting that

$F_{h_{1}(X,t_{0})}(z)=P((-X+b)^{\delta}1_{\\{X\leq
b\\}}\geq-z+t_{0})=P(-X+b\geq(-z+t_{0})^{1/\delta})=F_{X}(b-(t_{0}-z)^{1/\delta}).$

Similarly, we obtain the stated expression for $F_{h_{2}(X,t_{0})}(z)$.
Furthermore, it is clear that $\lambda_{F}^{\prime}(c)=1$, and application of
Theorem 3.2 concludes the proof. ∎

##  4 Simulation study

###  4.1 Aims of the simulation study, and simulation models

In this section we investigate the finite-sample behaviour of the estimator
$T_{n}^{*}$ of the generalized extremile
$e_{\boldsymbol{\tau},\delta}(X;D_{\boldsymbol{\tau}},\ell_{\delta})$. We
conduct five small simulation studies considering different pairs for
$(D_{\boldsymbol{\tau}},\ell_{\delta})$, and hence looking into estimating
different quantities. Tables 6 and 7 summarize the settings for the various
parts (simulation studies (St) 1–5). The various scenarios have the following
aims: investigating

  * St1

the performance of $T_{n}^{*}$ for different choices of $F_{X}$;

  * St2

the performance of $T_{n}^{*}$ for different choices of parameters
$(\tau,\delta)$;

  * St3

estimation of quantiles via two different pairs
$(D_{\boldsymbol{\tau}},\ell_{\delta})$;

  * St4

the asymptotic normality result for quantile estimation under censoring;

  * St5

the asymptotic normality result when using as loss function $-c\lvert
x\rvert+c^{2}/2$ (see Corollary 3.4).

In each simulation study we draw 500 samples of size $n$ from the simulation
model. We adapt various ways to present the results. We evaluate the finite-
sample performance of $T_{n}^{*}$ by reporting on the bias, variance and Mean
Squared Error (MSE) across the Monte Carlo runs. For illustration of the
sampling distribution of $T_{n}^{*}$ we present a kernel density estimate
based on the 500 values for $\sqrt{n}(T_{n}^{*}-t_{0})$.

Table 6: Model elements in Simulation studies 1–3 and 5. | Simulation study (St)
---|---
| St1 | St2 | St3 | St5
$\ell_{\delta}(x,c)$ | expectile | expectile | absolute value | quantile |
| loss | loss | loss | loss | loss
| $\left|\delta-1_{\\{x\leq c\\}}\right||x-c|^{2}$ | same as in St1 | $|x-c|$ | $\left|\delta-1_{\\{x\leq c\\}}\right||x-c|$ | $-c|x|+c^{2}/2$
| $\delta=0.9$ | $\delta=0.1,0.9,0.95$ |  | $\delta=0.01,0.05,0.1,$ |
|  |  |  | $\,0.5,0.9,0.95,0.99$ |
$D_{\boldsymbol{\tau}}(u)$ |  $K_{\tau}(u)$ in (9) | $D_{\tau}(u)$ |  $K_{\tau}(u)$ in (9) | $D_{\boldsymbol{\tau}}(u)=u$ | same as in St2
| $\tau=0.1,0.9,0.95$ | $\,\,=(1-\tau)^{-1}(u-\tau)\,1_{\\{u\geq\tau\\}}$ | $\tau=0.01,0.05,0.1,$ |  |
|  | $\tau=0.1,0.9,0.95$ | $\,0.5,0.9,0.95,0.99$ |  |
$F_{X}$ | $X\sim\mathcal{N}(0;1)$ | $X\sim\mathcal{N}(0;1)$ | $X\sim\mbox{Expo}(1)$ | same as in St1
|  or $X\sim\mbox{Expo}(1)$ |  |  |  |
$n$ | $n=50\,\mbox{or}\,800$ | same as in St1 | $n=50\,\mbox{or}\,400$ | same as in St1
Table 7: Model elements in Simulation study 4 (St4). $\ell_{\delta}(x,c)$ | $D_{\boldsymbol{\tau}}(u)$ | $F_{X}$ | $F_{C}$ | $Y$ | $n$
---|---|---|---|---|---
adapted quantile | $D_{\boldsymbol{\tau}}(u)=u$ | $X\sim\mbox{Expo}(1)$ | $C\sim\mbox{Expo}(\frac{p_{C}}{1-p_{C}})$ | $Y=\min(X,C)$ | $n=100\,\mbox{or}\,400$
loss; see (12) |  |  | $p_{C}=0.1,0.3,0.5$ |  |
$\delta=0.1,0.5$ |  |  |  |  |

###  4.2 Simulation results

#### Simulation study 1

Figure 2 shows a kernel density estimate of $\sqrt{n}(T_{n}^{*}-t_{0})$ for
$n=50,800$. The left (respectively right) panels are for samples drawn from
$X\sim\mathcal{N}(0,1)$ (respectively $X\sim\mbox{Expo}(1)$). Presented are
the results for sample size $n=50$ (blue lines) and $n=800$ (red lines),
together with the normal density with variance equal to the asymptotic
variance of Theorem 3.2. Note that with increasing $n$ the kernel density
estimates approach the asymptotically normal density. To be noted is also that
for values of $\tau$ closer to 1, the variance of the sampling distribution of
$T_{n}^{*}$ as well as the asymptotic variance is largest (see the scale of
the horizontale axis). Table S1 in the Supplementary Material provides the
estimated bias, variance and MSE of the estimator $T_{n}^{*}$.

Figure 2: Simulation study 1. Density estimates based on 500 values of
$\sqrt{n}(T_{n}^{*}-t_{0})$ for $n=50$ and $n=800$. Samples drawn from
$X\sim\mathcal{N}(0;1)$ (left panels); and from $X\sim\mbox{Expo}(1)$ (right
panels). Black lines: the normal density with variance equal to the asymptotic
variance of Theorem 3.2.

#### Simulation study 2

In this part, the function $D_{\tau}$ corresponds to the distortion function
of expected shortfall (see entry 5 in Table 3). Several values of the
parameters $(\tau,\delta)$ of respectively the distribution and the loss
function are considered. The simulation results are summarized in Figure 3 as
boxplots of $T_{n}^{*}-t_{0}$ (with $t_{0}$ the true generalized extremile)
over all Monte Carlo runs. The value of $t_{0}$ is indicated on the right-hand
side of each plot. Note that when passing from sample size $n=50$ to $n=800$
all boxplots become narrower, and more positioned around zero. For $n=50$, the
median is close to zero for the cases where $\delta=0.1$ and/or $\tau=0.1$. In
all cases the boxplots are relatively symmetric. As already seen from Figure
2, for fixed $\delta$, our estimates are more variable for larger values of
$\tau$. Similarly, for fixed $\tau$, the estimates are more variable for
larger $\delta$.

Figure 3: Simulation study 2. Finite-sample performance of the estimator
$T_{n}^{*}$. Boxplots of $T_{n}^{*}-t_{0}$ based on 500 samples of size
$n=50,800$ drawn from $X\sim\mathcal{N}(0;1)$ for different combination of
parameters $(\delta,\boldsymbol{\tau})$. The true value of $t_{0}$ is
indicated next to each boxplot.

#### Simulation study 3

As already mentioned at the start of Section 3.3 quantiles can be estimated
from two viewpoints, i.e. using two different pairs
$(D_{\boldsymbol{\tau}},\ell_{\delta})$. We present simulation results of the
two resulting estimators:

  * Estimator 1: based on using $D_{\boldsymbol{\tau}}(u)=u$ and the quantile loss function $\ell_{\delta}(x,c)=|\delta-1_{\\{x\leq c\\}}\,|x-c|$;

  * Estimator 2: based on using $D_{\boldsymbol{\tau}}=K_{\tau}$ and absolute value loss function $\ell_{\delta}(x,c)=|x-c|$.

By Corollaries 3.1 and 3.2, and Remarks 3.4 and 3.5 we know that both
estimators are asymptotically equivalent to the empirical quantile estimator.

Table 8 lists the mean squared error (MSE) for the two estimators, together
with their Monte Carlo approximated variance (between brackets). The results
for both estimators for sample size $n=50$ are identical, except for quantile
level $0.01$. When the sample size increases to $n=400$, the estimators are
indistinguishable when only considering the mean squared error. The difference
between the two estimators for quantile level $0.01$ is due to a larger bias
of estimator 1.

Table 8: Simulation study 3. Mean squared error (MSE) of two different quantile estimators ($\tau$th = $\delta$th quantile). The approximate variance of the estimators is given between brackets. | Estimator 2
---|---
$\boldsymbol{\tau}$ | $n=50$ | $n=400$
$0.01$ |  $1.01\cdot 10^{-3}$ $(5.65\cdot 10^{-4})$ | $3.68\cdot 10^{-5}(3.08\cdot 10^{-5})$
$0.05$ |  $1.99\cdot 10^{-3}$ $(1.49\cdot 10^{-3})$ |  $1.33\cdot 10^{-4}$ $(1.28\cdot 10^{-4})$
$0.1$ |  $2.53\cdot 10^{-3}$ $(2.35\cdot 10^{-3})$ |  $2.69\cdot 10^{-4}$ $(2.69\cdot 10^{-4})$
$0.5$ |  $1.98\cdot 10^{-2}$ $(1.96\cdot 10^{-2})$ |  $2.62\cdot 10^{-3}$ $(2.61\cdot 10^{-3})$
$0.9$ |  $1.66\cdot 10^{-1}$ $(1.66\cdot 10^{-1})$ |  $2.27\cdot 10^{-2}$ $(2.25\cdot 10^{-2})$
$0.95$ |  $3.08\cdot 10^{-1}$ $(2.69\cdot 10^{-1})$ |  $4.70\cdot 10^{-2}$ $(4.67\cdot 10^{-2})$
$0.99$ | 1.26 $(9.29\cdot 10^{-1})$ |  $1.97\cdot 10^{-1}$ $(1.97\cdot 10^{-1})$
| Estimator 1
$\boldsymbol{\tau}$ | $n=50$ | $n=400$
0.01 | $9.54\cdot 10^{-1}(6.75\cdot 10^{-4})$ | $3.68\cdot 10^{-5}(3.08\cdot 10^{-5})$
Other | Identical to results for Estimator 2

#### Simulation study 4

The adaptive loss function for quantile estimation in case of censoring was
briefly discussed in Section 2.4. This loss function is not convex, and hence
does not satisfy the conditions stated in Theorem 3.2. Since these are however
sufficient conditions, we might wonder whether the finite-sample distribution
can nevertheless be well approximated with the asymptotic normal distribution
as stated in the theorem.

In the right-random censoring setting, the variable of interest $X$ is
possibly censored by a variable $C$. We are interested in a $\delta$th
quantile of $X$, but only have data of the form $(Y,\Delta_{C})$, where
$Y=\min\\{X,C\\}$ and $\Delta_{C}=1_{\\{X\leq C\\}}$. Given a sample from
$(Y,\Delta_{C})$, we can use our estimator to estimate such a $\delta$th
quantile of $X$. Table 7 summarizes the various elements of the simulation
model.

When calculating the estimator we have to deal with the non-convexity of
$\lambda_{F}(c)$ and $\lambda_{\widehat{F}_{n}}^{*}(c)$. Globally minimizing
$\lambda_{\widehat{F}_{n}}^{*}(c)$ is not feasible, and instead we implemented
the following procedure. Based on a sample from $Y$, we construct a sequence
$\boldsymbol{c}=(c_{1},\cdots,c_{k})$ of values ranging from $\min(Y_{i})$ to
$\max(Y_{i})$. More precisely we take a equispaced grid of points, with an
increment of 0.01, i.e. for $m,m^{*}\in\mathbb{N}$, with $m^{*}>m$, we have
$c_{m^{*}}-c_{m}=(m^{*}-m)0.01$. We then search for the smallest value in this
sequence, say $c_{m}$, which satisfies the following inequalities

$\frac{\lambda_{\widehat{F}_{n}}^{*}(c_{m+j})-\lambda_{\widehat{F}_{n}}^{*}(c_{m})}{j\cdot
0.01}>0.01\quad\text{ for }j=1,\cdots,20.$

This requires difference quotients (discrete slopes) to be strictly positive
in an interval past a given point. In this manner, we search for a (local)
minimum. By considering multiple subsequent difference quotients, we avoid
selecting a value due to a small spike in the function
$\lambda_{\widehat{F}_{n}}^{*}(c)$.

Recall that $f_{X},F_{X}$ denote respectively the density and the cumulative
distribution function of $X$. Similarly, we use the notations $f_{C},F_{C}$
for these quantities for the censoring variable $C$. The asymptotic variance
stated in Theorem 3.2 equals in this case

$\frac{F_{X}(t_{0})(1-F_{X}(t_{0}))}{(f_{X}(t_{0})-(1-\delta)f_{C}(t_{0}))^{2}}.$
(26)

When there is no censoring (meaning that $F_{C}(t)=0$ for all $t$), this
expression reduces to the asymptotic variance of the empirical quantile
estimator.

Sander (1975) estimated a quantile for censored data relying on the Kaplan-
Meier estimator for a distribution function. We refer to this quantile
estimator as the Kaplan Meier-based (KM-based) estimator. The author
established asymptotic normality for the estimator (see Corollary 1 on p. 5 in
Sander (1975)). The asymptotic variance of this estimator equals

$\frac{(1-\delta)^{2}}{\left(f_{X}(t_{0})\right)^{2}}\int_{0}^{t_{0}}\frac{f_{X}(x)}{(1-F_{X}(x))^{2}(1-F_{C}(x))}\mathrm{d}x.$
(27)

Again, in case of no censoring, this asymptotic variance reduces to the one of
the empirical quantile estimator.

Table 9: Simulation study 4. Mean squared error (MSE) for the loss-based estimator and the KM-based estimator for different censoring proportions $p_{C}$, different quantile levels $\delta$ and different sample sizes $n$. Variances are given between brackets, and should be multiplied with the same factor $10^{-a}$ as appearing in the corresponding MSE-value.  |  | MSE Loss-based estimator | MSE KM-based estimator
---|---|---|---
$p_{C}$ | $\delta$ | $n=100$ | $n=400$ | $n=100$ | $n=400$
$0.1$ |  |  |  |  |  |  |  |  |
$0.1$ | $1.43\cdot 10^{-3}$ | ($1.39$) | $3.53\cdot 10^{-4}$ | ($3.52$) | $1.14\cdot 10^{-3}$ | ($1.13$) | $3.17\cdot 10^{-4}$ | ($3.17$)
$0.5$ | $1.20\cdot 10^{-2}$ | ($1.19$) | $2.90\cdot 10^{-3}$ | ($2.90$) | $9.90\cdot 10^{-3}$ | ($9.85$) | $2.51\cdot 10^{-3}$ | ($2.51$)
$0.3$ |  |  |  |  |  |  |  |  |
$0.1$ | $2.09\cdot 10^{-3}$ | ($2.01$) | $4.78\cdot 10^{-4}$ | ($4.76$) | $1.23\cdot 10^{-3}$ | ($1.23$) | $3.13\cdot 10^{-4}$ | ($3.12$)
$0.5$ | $1.74\cdot 10^{-2}$ | ($1.73$) | $4.24\cdot 10^{-3}$ | ($4.24$) | $1.15\cdot 10^{-2}$ | ($1.15$) | $2.76\cdot 10^{-3}$ | ($2.76$)
$0.5$ |  |  |  |  |  |  |  |  |
$0.1$ | $4.23\cdot 10^{-3}$ | ($3.98$) | $6.57\cdot 10^{-4}$ | ($6.40$) | $1.41\cdot 10^{-3}$ | ($1.37$) | $3.40\cdot 10^{-4}$ | ($3.38$)
$0.5$ | $3.34\cdot 10^{-2}$ | ($3.28$) | $8.37\cdot 10^{-3}$ | ($8.32$) | $1.69\cdot 10^{-2}$ | ($1.65$) | $3.64\cdot 10^{-3}$ | ($3.64$)
Figure 4: Simulation 4. Asymptotic variance expressions, i.e. (26) and (27)
for different values of $\delta$ and different censoring proportions $p_{C}$.
Distributions of $X$ and $C$ as in Table 7.

Figure S2 in the Supplementary Material depicts the density estimates of the
loss-based estimator $\sqrt{n}(T_{n}^{*}-t_{0})$, as well of the KM-based
estimator. The figure also superimposes the asymptotic normal distributions
with asymptotic variances as in (26) and (27) respectively. These results
confirm the appropriateness of the asymptotic variance (26), and seem to
indicate that the asymptotic normality result of Theorem 3.2 holds, even in
this case where the loss function does not satisfy the stipulated conditions.

Some findings from the simulation results are as follows.

  * $\bullet$

The KM-based estimator shows a smaller (finite-sample) variance compared to
the loss-based estimator.

  * $\bullet$

For the different censoring proportions, the asymptotic variance of both
estimators is larger when $\delta$ is $0.5$.

  * $\bullet$

The finite-sample variances of both estimators increase with increasing
censoring proportion.

Table 9 provides a summary of bias, variance and MSE for both estimators, and
confirms the visual findings from Figure S2.

Figure 4 presents the asymptotic variances in (26) and (27) for different
quantile levels $\delta$ and different censoring proportions $p_{C}$. As to be
expected the (asymptotic) variances of both estimators increase as the
censoring proportion increases. For small censoring proportion $p_{C}=0.1$ the
asymptotic variances are almost identical. As the censoring proportion
increases, so does the difference between the two asymptotic variances. For
larger censoring proportions $p_{C}$ it becomes more advantageous to use the
KM-based estimator. We end this simulation regarding censored quantiles by
noting that in De Backer et al. (2019), the authors established, in a linear
regression setting, asymptotic normality of an estimator using the adapted
quantile loss. They coped with non-convexity by using a majorize-minimize (MM)
algorithm.

#### Simulation study 5

With this simulation we aim to illustrate the result in Corollary 3.4. Figure
5 shows density estimates (based on 500 values) of $\sqrt{n}(T_{n}^{*}-t_{0})$
for $n=50,800$. The left panels are for samples drawn from
$X\sim\mathcal{N}(0;1)$. The right panels are for samples drawn from
$X\sim\mbox{Expo}(1)$. The normal density with variance equal to the
asymptotic variance is presented as the black lines.

Figure 5: Simulation study 5. Finite-sample distribution of the estimator
$T_{n}^{*}$. Density estimates of 500 values of $\sqrt{n}(T_{n}^{*}-t_{0})$
for $n=50$ and $n=800$. Samples drawn from $X\sim\mathcal{N}(0,1)$ (left
panels); and from $X\sim\mbox{Expo}(1)$ (right panels). In black, the normal
density with variance equal to the asymptotic variance of Theorem 3.2.

Note that even for small sample size $n=50$, there is a good correspondence
between the finite-sample distribution and the asymptotic disribution. Recall
from Section S2.2 that the equicontinuity assumption of Theorem 3.2 is
satisfied under the sufficient condition that $d_{\tau}$ is Lipschitz
continuous. Despite the fact that this not holds for $d_{\tau}$ in this
simulation model, the asymptotic normality result still seems to hold.

##  5 General considerations for risks of distorted random variables

In this section we establish some general properties about risk measures
$\rho(X_{{\scriptsize D}_{\boldsymbol{\tau}}})$. In the real data example in
Section 6 we refer to these results. For notational simplicity we simply write
$X_{{\scriptsize D}}$ instead of $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$ in
this section. Consider a random variable $X$ defined on the probability space
$(\Omega,\mathcal{F},P)$, with $\mathcal{F}$ the sigma-algebra, and $P$ the
probability measure. Denote by $\mathcal{X}$ the linear space of random
variables defined on $(\Omega,\mathcal{F},P)$, and for a given
$D\in\mathbb{D}$ consider a risk measure

$\begin{array}[]{ll}\rho:&\mathcal{X}\to\mathbb{R}\\\
&X\to\rho(X_{{\scriptsize D}}).\end{array}$

Before stating the main result of this section, we recall some basic
properties of a risk measure $\rho$.

  * $\star$

A risk measure $\rho$ is called monotone if: $\forall
X,Y\in\mathcal{X}:X\stackrel{{\scriptstyle
a.s.}}{{\leq}}Y\Longrightarrow\rho(X)\leq\rho(Y)$.

  * $\star$

A risk measure $\rho$ is called translation invariant if: $\forall
X\in\mathcal{X}$, $c\in\mathbb{R}:\rho(X+c)=\rho(X)+c$.

  * $\star$

A risk measure $\rho$ is called positive homogeneous if: $\forall
X\in\mathcal{X}$, $a\in[0,+\infty)$, such that
$aX\in\mathcal{X}:\rho(aX)=a\rho(X)$.

  * $\star$

A risk measure $\rho$ is called subadditive if: $\forall
X,Y,X+Y\in\mathcal{X}$ it holds that $\rho(X+Y)\leq\rho(X)+\rho(Y)$.

A risk measure $\rho$ is coherent when it is monotone, translation invariant,
positive homogeneous and subadditive.

Theorem 5.1 investigates risk measures for the distorted random variable
$X_{{\scriptsize D}}$. In establishing some of these statements, we can rely
on results of Liu et al. (2021). The proof of the theorem is provided in
Appendix C.

######  Theorem 5.1.

Denote by $D$ a distortion function and by $\rho$ a risk measure defined on
$\mathcal{X}$, the linear vector space of random variables. For two random
variables $X$ and $Y$ in the domain of $\rho$ we then have:

  * (i)

If $\rho$ is positive homogeneous, then also $\rho((aX)_{{\scriptsize
D}})=a\rho(X_{{\scriptsize D}})$ for $a\in[0,\infty)$.

  * (ii)

If $\rho$ is translation invariant, then also $\rho((X+c)_{{\scriptsize
D}})=\rho(X_{{\scriptsize D}})+c$ for $c\in\mathbb{R}$.

  * (iii)

If $\rho$ is monotone and $X\stackrel{{\scriptstyle a.s.}}{{\leq}}Y$ then
$\rho(X_{{\scriptsize D}})\leq\rho(Y_{{\scriptsize D}})$.

  * (iv)

Suppose $\rho_{\delta}$ is coherent. If $D$ is convex, then the composition
(distortion) risk measure is also coherent (and thus in particular
subadditive), i.e. for all $X$ and $Y$, we have $\rho((X+Y)_{{\scriptsize
D}})\leq\rho(X_{{\scriptsize D}})+\rho(Y_{{\scriptsize D}})$.

  * (v)

Denote by $D_{1}$ and $D_{2}$ two distortion functions such that $D_{1}(u)\leq
D_{2}(u)$ for all $u\in[0,1]$. If $\rho$ is monotone, then
$\rho(X_{{\scriptsize D}_{2}})\leq\rho(X_{{\scriptsize D}_{1}})$. Specifically
for $D_{1}(u)\leq u\leq D_{2}(u)$ we have $\rho(X_{{\scriptsize
D}_{2}})\leq\rho(X)\leq\rho(X_{{\scriptsize D}_{1}})$.

  * (vi)

Suppose that the risk measure $\rho$ depends on a parameter $\gamma$, and
denote the risk measure by $\rho_{\gamma}$. If $\rho_{\gamma}$ is monotone
increasing in $\gamma$, then so is the composition risk measure
$\rho_{\gamma}(X_{{\scriptsize D}})$.

Risk measures defined as the minimizer of an expected loss minimization
procedure as in (6) avoid unjustified risk-loading under general conditions.
If $\ell$ is such that $\arg\min_{c\in\mathbb{R}}\ell(x,c)=x$ for all
$x\in\mathbb{R}$ then we have for an almost surely constant random variable
$X\sim\mathds{1}_{[b,\infty)}$, i.e., $X\stackrel{{\scriptstyle a.s.}}{{=}}b$
for some $b\in\mathbb{R}$, that
$\rho(X)=\arg\min_{c\in\mathbb{R}}\mathbb{E}[\ell(X,c)]=\arg\min_{c\in\mathbb{R}}\ell(b,c)=b$.
This property also carries over to the distorted version $X_{{\scriptsize
D}}$, where we have ${\scriptsize
D}\circ\mathds{1}_{[b,\infty)}=\mathds{1}_{[b,\infty)}$ due to $D(0)=0$ and
$D(1)=1$. As such we have $X_{{\scriptsize D}}\stackrel{{\scriptstyle
a.s.}}{{=}}b$ and hence also $\rho(X_{{\scriptsize D}})=b$.

##  6 Real data example: US disaster data 1980-2023

As an illustration we include some analysis of a dataset from NOAA National
Centers for Environmental Information (2023) (NCEI). The data concern
measurements on weather and climate disasters in the U.S. since 1980,
restricted to disasters that led to an overall damage/costs of more than or
equal to $1 billion (Consumer Price Index (CPI) adjusted to 2023). There are
360 such disasters amounting to a total cost of $2575,7 billion. Each disaster
event has been classified into one of seven classes. Some summary statistics
regarding the different (disaster) events are in Table 10. Our analysis mainly
focusses on the three disaster types with the largest number of observations:
severe storms, tropical cyclones and flooding. Per disaster type, we treat the
entries as outcomes of i.i.d. nonnegative random variables
$X_{1},\cdots,X_{n}$. The left panels of Figure 6 display histograms and
scatter plots of the data.

Table 10: US disaster data. Summary statistics. Columns indicated with ∗:
numbers are in billions of US dollars.

Disaster Type | Events | Events/Year | Total Costs∗ | Cost/Event∗ | Cost/Year∗ | Variance/Event∗
---|---|---|---|---|---|---
Drought | 30 | 0.7 | $334,8 | $11,2 | $7,6 | 12,2
Flooding | 41 | 0.9 | $190,2 | $4,6 | $4,3 | 7,2
Freeze | 9 | 0.2 | $36,0 | $4,0 | $0,8 | 2,2
Severe Storm | 177 | 4.0 | $423,1 | $2,4 | $9,6 | 1,9
Tropical Cyclone | 60 | 1.4 | $1359,0 | $22,7 | $30,9 | 38,5
Wildfire | 21 | 0.5 | $135,5 | $6,5 | $3,1 | 7,7
Winter Storm | 22 | 0.5 | $97,1 | $4,4 | $2,2 | 5,6
All Disasters | 360 | 8.2 | $2575,7 | $7,2 | $58,5 | 17,9

Figure 6: US disaster data. Left: histograms and scatter plots of different
disaster types. Right: estimates of the mean and various
($\delta=0.25,0.5,0.75$) quantiles of $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$ with $D_{\boldsymbol{\tau}}$ corresponding to
expected shortfall.

Figure 7: US disaster data. Estimated $\mbox{ES}_{\boldsymbol{\tau}}$ and
$0.75$th quantile curves, together with 95% asymptotic pointwise confidence
intervals (in respectively dotted and dashed lines). Left panel: for severe
storms. Right panel: for all disasters.

Figure 8: US disaster data. Severe storms. Left: estimates of various
($\delta=0.25,0.75$) expectiles of $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$
with $D_{\boldsymbol{\tau}}$ corresponding to expected shortfall. Right:
estimates of various ($\delta=0.25,0.5,0.75$) expectiles of $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$ with $D_{\boldsymbol{\tau}}=K_{\boldsymbol{\tau}}$,
i.e. corresponding to extremiles. In dotted line, estimates of the usual
expectiles.

We firstly consider the distorted random variable $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$ using
$D_{\tau}(u)=(1-\tau)^{-1}(u-\tau)1_{\\{u\geq\tau\\}}$, the Expected Shortfall
distortion function (see entry 5 in Table 3). We evaluate the risks
$\rho_{\delta}$ attributed to $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$, using
(i) the mean risk (denoted by $\mbox{ES}_{\tau}$ in the figures)–using square
loss; and (ii) the risk associated to the $\delta$th quantile –using quantile
loss. The right panels of Figure 6 display estimates of these risks, as a
function of $\tau$. Note that the estimates break down, i.e., become zero, at
$\tau=n/(n+1)$. This is to be expected from the fact that $D_{\tau}$ is the
cumulative distribution function corresponding to $\mbox{U}[\tau,1]$, and the
lack of ranked data beyond $n(n+1)^{-1}$. For the three different disaster
types the estimates of $\text{Median}\left(X_{{\scriptsize
D}_{\boldsymbol{\tau}}}\right)$ lie below the estimates of
$\text{Mean}\left(X_{{\scriptsize D}_{\boldsymbol{\tau}}}\right)$ (denoted
$\mbox{ES}_{\boldsymbol{\tau}}$ on the plots) when $\boldsymbol{\tau}\leq
0.9$. This means that the distribution of $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$ is right-skewed. For the cases of floodings and
severe storms, the estimates of the $\delta=0.75$ quantiles are close to the
estimates of $\text{ES}_{\boldsymbol{\tau}}$, although floodings are roughly
two and a half times as risky/costly.

To illustrate the uncertainty that comes with the estimates, we calculated
also approximate 95% pointwise confidence intervals for the severe storm
category, relying on the asymptotic results established in Sections 3.2 and
3.3. In Figure 7 (left panel) we depict these approximate pointwise 95%
confidence intervals, for both $\mbox{ES}_{\tau}$ and the 0.75th quantile. The
construction of these approximate confidence intervals relies on estimators
for the asymptotic variance. For the estimate of $\mbox{ES}_{\tau}$, we obtain
the approximate variance by substituting the empirical cumulative distribution
function into expression (16). For the asymptotic variance of the estimated
$\delta$th quantile, we use expression (24). In our implementation we apply
the kernel density estimator from the R package stats in combination with the
Sheather & Jones bandwidth selection method. More specifically, we use the
command density with bandwidth selector bw.SJ. To prevent division by zero in
expression (24), we need to ensure that $\widehat{f}_{X}(T_{n}^{*})\not=0$. To
achieve this, we increase the Sheather & Jones bandwidth at each point where
we estimate the density, ensuring that the neighbourhood determined by the
bandwidth contains at least a fixed proportion of observations. We set the
threshold at 10% of the total number of observations. For the severe storm
category, this corresponds to a minimum of 18 observations. Similar plots for
the other two values of $\delta$ (quantile curves) are to be found in Figure
S3 in Section S5 of the Supplementary Material. In the right panel of Figure 7
we display the estimated $\mbox{ES}_{\tau}$ and 0.75th quantile curves
together with the approximate pointwise 95% confidence intervals, but now for
all disasters together. A histogram and scatter plot for data on all disasters
(all seven categories) is depicted in the left panel of Figure 9. The
discrepancy between the estimated $\mbox{ES}_{\boldsymbol{\tau}}$ curve and
the quantile curve, in Figure 7, is less pronounced when considering all
disasters together.

It is also interesting to evaluate the estimated risk curves keeping in mind
the theoretical results established in Theorem 5.1. Note that since
$D_{\boldsymbol{\tau}}$ is decreasing in $\boldsymbol{\tau}$, the risk
$\rho_{\delta}$ is expected to be an increasing function in $\tau$, according
to Theorem 5.1(v). As can be seen from Figure 6 (right panels) there is no
guarantee that the estimated curves are increasing (although the overall
increasing trend is clear). Furthermore, since the quantile risk is monotone
increasing in $\delta$, Theorem 5.1(vi) ensures that this is also the case for
the risk applied to the distorted random variable $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$. From visual inspection of the estimated curves in
Figure 6 it seems that there is no issue of crossing quantile curves in this
example (at least not before the occurence of the breakdown point). Developing
estimation procedures that would guarantee that estimators also present the
associated theoretical properties is an open research question.

Secondly, we estimate expectile risk measures associated to $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$, meaning that we take the expectile loss function. In
the left panel of Figure 8, we display estimates of various expectiles
($\delta=0.25,0.75$) of $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$. In this
figure and in Figure 9, we display the black curve corresponding to the
estimate $\mbox{ES}_{\boldsymbol{\tau}}$ as a reference curve. This in fact
corresponds to the estimated risk curve for the $\delta=0.5$th expectile risk.
Within each disaster type the curves exhibit a high degree of similarity. The
parameter $\delta$ allows us to obtain more (or less) conservative risk
estimates. As in our previous risk analysis, there is a similar behaviour for
the disasters of floodings and severe storms. We only present the plots for
the disaster severe storms here. Similar plots for tropical cyclone and
flooding disaster categories are displayed in Section S5 of the Supplementary
Material. By Theorem 5.1(v), we know that the population risk $\rho_{\delta}$
is an increasing function in $\tau$, for each expectile level $\delta$. The
estimated risk curves also clearly exhibit this increasing behaviour. Theorem
5.1(vi) also states that there should be no crossings of the population risk
curves for different $\delta$-values. Also the estimated risk curves do not
show crossings (before the breakdown point).

Figure 9: US disaster data. Top left: histogram and scatterplot of all
disasters. Top right: estimates of various ($\delta=0.25,0.5,0.75$) expectiles
of $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$ with
$D_{\boldsymbol{\tau}}=K_{\boldsymbol{\tau}}$, i.e. corresponding to
extremiles. In dotted line, estimates of the usual expectiles.

Thirdly, we look into extremiles as risk measures. Therefore we take
$D_{\boldsymbol{\tau}}=K_{\boldsymbol{\tau}}$, and consider expectile risks of
the distorted random variable $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$. The
right panel of Figure 8 displays estimates of different expectile risk curves
($\delta=0.25,0.5,0.75$) of $X_{{\scriptsize D}_{\boldsymbol{\tau}}}$ with
$D_{\boldsymbol{\tau}}=K_{\boldsymbol{\tau}}$, which gives estimated extremile
risk curves for $X$. We only present the estimates for the disaster category
severe storms. In black, we also display estimates of the usual expectile
curve (as function of $\tau$). In contrast to all previous estimated curves in
this example, we now observe smooth estimated curves. In addition, there is no
breakdown point. We note that the $\delta=0.5$th expectile of $X_{{\scriptsize
D}_{\boldsymbol{\tau}}}$ coincides with the usual extremile. Note again that
the estimated curves exhibit a high shape similarity. For low $\tau$-values
the estimates of the usual expectiles (in black) and the usual extremiles
almost coincide. For larger $\boldsymbol{\tau}$-values, expectiles are more
conservative than extremiles. The cumulative distribution function
$D_{\tau}=K_{\tau}$ is decreasing in $\tau$. Hence, again by Theorem 5.1(v),
we know that the population risk $\rho_{\delta}$ is an increasing function in
$\tau$. The estimated risk curves also show this monotone behaviour. Also the
non-crossingness property stated in Theorem 5.1(vi) seems to hold for the
estimated curves. Figure S4 in the Supplementary Material presents the
estimated expectile and extremile curves for floodings, tropical storms as
well as for all disasters together.

Table 11: Estimated risks for severe storms, floodings and tropical storms, for various risk measures. | $\tau=0.85$ | $\tau=0.95$
---|---|---
pair $(D_{\boldsymbol{\tau}},\ell_{\delta})$ | Severe | Tropical | Flooding | All | Severe | Tropical | Flooding | All
| storm | cyclone |  | disasters | storm | cyclone |  | disasters
$D_{\tau}\sim U[\tau,1]$ |  |  |  |  |  |  |  |
Squared loss | 5.73 | 100.20 | 16.53 | 32.61 | 9.21 | 154.23 | 29.49 | 68.23
Quantile loss $\delta=0.25$ | 3.97 | 59.91 | 10.10 | 11.88 | 5.74 | 127.20 | 17.06 | 35.99
Quantile loss $\delta=0.5$ | 4.38 | 95.48 | 12.89 | 17.30 | 8.38 | 153.75 | 41.92 | 48.63
Quantile loss $\delta=0.75$ | 5.39 | 127.20 | 13.66 | 31.39 | 12.59 | 181.75 | 41.92 | 83.43
Expectile loss $\delta=0.25$ | 4.81 | 81.65 | 12.80 | 22.12 | 7.56 | 138.60 | 21.92 | 51.64
Expectile loss $\delta=0.75$ | 7.32 | 122.84 | 23.56 | 49.81 | 10.85 | 170.06 | 37.07 | 91.34
$D_{\tau}=K_{\tau}$ |  |  |  |  |  |  |  |
Expectile loss $\delta=0.25$ | 3.39 | 41.36 | 7.39 | 12.40 | 5.36 | 89.60 | 14.82 | 29.83
Expectile loss $\delta=0.5$ | 4.28 | 63.08 | 10.86 | 20.72 | 6.71 | 114.35 | 20.41 | 45.24
Expectile loss $\delta=0.75$ | 5.63 | 90.73 | 16.43 | 35.04 | 8.74 | 140.32 | 29.13 | 68.45
$D_{\tau}\sim U[0,1]$ |  |  |  |  |  |  |  |
Expectile loss $\delta=0.95$ | 5.46 | 88.48 | 16.84 | 37.02 | 5.46 | 88.48 | 16.84 | 37.02

In a final analysis, we again combine the data of all seven classes of
disasters. In Figure 9, and Figure S4 (bottom panel) in the Supplementary
Material, we display similar estimated risk curves, now based on all 360
observations. Table 10 shows that over half of the total cost is due to
tropical cyclones. Nevertheless, the shape of the estimated risk curves of all
disasters together appear to be somewhat similar to the estimated risk curves
for floodings and severe storms. To compare various risks estimates, we
present in Table 11 the values of the estimated risks for each of the separate
disaster categories, as well as the estimated risks for all disasters
together. We quantify the various risks for two specific values of $\tau$,
namely $\tau=0.85$, and $\tau=0.95$. Note that, for all risk estimates (i.e.,
looking row by row), the risks are systematically highest for tropical
cyclones, and lowest for severe storms. The estimated risks for flooding are
about a factor 3 larger than these for severe storms, and a factor 6 to 7
smaller than these for tropical cyclones. Our framework not only allows to
estimate the various risks, it also provides approximate confidence intervals
for the risks. In Table 12 we provide this information for some of the risk
measures for severe storms. As expected the approximate CIs are wider for
larger values of $\tau$. For a given $\tau$, the CIs are wider for higher-
level risk measures (i.e. higher values of $\delta$).

Table 12: Some estimated risks for severe storms, with the approximate 95% confidence interval (CI) and its length. | $\tau=0.85$ | $\tau=0.95$
---|---|---
pair $(D_{\boldsymbol{\tau}},\ell_{\delta})$ | risk |  |  | risk |  |
$\quad D_{\tau}\sim U[\tau,1]$ | estimate | CI | length CI | estimate | CI | length CI
Squared loss | 5.73 | $[4.35,7.12]$ | 2.77 | 9.21 | $[6.04,12.37]$ | 6.33
Quantile loss $\delta=0.25$ | 3.97 | $[3.39,4.55]$ | 1.16 | 5.74 | $[4.51,6.97]$ | 2.46
Quantile loss $\delta=0.5$ | 4.38 | $[3.65,5.11]$ | 1.46 | 8.38 | $[4.83,11.94]$ | 7.11
Quantile loss $\delta=0.75$ | 5.39 | $[4.32,6.45]$ | 2.22 | 12.59 | $[7.73,17.46]$ | 9.73

##  7 Concluding remarks and further discussion

This paper studies general functionals, called generalized extremiles, that
are solutions to an optimization problem involving as crucial ingredient a
pair $(D_{\boldsymbol{\tau}},\ell_{\delta})$ with $D_{\boldsymbol{\tau}}$ a
cumulative distribution function with support $[0,1]$, and with
$\ell_{\delta}$ a loss function. The main contribution in the paper consists
of providing estimators for the generalized extremile, and proving their
consistency and asymptotic normality. By allowing for a general pair
$(D_{\boldsymbol{\tau}},\ell_{\delta})$ several functionals studied in the
literature fall under this framework, and hence we also provide statistical
inference for all these. Cases for which estimators and asymptotic normality
results for them are available in the literature are limited, and for these
cases we showed that our general theorem includes these results as special
cases.

Several interesting research questions can be raised. A first question is that
one can wonder under which conditions the pair
$(D_{\boldsymbol{\tau}},\ell_{\delta})$ would lead to a coherent risk measure
itself. Some partial answers are to be found in Section 2.2. Furthermore, in
Section 5 we derive properties such as e.g. monotonicity or coherence of
distorted risk measures based on properties of the base risk measure and the
utilized distortion function. We provide an expression for the asymptotic
variance for the estimator of the generalized extremile. Such a result allows
to construct approximate confidence intervals for it. This however requires an
estimator for the asymptotic variance. Although plug-in estimators are
straightforward, it still remains to be established which are sufficient
conditions for such a plug-in estimator to be consistent. Another interesting
aspect is the behaviour of the asymptotic variance as a function of the
parameters $\boldsymbol{\tau}$ and $\delta$. For certain functionals it would
be interesting to look into such behaviour. Lastly regarding the asymptotic
normality result, this has been established under a set of sufficient
conditions. From the simulation study however we know that the result
continues to hold in cases even when some of these conditions are violated. It
would be interesting to look for a minimal set of conditions. A final issue
concerns the qualitative properties to be expected from a generalized
extremile, such as monotonicity as a function of $\boldsymbol{\tau}$, or non-
crossingness for various values of $\delta$. See Section 6 and the discussion
therein. An interesting open research question involves estimation of the
generalized extremile which would guarantee these properties to hold for the
estimators.

Acknowledgement This work was initiated while the third author was visiting
the KU Leuven. The first and second author gratefully acknowledge support from
the Research Fund KU Leuven [C16/20/002 project]. The third author
acknowledges support from NSERC through Discovery Grant RGPIN-2020-05784.

## References

  * Aigner et al. (1967) D.J. Aigner, T. Amemiya, and D.J. Poirier.  On the estimation of production frontiers: maximum likelihood estimation of the parameters of a discontinuous density function.  _International Economic Review_ , 17(2):377–396, 1967.
  * Artzner et al. (1999) P. Artzner, F. Delbaen, J. M. Eber, and D. Heath.  Coherent measures of risk.  _Mathematical Finance_ , 9(3):203–228, 1999.
  * Bellini and Di Bernardina (2017) F. Bellini and E. Di Bernardina.  Risk management with expectiles.  _The European Journal of Finance_ , 23(6):487–506, 2017.  doi: 10.1080/1351847X.2015.1052150.
  * Bellini et al. (2014) F. Bellini, B. Klar, A. Müller, and E. R. Gianin.  Generalized quantiles as risk measures.  _Insurance: Mathematics and Economics_ , 54:41–48, 2014\.
  * Boyd and Vandenberghe (2004) S. Boyd and L. Vandenberghe.  _Convex Optimization_.  Cambridge University Press, 2004.  doi: 10.1017/CBO9780511804441.
  * Cai and Wang (2019) J. Cai and Y. Wang.  Reinsurance premium principles based on weighted loss functions.  _Scandinavian Actuarial Journal_ , 2019(10):903–923, 2019.  doi: 10.1080/03461238.2019.1628101.
  * Chen (1996) Z. Chen.  Conditional $l_{p}$-quantiles and their application to the testing of symmetry in non-parametric regression.  _Statistics & Probability Letters_, 29(2):107–115, 1996.  ISSN 0167-7152.  doi: 10.1016/0167-7152(95)00163-8.
  * Cherny and Madan (2017) A. Cherny and D. Madan.  New measures for performance evaluation.  _The Review of Financial Studies_ , 22(7):2571–2606, 2017.  URL https://doi.org/10.1093/rfs/hhn081.
  * Daouia et al. (2019) A. Daouia, I. Gijbels, and G. Stupfler.  Extremiles: A new perspective on asymmetric least squares.  _Journal of the American Statistical Association_ , 114(527):1366–1381, 2019.  doi: 10.1080/01621459.2018.1498348.
  * De Backer et al. (2019) M. De Backer, A. El Ghouch, and I. Van Keilegom.  An adapted loss function for censored quantile regression.  _Journal of the American Statistical Association_ , 114(527):1126–1137, 2019.
  * Dickson (2005) D. Dickson.  _Principles of premium calculation_ , page 38–51.  International Series on Actuarial Science. Cambridge University Press, 2005.  doi: 10.1017/CBO9780511624155.004.
  * Embrechts et al. (2021) P. Embrechts, T. Mao, Q. Wang, and R. Wang.  Bayes risk, elicitability, and the expected shortfall.  _Mathematical Finance_ , 31(4):1190–1217, 2021\.
  * Feinstein (2007) J. F. Feinstein.  Convergence from below suffices.  _Bulletin of the Irish Mathematical Society_ , 59:65–70, 2007.
  * Ferguson (1967) T.S. Ferguson.  _Mathematical Statistics: A Decision Theoretic Approach_.  Probability and Mathematical Statistics. Academic Press, 1967.  ISBN 9780122537509.
  * Föllmer and Schied (2002) H. Föllmer and A. Schied.  Convex measures of risk and trading constraints.  _Finance and Stochastics_ , 6:429–447, 2002.
  * Furman and Zitikis (2009) E. Furman and R. Zitikis.  Weighted pricing functionals with applications to insurance.  _North American Actuarial Journal_ , 13(4):483–496, 2009.  doi: 10.1080/10920277.2009.10597570.
  * Gneiting (2011) T. Gneiting.  Making and evaluating point forecasts.  _Journal of the American Statistical Association_ , 106(494):746–762, 2011.  doi: 10.1198/jasa.2011.r10138.
  * Hampel et al. (1986) F. Hampel, E. Ronchetti, P. Rousseeuw, and W. Stahel.  _Robust Statistics: The Approach Based on Influence Functions_.  Wiley, 1986.
  * Heilmann (1989) W. R. Heilmann.  Decision theoretic foundations of credibility theory.  _Insurance: Mathematics and Economics_ , 8(1):77–95, 1989.  doi: 10.1016/0167-6687(89)90050-4.
  * Herrmann et al. (2018) K. Herrmann, M. Hofert, and M. Mailhot.  Multivariate geometric expectiles.  _Scandinavian Actuarial Journal_ , 2018(7):629–659, 2018.  doi: 10.1080/03461238.2018.1426038.
  * Herrmann et al. (2020) K. Herrmann, M. Hofert, and M. Mailhot.  Multivariate geometric tail- and range-value-at-risk.  _ASTIN Bulletin: The Journal of the IAA_ , 50(1):265–292, 2020.  doi: 10.1017/asb.2019.31.
  * Holzmann and Klar (2016) H. Holzmann and B. Klar.  Expectile asymptotics.  _Electronic Journal of Statistics_ , 10(2):2355 – 2371, 2016.  doi: 10.1214/16-EJS1173.
  * Hosseini (2009) M. Hosseini.  _Statistical models for agroclimate risk analysis_.  PhD thesis, University of British Columbia, 2009.
  * Huber (1964) P. J. Huber.  Robust estimation of a location parameter.  _The Annals of Mathematical Statistics_ , 35(1):73 – 101, 1964.  doi: 10.1214/aoms/1177703732.
  * Hürlimann (2006) W. Hürlimann.  A note on generalized distortion risk measures.  _Finance Research Letters_ , 3(4):267–272, 2006\.  doi: 10.1016/j.frl.2006.07.001.
  * Jiang et al. (2021) Y. Jiang, F. Lin, and Y. Zhou.  The kth power expectile regression.  _Annals of the institute of statistical mathematics_ , 73(1):83–113, 2021.  doi: 10.1007/s10463-019-00738-y.
  * Jones and Zitikis (2003) B. L. Jones and R. Zitikis.  Empirical estimation of risk measures and related quantities.  _North American Actuarial Journal_ , 7(4):44–54, 2003.  doi: 10.1080/10920277.2003.10596117.
  * Junike (2019) G. Junike.  Representation of concave distortions and applications.  _Scandinavian Actuarial Journal_ , 2019(9):768–783, 2019.  doi: 10.1080/03461238.2019.1615543.
  * Klenke (2013) A. Klenke.  _Probability Theory: A Comprehensive Course_.  Springer Science & Business Media, 2013.
  * Koenker and Bassett (1978) R. Koenker and G. Bassett.  Regression quantiles.  _Econometrica_ , 46(1):33–50, 1978.  doi: 10.2307/1913643.
  * Liu et al. (2021) P. Liu, A. Schied, and R. Wang.  Distributional transforms, probability distortions, and their applications.  _Mathematics of Operations Research_ , 46(4):1490–1512, 2021.
  * Mao and Cai (2018) T. Mao and J. Cai.  Risk measures based on behavioural economics theory.  _Finance and Stochastics_ , 22:367–393, 2018.  doi: 10.1007/s00780-018-0358-6.
  * Mao et al. (2023) T. Mao, G. Stupfler, and F. Yang.  Asymptotic properties of generalized shortfall risk measures for heavy-tailed risks.  _Insurance: Mathematics and Economics_ , 111:173–192, 2023\.  ISSN 0167-6687.  doi: 10.1016/j.insmatheco.2023.05.001.
  * Newey and McFadden (1986) W. K. Newey and D. McFadden.  Large sample estimation and hypothesis testing.  In R. F. Engle and D. McFadden, editors, _Handbook of Econometrics_ , volume 4, chapter 36, pages 2111–2245. Elsevier, 1 edition, 1986\.
  * Newey and Powell (1987) W. K. Newey and J. L. Powell.  Asymmetric least squares estimation and testing.  _Econometrica_ , 55(4):819–847, 1987.  doi: 10.2307/1911031.
  * Niculescu and Persson (2006) C. Niculescu and L.-E. Persson.  _Convex Functions and Their Applications_ , volume 23.  Springer, 2006.
  * NOAA National Centers for Environmental Information (2023) (NCEI) NOAA National Centers for Environmental Information (NCEI).  U.S. Billion-Dollar Weather and Climate Disasters.  https://www.ncei.noaa.gov/access/billions/, 2023.
  * Osband (1985) K. H. Osband.  _Providing Incentives for Better Cost Forecasting_.  PhD thesis, University of California at Berkeley, December 1985.
  * Pakes and Pollard (1989) A. Pakes and D. Pollard.  Simulation and the asymptotics of optimization estimators.  _Econometrica_ , 57(5):1027–1057, 1989.  doi: 10.2307/1913622.
  * Sander (1975) J.M. Sander.  The weak convergence of quantiles of the product-limit estimator.  Technical Report 5, Division of Biostatistics, Stanford University, 1975\.
  * Serfling (1980) R. J. Serfling.  _Approximation Theorems of Mathematical Statistics_.  Wiley series in probability and mathematical statistics. Wiley, 1980.  ISBN 978-0471024033.
  * Shorack and Wellner (2009) G. R. Shorack and J. A. Wellner.  _Empirical Processes with Applications to Statistics_.  Society for Industrial and Applied Mathematics, 2009.  doi: 10.1137/1.9780898719017.
  * Taylor (2008) J. W. Taylor.  Estimating value at risk and expected shortfall using expectiles.  _Journal of Financial Econometrics_ , 6:231–252, 2008.
  * Van Heerwaarden et al. (1989) A.E. Van Heerwaarden, R. Kaas, and M.J. Goovaerts.  Properties of the esscher premium calculation principle.  _Insurance: Mathematics and Economics_ , 8(4):261–267, 1989.  doi: 10.1016/0167-6687(89)90001-2.
  * Wang (1996) S. Wang.  Premium calculation by transforming the layer premium density.  _Astin Bulletin_ , 26(1):71–92, 1996.
  * Wang et al. (1997) S.S. Wang, V. R. Young, and H. H. Panjer.  Axiomatic characterisation of insurence prices.  _Insurance: Mathematics and Economics_ , 21(2):173–183, 1997.
  * Wellner (1977) J.A. Wellner.  A Glivenko-Cantelli theorem and strong laws of large numbers for functions of order statistics.  _The Annals of Statistics_ , 5(3):473–480, 1977\.  doi: 10.2307/2958898.
  * Wozabal (2009) N. Wozabal.  Uniform limit theorems for functions of order statistics.  _Statistics & Probability Letters_, 79(12):1450–1455, 2009.  doi: 10.1016/j.spl.2009.03.007.
  * Yin and Zhu (2018) C. C. Yin and D. Zhu.  New class of distortion risk measures and their tail asymptotics with emphasis on VaR.  _Journal of Financial Risk Management_ , 7(1):12–38, 2018.  doi: 10.4236/jfrm.2018.71002.
  * Ziemer and Torres (2017) W. P. Ziemer and M. Torres.  _Modern Real Analysis_ , volume 278.  Springer, 2017.

Appendices

##  A Proofs of propositions in Section 2.2

### Proof of Proposition 2.3

We have $\widetilde{d}_{\boldsymbol{\tau}}(t)=d_{\boldsymbol{\tau}}(1-t)$ and
by continuity of $F_{X}$ also $F_{X}(y)=$ $1-F_{-X}(-y)$ for all
$y\in\mathbb{R}$. Defining

$L_{\boldsymbol{\tau},\delta}\left(c;Y,D_{\boldsymbol{\tau}},\ell_{\delta}\right)=\mathbb{E}\left[d_{\boldsymbol{\tau}}\left(F_{Y}(Y)\right)(\ell_{\delta}(Y,c)-\ell_{\delta}(Y,0))\right]$

and setting $Y=-X$ we have

$\displaystyle
L_{\boldsymbol{\tau},\delta}\left(c;Y,\widetilde{D}_{\boldsymbol{\tau}},\ell_{\delta}\right)$
$\displaystyle=\mathbb{E}\left[\widetilde{d}_{\boldsymbol{\tau}}\left(F_{-X}(-X)\right)(\ell_{\delta}(-X,c)-\ell_{\delta}(-X,0))\right]$
$\displaystyle=\mathbb{E}\left[d_{\boldsymbol{\tau}}\left(1-F_{-X}(-X)\right)(\ell_{\delta}(X,-c)-\ell_{\delta}(X,0))\right]$
$\displaystyle=\mathbb{E}\left[d_{\boldsymbol{\tau}}\left(F_{X}(X)\right)(\ell_{\delta}(X,-c)-\ell_{\delta}(X,0))\right]$
$\displaystyle=L_{\boldsymbol{\tau},\delta}\left(-c;X,D_{\boldsymbol{\tau}},\ell_{\delta}\right).$

The minimizer
$e_{\boldsymbol{\tau},\delta}\left(-X;\widetilde{D}_{\boldsymbol{\tau}},\ell_{\delta}\right)$
of