# M-ESTIMATION METHOD BASED ASYMMETRIC OBJECTIVE FUNCTION 111

Mehmet Niyazi Çankaya –  Faculty of Arts and Science, Department of
Statistics, Uşak University, Uşak, Turkey<EMAIL_ADDRESS>
Olcay Arslan –  Faculty Science, Department of Statistics, Ankara University,
Ankara, Turkey<EMAIL_ADDRESS>
###### Abstract

The asymmetric $\rho$ function is proposed as an alternative to Huber $\rho$
function to model skewness and obtain robust estimators for the location,
scale and skewness parameters. The robustness and asymptotic properties of the
asymmetric M-estimators are explored. A simulation study and real data
examples are given to illustrate the performance of proposed asymmetric
M-estimation method over the symmetric M-estimation method. It is observed
from the simulation results that the asymmetric M-estimators perform better
than Huber M-estimators when the data have skewness. The application on
regression is also considered.

Key-Words:

  * •

Asymmetric Huber; M-estimation; Robustness.

##  1 Introduction

The robust estimation method for the location parameter was proposed by [12].
In the robustness, there are different influence functions have been
considered. [2, 11] gave the influence functions to estimate the location and
scale parameters robustly. It can be observed that these functions are
symmetric. Then, it is not possible to model the asymmetry in data set. In our
proposal, we will consider the asymmetric form of influence functions. The
asymmetric form of Huber M-estimation will be proposed. The benefit of our
asymmetric Huber M-estimation is that the location, scale and skewness
parameters can be estimated when the asymmetrically data set does not come
from a distribution. In other words, it is well known that the distribution
assumption on data set can be a restrictive. In such a case, the location,
scale and skewness parameters can be estimated by means of the function we
proposed. These approaches for estimating the parameters are in the robust
methods.

To get the asymmetric objective function denoted by $\rho_{ESH}$, the
distributions proposed by [16, 17, 18, 7] will be used. A family of these
distributions are proposed by [7]. The some special values of parameters in
distribution proposed by [7] give the epsilon-skew normal (ESN) and epsilon
skew Laplace distributions (ESL). The details of how one can get ESN and ESL
are given by [7]. We will use these distributions to get the asymmetric
objective function. In this context, we will give the following approach:

Let $f$ be a probability density function. $\rho=-log(f)$ is known to be
objective function in the robustness. The normalizing constant in $f$ can be
removed. Let

  * •

$\rho_{ESN}(u)=\frac{u^{2}}{2(1-sign(u)\varepsilon)^{2}}$

and

  * •

$\rho_{ESL}(u)=\frac{|u|}{2^{1/2}(1-sign(u)\varepsilon)}$

be an objective functions of ESN and ESL distributions.

Huber’s $\rho$ function is given by the following form:

(1.1) $\rho(u)=\left\\{\begin{array}[]{ll}u^{2}{}&,|u|\leq k;\\\
2k|u|-k^{2}&,|u|>k,\end{array}\right.$

and $\rho^{{}^{\prime}}(u)=2\psi(u)$, that is, the function $\psi$ is a
derivative of $\rho$.

(1.2) $\psi(u)=\left\\{\begin{array}[]{ll}u&,|u|\leq k;\\\
sign(u)k&,|u|>k.\end{array}\right.$

##  2 M-estimation based on Asymmetric Objective Function

We used the Huber’s $\rho$ function to propose asymmetric Huber M-estimation.
The parameter $k$ in Huber is the tuning parameter to get the robust
estimators. In our case, asymmetric Huber M-estimation has $c_{1}$ and $c_{2}$
that are the tuning parameters due to fact that we will propose the asymmetric
form of Huber’s $\rho$ function.

As it is given, the asymmetric Huber M-estimation can be considered as follow:

(2.1)
$\rho^{*}(u)=\left\\{\begin{array}[]{ll}\frac{u}{2^{1/2}(1+\varepsilon)},&(-\infty,c_{1});\\\
\frac{u^{2}}{2(1+\varepsilon)^{2}},&[c_{1},0);\\\
\frac{u^{2}}{2(1-\varepsilon)^{2}},&[0,c_{2}];\\\
\frac{u}{2^{1/2}(1-\varepsilon)},&(c_{2},\infty).\end{array}\right.$

The proposed asymmetric $\rho^{*}$ function is not continuous at points
$c_{1}$ and $c_{2}$. After the required regularization on function $\rho^{*}$
is done, the following new $\rho^{**}$ function can be proposed as follow:

(2.2)
$\rho^{**}(u)=\left\\{\begin{array}[]{ll}\frac{c_{1}u}{(1+\varepsilon)^{2}}-\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}},&(-\infty,c_{1});\\\
\frac{u^{2}}{2(1+\varepsilon)^{2}},&[c_{1},0);\\\
\frac{u^{2}}{2(1-\varepsilon)^{2}},&[0,c_{2}];\\\
\frac{c_{2}u}{(1-\varepsilon)^{2}}-\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}},&(c_{2},\infty).\end{array}\right.$

where $c_{1}$ and $c_{2}$ are the tuning parameters and the continuity of
$\rho^{*}$ is guaranteed via these parameters. Here, $\varepsilon$ is a
skewness parameter to model the asymmetry.

######  Definition 2.1.

The function $\rho^{**}$ in equation (2.2) is defined to be asymmetric
$\rho_{ESH}$ function.

$\rho_{ESH}$ is used to show the asymmetric $\rho^{**}$ function we proposed.
When $c_{1}=c_{2}$ and $\varepsilon=0$, $\rho_{H}$ in equation (1.1) can be
obtained.

The function $\psi$ is a derivative of function $\rho_{ESH}$. It can be given
in the following form:

(2.3)
$\psi(u)=\left\\{\begin{array}[]{ll}\frac{c_{1}}{(1+\varepsilon)^{2}},&(-\infty,c_{1});\\\
\frac{u}{(1+\varepsilon)^{2}},&[c_{1},0);\\\
\frac{u}{(1-\varepsilon)^{2}},&[0,c_{2}];\\\
\frac{c_{2}}{(1-\varepsilon)^{2}},&(c_{2},\infty).\end{array}\right.$

The estimators of parameters $\theta$, $\sigma$ ve $\varepsilon$ can be
obtained by means of asymmetric objective function given in equation (2.2).
The functions $\psi_{ESH}$, $\psi_{ESN}$ ve $\psi_{ESL}$ can be obtained from
the objective functions $\rho_{ESH}$, $\rho_{ESN}$ ve $\rho_{ESL}$. When
$\varepsilon=0$, the influence functions ($\psi$), the symmetric influence
functions $\psi_{H},\psi_{N}$ and $\psi_{L}$ are obtained.

###  2.1 M-estimators generated by asymmetric M-objective function

Suppose that the random variables $X_{1},X_{2},...,X_{n}$ are distributed as a
probability density function $f$. The parameters $\theta$, $\sigma$ and
$\varepsilon$ in function $f$ exists and they are location, scale and skewness
parameters, respectively. There are other parameters in the considered $f$,
however we are not interested in other parameters.

In our proposal, our aim is to estimate the parameters $\theta$, $\sigma$ and
$\varepsilon$ for the random sample $X_{n}=\\{x_{1},x_{2},...,x_{n}\\}$. The
random sample is supposed to be asymmetrically distributed. Owing to the fact
that the probability density function is not known, using the maximum
likelihood estimation (MLE) method is not possible. In such a case, the
function $Q$ given in the following form is proposed to estimate the
parameters interested.

(2.4) $\displaystyle Q(\theta,\sigma,\varepsilon;X_{n})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\rho_{ESH}\bigg{(}\frac{x_{i}-\theta}{\sigma(1-sign(x_{i}-\theta)\varepsilon)}\bigg{)}+nlog(\sigma)$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{n}log(1-sign(x_{i}-\theta)\varepsilon)$

The function in equation (2.4) is minimized. To get the estimators of
parameters $\theta$, $\sigma$ and $\varepsilon$, we will take the derivative
of parameters interested.

Let $u_{i}$ be $\frac{x_{i}-\theta}{\sigma(1-sign(x_{i}-\theta)\varepsilon)}$.
Then, the derivative of $Q(\theta,\sigma,\varepsilon;X_{n})$ with respect to
$\theta$ and setting it to zero will produce the following equation.

(2.5)
$\frac{\partial}{\partial\theta}Q(\theta,\sigma,\varepsilon;X_{n})=\sum_{i=1}^{n}\psi_{\theta}\bigg{(}\frac{x_{i}-\theta}{\sigma(1-sign(x_{i}-\theta)\varepsilon)}\bigg{)}\frac{-1}{\sigma(1-sign(x_{i}-\theta)\varepsilon)}=0.$

The weight function is defined to be $w(u_{i})=\psi_{\theta}(u_{i})/u_{i}$.
Then, the M-estimator of location parameter $\theta$ will be

(2.6)
$\hat{\theta}=\sum_{i=1}^{n}w_{i}\frac{x_{i}}{(\hat{\sigma}(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon}))^{2}}/\sum_{i=1}^{n}w_{i}\frac{1}{(\hat{\sigma}(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon}))^{2}}.$

where
$w_{i}=w(\frac{x_{i}-\hat{\theta}}{\hat{\sigma}(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon})})$.
The derivative of $Q(\theta,\sigma,\varepsilon;X_{n})$ with respect to
$\sigma$ and setting it to zero will produce the following equation.

(2.7)
$\frac{\partial}{\partial\sigma}Q(\theta,\sigma,\varepsilon;X_{n})=-\sum_{i=1}^{n}\psi_{\sigma}\bigg{(}\frac{x_{i}-\theta}{\sigma(1-sign(x_{i}-\theta)\varepsilon)}\bigg{)}\frac{x_{i}-\theta}{\sigma^{2}(1-sign(x_{i}-\theta)\varepsilon)}+\frac{n}{\sigma}=0.$

The weight function is defined to be $w(u_{i})=\psi_{\sigma}(u_{i})/u_{i}$.
Then, the M-estimator of scale parameter $\sigma$ will be

(2.8)
$\hat{\sigma}^{2}=\frac{1}{n}\sum_{i=1}^{n}w_{i}\frac{(x_{i}-\hat{\theta})^{2}}{(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon})^{2}}.$

where
$w_{i}=w(\frac{x_{i}-\hat{\theta}}{\hat{\sigma}(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon})})$.
The derivative of $Q(\theta,\sigma,\varepsilon;X_{n})$ with respect to
$\varepsilon$ and setting it to zero will produce the following equation.

(2.9)
$\displaystyle\frac{\partial}{\partial\varepsilon}Q(\theta,\sigma,\varepsilon;X_{n})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\psi_{\varepsilon}\bigg{(}\frac{x_{i}-\theta}{\sigma(1-sign(x_{i}-\theta)\varepsilon)}\bigg{)}\frac{(x_{i}-\theta)sign(x_{i}-\theta)}{\sigma(1-sign(x_{i}-\theta)\varepsilon)^{2}}$
$\displaystyle-$
$\displaystyle\sum_{i=1}^{n}\frac{sign(x_{i}-\theta)}{(1-sign(x_{i}-\theta)\varepsilon)}=0.$

The weight function is defined to be
$w(u_{i})=\psi_{\varepsilon}(u_{i})/u_{i}$. Then, the M-estimator of skewness
parameter $\varepsilon$ will be

(2.10)
$\hat{\varepsilon}=\sum_{i=1}^{n}\bigg{[}\frac{sign(x_{i}-\hat{\theta})}{(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon})^{2}}-w_{i}\frac{(x_{i}-\hat{\theta})^{2}sign(x_{i}-\hat{\theta})}{\hat{\sigma}^{2}(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon})^{3}}\bigg{]}/\sum_{i=1}^{n}\frac{1}{(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon})^{2}}.$

The weight function is defined to be
$w_{i}=w(\frac{x_{i}-\hat{\theta}}{\hat{\sigma}(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon})})$.

The weight function of these parameters will be given as follow:

(2.11)
$w(u)=\left\\{\begin{array}[]{ll}\frac{c_{1}}{(1+\varepsilon)^{2}u},&(-\infty,c_{1}];\\\
\frac{1}{(1+\varepsilon)^{2}},&[c_{1},0);\\\
\frac{1}{(1-\varepsilon)^{2}},&[0,c_{2}];\\\
\frac{c_{2}}{(1-\varepsilon)^{2}u},&[c_{2},\infty).\end{array}\right.$

As a result, the estimators of parameters $\theta$, $\sigma$ and $\varepsilon$
are gotten. The weight function in equation (2.11) can give the different
weights in data set that is negative and positive sides of axis. Thus, the
estimators can model the asymmetry in the data set.

###  2.2 The Computation Steps of Estimators

The random sample is $X_{n}=\\{x_{1},x_{2},...,x_{n}\\}$ and
$k\in\mathbb{N}^{+}$ is the iteration number. Then iterative reweighting
algortihm (IRA) will be given in the following form:

1\. Step $\theta^{(1)}$, $\sigma^{(1)}$ ve $\varepsilon^{(1)}$ are the initial
values to start the algorithm.

2\. Step The weight function $w$ in equation (2.11) is computed by using the
following form:

$\displaystyle u_{i}^{(k)}$ $\displaystyle=$
$\displaystyle\frac{x_{i}-\hat{\theta}^{(k)}}{\hat{\sigma}^{(k)}(1-sign(x_{i}-\hat{\theta}^{(k)})\hat{\varepsilon}^{(k)})}$

3\. Step The estimated value of parameter $\theta$ is computed by

$\displaystyle\hat{\theta}^{(k+1)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}w_{i}^{(k)}\frac{x_{i}}{(\hat{\sigma}^{{(k)}})^{2}(1-sign(x_{i}-\hat{\theta}^{(k)})\hat{\varepsilon}^{(k)})^{2}}$
$\displaystyle/\sum_{i=1}^{n}w_{i}^{(k)}\frac{1}{(\hat{\sigma}^{{(k)}})^{2}(1-sign(x_{i}-\hat{\theta}^{(k)})\hat{\varepsilon}^{(k)})^{2}}.$

4\. Step The estimated value of parameter $\sigma$ is computed by

$\displaystyle(\hat{\sigma}^{2})^{(k+1)}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}w_{i}^{(k)}\frac{(x_{i}-\hat{\theta}^{(k+1)})^{2}}{(1-sign(x_{i}-\hat{\theta}^{(k+1)})\hat{\varepsilon}^{(k)})^{2}}$

5\. Step The estimated value of parameter $\varepsilon$ is computed by

$\displaystyle\hat{\varepsilon}^{(k+1)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\bigg{[}\frac{sign(x_{i}-\hat{\theta}^{(k+1)})}{(1-sign(x_{i}-\hat{\theta})\hat{\varepsilon}^{(k)})^{2}}-w_{i}^{(k+1)}\frac{(x_{i}-\hat{\theta}^{(k+1)})^{2}sign(x_{i}-\hat{\theta}^{(k+1)})}{(\hat{\sigma}^{(k+1)})^{2}(1-sign(x_{i}-\hat{\theta}^{(k+1)})\hat{\varepsilon}^{(k)})^{3}}\bigg{]}~{}~{}$
$\displaystyle/\sum_{i=1}^{n}\frac{1}{(1-sign(x_{i}-\hat{\theta}^{(k+1)})\hat{\varepsilon}^{(k)})^{2}}$

where $w_{i}^{(k+1)}=w(u_{i}^{(k+1)})$. Then, the weight function $w$ in
second step is computed by using the estimates $\hat{\theta}^{(k+1)}$,
$\hat{\sigma}^{(k+1)}$ and $\hat{\varepsilon}^{(k)}$.

6\. Step If the norm of vector
$(\hat{\theta}^{(k+1)}-\hat{\theta}^{(k)},\hat{\sigma}^{(k+1)}-\hat{\sigma}^{(k)},\hat{\varepsilon}^{(k+1)}-\hat{\varepsilon}^{(k)})^{T}$
is bigger than the prescribed value $\epsilon>0$, the steps are repeated until
the prescribed value $\epsilon>0$ is guaranteed. Finally, the values at last
steps are assigned to be estimates of parameters.

##  3 Robustness Properties of Estimators

In this section, the robustness properties of estimators of parameters
$\theta,\sigma$ and $\varepsilon$ will be examined. In this context, the
influence function that is an indicator for the local robustness and gross
error sensitivity that is an indicator of global robustness are considered for
the estimators of $\theta,\sigma$ and $\varepsilon$.

###  3.1 The Influence Function of Estimators

The function $\rho$ in equation (2.2) is used to get the following functions
$\psi_{\theta}(x)=\frac{\partial}{\partial\theta}\rho_{ESH}(\frac{x-\theta}{\sigma(1-sign(x)\varepsilon)}),\psi_{\sigma}(x)=\frac{\partial}{\partial\sigma}\rho_{ESH}(\frac{x-\theta}{\sigma(1-sign(x)\varepsilon)})$
and
$\psi_{\varepsilon}(x)=\frac{\partial}{\partial\varepsilon}\rho_{ESH}(\frac{x-\theta}{\sigma(1-sign(x)\varepsilon)})$.
For $\theta=0$ and $\sigma=1$, these functions are given by

(3.1)
$\psi_{\theta}(x)=\left\\{\begin{array}[]{ll}\frac{-c_{1}}{(1+\varepsilon)^{3}},&(-\infty,c_{1}(1+\varepsilon)];\\\
\frac{-x}{(1+\varepsilon)^{4}},&[c_{1}(1+\varepsilon),0);\\\
\frac{-x}{(1-\varepsilon)^{4}},&[0,c_{2}(1-\varepsilon)];\\\
\frac{-c_{2}}{(1-\varepsilon)^{3}},&[0+c_{2}(1-\varepsilon),\infty).\end{array}\right.$
(3.2)
$\psi_{\sigma}(x)=\left\\{\begin{array}[]{ll}\frac{-c_{1}x}{(1+\varepsilon)^{3}},&(-\infty,c_{1}(1+\varepsilon)];\\\
\frac{-x^{2}}{(1+\varepsilon)^{4}},&[c_{1}(1+\varepsilon),0);\\\
\frac{-x^{2}}{(1-\varepsilon)^{4}},&[0,c_{2}(1-\varepsilon)];\\\
\frac{-c_{2}x}{(1-\varepsilon)^{3}},&[c_{2}(1-\varepsilon),\infty).\end{array}\right.$
(3.3)
$\psi_{\varepsilon}(x)=\left\\{\begin{array}[]{ll}\frac{-3c_{1}x}{(1+\varepsilon)^{4}}+\frac{c_{1}^{2}}{(1+\varepsilon)^{3}},&(-\infty,c_{1}(1+\varepsilon)];\\\
\frac{-2x^{2}}{(1+\varepsilon)^{5}},&[c_{1}(1+\varepsilon),0);\\\
\frac{2x^{2}}{(1-\varepsilon)^{5}},&[0,c_{2}(1-\varepsilon)];\\\
\frac{3c_{2}x}{(1-\varepsilon)^{4}}-\frac{c_{2}^{2}}{(1-\varepsilon)^{3}},&[c_{2}(1-\varepsilon),\infty).\end{array}\right.$

$\underset{x\rightarrow-\infty}{lim}\psi_{\theta}(x)=\frac{-c_{1}}{(1+\varepsilon)^{3}}<\infty$,
however $\underset{x\rightarrow-\infty}{lim}\psi_{\sigma}(x)=-\infty$ and
$\underset{x\rightarrow-\infty}{lim}\psi_{\varepsilon}(x)=-\infty$. Then, they
are not finite. When the parameters $\sigma$ and $\varepsilon$ are known, the
influence function of estimator of parameter $\theta$ is finite. However, the
influence function of estimators of three parameters are not finite at the
same time as it is proved by the tools given by the following forms:

The influence function of estimators of three parameters is

(3.4) $IF(x;\hat{\theta},\hat{\sigma},\hat{\varepsilon})=-B^{-1}\Psi(x)$

where $\Psi(x)=(\psi_{\theta}(x),\psi_{\sigma}(x),\psi_{\varepsilon}(x))^{T}$
and

(3.5)
$B=\begin{bmatrix}E_{ESN}[\frac{\partial}{\partial\theta}\psi_{\theta}(X)]&E_{ESN}[\frac{\partial}{\partial\sigma}\psi_{\theta}(X)]&E_{ESN}[\frac{\partial}{\partial\varepsilon}\psi_{\theta}(X)]\\\\[3.00003pt]
&E_{ESN}[\frac{\partial}{\partial\sigma}\psi_{\sigma}(X)]&E_{ESN}[\frac{\partial}{\partial\varepsilon}\psi_{\sigma}(X)]\\\\[3.00003pt]
&&E_{ESN}[\frac{\partial}{\partial\varepsilon}\psi_{\varepsilon}(X)]\end{bmatrix}$

$ESN$ shows that the underlying distribution is taken as $ESN$ to get the
integral values. It should be noted that $det(B)\neq 0$. Then, the matrix
$B^{-1}$ exists. Thus, the influence function of estimators of three
parameters exists.

The equation (3.4) can be rewritten as the following form:

(3.6)
$~{}~{}~{}~{}~{}~{}~{}IF(x;\hat{\theta},\hat{\sigma},\hat{\varepsilon})=\begin{bmatrix}T_{11}\psi_{\theta}(x)+T_{12}\psi_{\sigma}(x)+T_{13}\psi_{\varepsilon}(x)\\\
T_{21}\psi_{\theta}(x)+T_{22}\psi_{\sigma}(x)+T_{23}\psi_{\varepsilon}(x)\\\
T_{31}\psi_{\theta}(x)+T_{32}\psi_{\sigma}(x)+T_{33}\psi_{\varepsilon}(x)\end{bmatrix}=\begin{bmatrix}IF_{1}(x;\hat{\theta},\hat{\sigma},\hat{\varepsilon})\\\
IF_{2}(x;\hat{\theta},\hat{\sigma},\hat{\varepsilon})\\\
IF_{3}(x;\hat{\theta},\hat{\sigma},\hat{\varepsilon})\end{bmatrix}$

where $T_{ij}$ represents the row $i.$ and column $j.$ of matrix $B^{-1}$
($i,j=1,2,3$). Here, the components $IF_{1},IF_{2}$ and $IF_{3}$ of the
influence function $(IF)$ are not finite, because $\psi_{\sigma}(x)$ and
$\psi_{\varepsilon}(x)$ are not finite. Thus, the influence function of the
estimators is not finite. It is known that the norm of influence function is
defined to be the gross error sensitivity. Then, the gross error sensitivity
is given by the following form:

(3.7)
$GES(\hat{\theta},\hat{\sigma},\hat{\varepsilon},\rho_{ESH})=\\{(IF_{1})^{2}+(IF_{2})^{2}+(IF_{3})^{2}\\}^{1/2}.$

The components $IF_{1},IF_{2}$ and $IF_{3}$ are not finite. Then,
$GES(\hat{\theta},\hat{\sigma},\hat{\varepsilon},\rho_{ESH})$ will not be
finite.

###  3.2 Breakdown Point of Estimator for Location Parameter

[14] and [25] give the assumption for the breakdown properties of location
M-estimator. The convexity in asymmetric case is satisfied. Then, these
assumptions given below can be used to satisfy the M-estimator generated from
the asymmetric objective function.

  1. 1.

$\rho(0)=0$ [14] and [25].

  2. 2.

$\underset{|u|\rightarrow\infty}{lim}\rho(u)=\infty$ [14] and [25].

  3. 3.

$\psi(u)=\frac{d}{du}\rho(u)$ is continuous for every point of $u$. [14].

  4. 4.

Let $u_{0}$ exist when $\psi(u)$ is nondecreasing for $0<u\leq u_{0}$ and
nonincreasing for $u_{0}<u<\infty$ for monotone $\psi$ functions [25].

Let us check these assumptions for asymmetric objective and influence
functions.

  1. 1.

$\rho_{ESH}(0)=0,$

  2. 2.

$\underset{|u|\rightarrow\infty}{lim}\rho_{ESH}(u)=\infty,$

  3. 3.

The function $\psi$ is continuous for every point of $u$.

  4. 4.

The function $\psi(u)$ is increasing for the point $c_{2}$ given arbitrarily
for $[0,c_{2}]$ and it is constant for $(c_{2},\infty)$

These assumptions given above are satisfied. Then, the location estimator
obtained from the asymmetric objective function $\rho_{ESH}$ has a global
robustness that is breakdown point. The value of breakdown point is $1/2$.

##  4 Asymptotic Properties

The asymptotic properties that are consistency and asymptotic normality of
estimators for the parameters $\theta,\sigma$ ve $\varepsilon$ will be
examined in this section. The function $Q$ in equation (2.4) is used to show
the asymptotic property. The following equations can be obtained after taking
the derivatives with respect to parameters and setting them to zero. The
explicit forms of the following equations were given by the equations
(2.5),(2.7) and (2.9).

(4.1)
$\displaystyle\sum_{i=1}^{n}\psi_{\theta}(x_{i};\theta,\sigma,\varepsilon)$
$\displaystyle=$ $\displaystyle 0$ (4.2)
$\displaystyle\sum_{i=1}^{n}\psi_{\sigma}(x_{i};\theta,\sigma,\varepsilon)$
$\displaystyle=$ $\displaystyle 0$ (4.3)
$\displaystyle\sum_{i=1}^{n}\psi_{\varepsilon}(x_{i};\theta,\sigma,\varepsilon)$
$\displaystyle=$ $\displaystyle 0$

The simultaneous estimations of the parameters $\theta$, $\sigma$ and
$\varepsilon$

(4.4)
$\sum_{i=1}^{n}\Psi(x_{i};\hat{\theta},\hat{\sigma},\hat{\varepsilon})=\mathbf{0}.$

where $\Psi=(\psi_{\theta},\psi_{\sigma},\psi_{\varepsilon})$. The approach
given by [24] is adapted into the asymmetric M-estimation. Then, there is an
one solution of equation (4.4) at least.

Suppose that $\hat{\theta}$ exists for each of $\hat{\sigma}$. Then,

(4.5) $\sum_{i=1}^{n}\psi_{\theta}(x_{i};\theta,\sigma,\varepsilon)=0.$

the location estimation $\hat{\theta}$

$\underset{1\leq i\leq
n~{}~{}~{}~{}~{}~{}}{min(x_{i})}\leq\hat{\theta}\leq\underset{1\leq i\leq
n~{}~{}~{}~{}~{}~{}}{max(x_{i})}$

is satisfied. Thus, at least one solution can exist for the location
estimation. When the $\hat{\sigma}$ changes from $0$ to $\infty$. The term

(4.6) $\sum_{i=1}^{n}\psi_{\sigma}(x_{i};\theta,\sigma,\varepsilon)=0.$

changes from
$sup\\{\psi_{\sigma}(x_{i};\theta,\sigma,\varepsilon):x\in\mathbb{R}\\}$ to
$0$. The estimation of parameter $\varepsilon$ will be solution of the
following equation.

(4.7) $\sum_{i=1}^{n}\psi_{\varepsilon}(x_{i};\theta,\sigma,\varepsilon)=0.$

The solution is in interval $(-1,1)$, because $\varepsilon\in(-1,1)$. Thus, at
least one solution of equation (4.4) can be given.

To guarantee the uniqueness of solution, the following two conditions are
satisfied [19]:

  1. 1.

The function $\rho_{ESH}$ is differentiable.

  2. 2.

The Jacobian of equation (4.4) exists and upper-left corner principal minors
of matrix is non-zero.

We will examine whether the conditions are satisfied.

  1. 1.

The proposed function $\rho_{ESH}$ is differentiable at each points of the
interval $[c_{1},c_{2}]$. However, the derivative of $\rho_{ESH}$ is zero at
the intervals $(-\infty,c_{1})$ and $(c_{2},\infty)$.

  2. 2.

To construct the equation (4.4)

$\lambda(\tau)=E_{ESN}\Psi(X,\tau),~{}~{}~{}~{}\tau=(\theta,\sigma,\varepsilon)$

then

$B_{jk}=\frac{\partial\lambda_{j}}{\partial\tau_{k}},~{}~{}~{}~{}~{}~{}~{}j,k=1,2,3$

exists each term of matrix $B$

Constructing the matrix $B$ in second condition was proposed by [13, 21].

This matrix will be given when the asymptotic normality of estimators. Left-
upper three corners of the matrix are given as follows:

$K_{1}=|B_{11}|$, $K_{2}=\begin{vmatrix}B_{11}&B_{12}\\\ B_{21}&B_{22}\\\
\end{vmatrix}$, $K_{3}=\begin{vmatrix}B_{11}&B_{12}&B_{13}\\\
B_{21}&B_{22}&B_{23}\\\ B_{31}&B_{32}&B_{33}\end{vmatrix}$

$K_{1},K_{2}$ and $K_{3}$ should be non-zero. Then, the solutions of equation
(4.4) has an unique solution, because the conditions $1.$ and $2.$ are
satisfied.

Uniqueness of solution was guaranteed. Then, we will examine whether the
estimators $\hat{\theta},\hat{\sigma}$ ve $\hat{\varepsilon}$ are consistent.
The convexity can hold when the asymmetric form of objective function is
proposed. Then, we can use the assumptions considered by [10].

  1. 1.

$E_{ESN}[\rho_{ESH}(X)]<\infty$

  2. 2.

$E_{ESN}[\psi_{\theta}(X)]<\infty$, $E_{F}[\psi_{\sigma}(X)]<\infty$,
$E_{F}[\psi_{\varepsilon}(X)]<\infty$

The result of $E_{ESN}[\rho_{ESH}(X)]$ is given by

$\displaystyle E[\rho(X)]$ $\displaystyle=$
$\displaystyle\frac{-c_{1}}{\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})-\frac{c_{1}^{2}}{4\sqrt{\pi}(1+\varepsilon)}\cdot\Gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{(1+\varepsilon)}{2\sqrt{\pi}}$
$\displaystyle\cdot\gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{(1-\varepsilon)}{2\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{c_{2}}{\sqrt{2\pi}}$
$\displaystyle\cdot\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})-\frac{c_{2}^{2}}{4\sqrt{\pi}(1-\varepsilon)}\cdot\Gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}).$

The results of $E_{ESN}[\psi_{\theta}(X)]$, $E_{ESN}[\psi_{\sigma}(X)]$,
$E_{ESN}[\psi_{\varepsilon}(X)]$ are given by

$\displaystyle E[\psi_{\theta}(X)]$ $\displaystyle=$
$\displaystyle\frac{-c_{1}}{\sigma(1+\varepsilon)^{2}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{1}{\sigma(1+\varepsilon)\sqrt{2\pi}}\cdot\gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle-\frac{1}{\sigma(1-\varepsilon)\sqrt{2\pi}}\cdot\gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})-\frac{c_{2}}{\sigma(1-\varepsilon)^{2}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\psi_{\sigma}(X)]$ $\displaystyle=$
$\displaystyle\frac{c_{1}}{\sigma\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})-\frac{(1+\varepsilon)}{\sigma\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle-\frac{(1-\varepsilon)}{\sigma\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})-\frac{c_{2}}{\sigma\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\psi_{\varepsilon}(X)]$ $\displaystyle=$
$\displaystyle\frac{3c_{1}}{(1+\varepsilon)\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{c_{1}^{2}}{(1+\varepsilon)^{2}2\sqrt{\pi}}\Gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle-\frac{2}{\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{2}{\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})$
$\displaystyle+\frac{3c_{2}}{(1-\varepsilon)\sqrt{2\pi}}\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})-\frac{c_{2}^{2}}{(1-\varepsilon)^{2}2\sqrt{\pi}}\Gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}).$

These results will be finite when the $c_{1}$, $c_{2}$, $\sigma$ and
$\varepsilon$ are finite. Then, the conditions $1.$ and $2.$ were satisfied.
The consistency of estimators obtained simultaneously was been examined. Then,
it can examine the asymptotic normality of estimators. Since the estimators
are not explicit form, the Taylor expansion of influence function will be
consider as follow:

(4.12)
$\Psi(x_{i},\hat{\tau})=\Psi(x_{i},\tau)+(\hat{\tau}-\tau)\dot{\Psi}(x_{i},\tau)+\mathbf{R}_{n}^{*}$

the summation of both sides are taken and multiplied by $1/n$. Then,

$\mathbf{0}=\frac{1}{n}\sum_{i=1}^{n}\Psi(x_{i},\tau)+(\hat{\tau}-\tau)\frac{1}{n}\sum_{i=1}^{n}\dot{\Psi}(x_{i},\tau)+\mathbf{R}_{n}$

was gotten, where
$\dot{\Psi}(x_{i},\tau)=\frac{\partial\Psi(x_{i},\tau)}{\partial\tau^{T}}$. If
$[\frac{1}{n}\sum_{i=1}^{n}\dot{\Psi}(x_{i},\tau)]^{-1}$ exists,

(4.13)
$\displaystyle-(\hat{\tau}-\tau)\frac{1}{n}\sum_{i=1}^{n}\dot{\Psi}(x_{i},\tau)$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\Psi(x_{i},\tau)+\mathbf{R}_{n}$
$\displaystyle\sqrt{n}(\hat{\tau}-\tau)$ $\displaystyle=$ $\displaystyle
B_{n}^{-1}\sqrt{n}A_{n}+\sqrt{n}\mathbf{R}_{n}$

where $\sqrt{n}\mathbf{R}_{n}\overset{P}{\longrightarrow}\mathbf{0}$. Under
the regularity of conditions, when $n\rightarrow\infty$, the weak of large
numbers

(4.14)
$B_{n}=\frac{1}{n}\sum_{i=1}^{n}(-\dot{\Psi}(x_{i},\tau))\overset{P}{\longrightarrow}E[-\dot{\Psi}(X,\tau)]=B$

can be obtained. By means of central limit theorem,

(4.15)
$\sqrt{n}A_{n}\overset{D}{\longrightarrow}N_{3}(0,A),~{}~{}~{}~{}A=E[\Psi(X,\tau)\Psi(X,\tau)^{T}]$

can be obtained. Here, $\Psi$ is three-dimensional. Thus, by means of the
Slutsky’s multivariate lemma,

(4.16)
$\sqrt{n}(\hat{\tau}-\tau)\overset{D}{\longrightarrow}N_{3}(0,B^{-1}A(B^{T})^{-1})$

$N_{3}$ shows the $3-$dimensional asymptotic normally distributed.

Then, the matrices $A$ and $B$ exist and the inverse of matrix $B$ exists as
well. These matrices are obtained when the underlying distribution is chosen
as $ESN$.

$A=\begin{bmatrix}E[\psi_{\theta}^{2}(X)]&E[\psi_{\sigma}(X)\psi_{\theta}(X)]&E[\psi_{\varepsilon}(X)\psi_{\theta}(X)]\\\
&E[\psi_{\sigma}^{2}(X)]&E[\psi_{\varepsilon}(X)\psi_{\sigma}(X)]\\\
&&E[\psi_{\varepsilon}^{2}(X)]\end{bmatrix}$

The elements of matrix $A$ are

$\displaystyle E[\psi_{\theta}^{2}(X)]$ $\displaystyle=$
$\displaystyle\frac{c_{1}^{2}}{\sigma^{2}(1+\varepsilon)^{5}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{1}{\sigma^{2}(1+\varepsilon)^{3}\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle+\frac{1}{\sigma^{2}(1-\varepsilon)^{3}\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{c_{2}^{2}}{\sigma^{2}(1-\varepsilon)^{5}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\psi_{\sigma}(X)\psi_{\theta}(X)]$ $\displaystyle=$
$\displaystyle\frac{-c_{1}^{2}}{\sigma^{2}(1+\varepsilon)^{3}\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})-\frac{2}{\sigma^{2}(1+\varepsilon)\sqrt{2\pi}}\cdot\gamma(2,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle+\frac{2}{\sigma^{2}(1-\varepsilon)\sqrt{2\pi}}\cdot\gamma(2,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{c_{2}^{2}}{\sigma^{2}(1-\varepsilon)^{3}\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\psi_{\varepsilon}(X)\psi_{\theta}(X)]$ $\displaystyle=$
$\displaystyle\frac{-3c_{1}^{2}}{\sigma(1+\varepsilon)^{4}\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})-\frac{c_{1}^{3}}{\sigma(1+\varepsilon)^{5}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle-\frac{2\sqrt{2}}{\sigma(1+\varepsilon)^{2}\sqrt{\pi}}\cdot\gamma(2,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})-\frac{2\sqrt{2}}{\sigma(1-\varepsilon)^{2}\sqrt{\pi}}\cdot\gamma(2,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})$
$\displaystyle-\frac{3c_{2}^{2}}{\sigma(1-\varepsilon)^{4}\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{c_{2}^{3}}{\sigma(1-\varepsilon)^{5}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\psi_{\sigma}^{2}(X)]$ $\displaystyle=$
$\displaystyle\frac{c_{1}^{2}}{\sigma^{2}(1+\varepsilon)\sqrt{\pi}}\cdot\Gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{2(1+\varepsilon)}{\sigma^{2}\sqrt{\pi}}\cdot\gamma(\frac{5}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle+\frac{2(1-\varepsilon)}{\sigma^{2}\sqrt{\pi}}\cdot\gamma(\frac{5}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{c_{2}^{2}}{\sigma^{2}(1-\varepsilon)\sqrt{\pi}}\cdot\Gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\psi_{\varepsilon}(X)\psi_{\sigma}(X)]$ $\displaystyle=$
$\displaystyle\frac{3c_{1}^{2}}{\sigma(1+\varepsilon)^{2}\sqrt{\pi}}\cdot\Gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{c_{1}^{3}}{\sigma(1+\varepsilon)^{3}\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle+\frac{4}{\sigma\sqrt{\pi}}\cdot\gamma(\frac{5}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})-\frac{4}{\sigma\sqrt{\pi}}\cdot\gamma(\frac{5}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})$
$\displaystyle-\frac{3c_{2}^{2}}{\sigma(1-\varepsilon)^{2}\sqrt{\pi}}\Gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{c_{2}^{3}}{\sigma(1-\varepsilon)^{3}\sqrt{2\pi}}\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\psi_{\varepsilon}^{2}(X)]$ $\displaystyle=$
$\displaystyle\frac{9c_{1}^{2}}{(1+\varepsilon)^{3}\sqrt{\pi}}\cdot\Gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{3\sqrt{2}c_{1}^{3}}{(1+\varepsilon)^{4}\sqrt{\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle+\frac{c_{1}^{4}}{(1+\varepsilon)^{5}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{8}{(1+\varepsilon)\sqrt{\pi}}\cdot\gamma(\frac{5}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle+\frac{8}{(1-\varepsilon)\sqrt{\pi}}\cdot\gamma(\frac{5}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{9c_{2}^{2}}{(1-\varepsilon)^{3}\sqrt{\pi}}\cdot\Gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})$
$\displaystyle-\frac{3\sqrt{2}c_{2}^{3}}{(1-\varepsilon)^{4}\sqrt{\pi}}\cdot\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{c_{2}^{4}}{(1-\varepsilon)^{5}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}).$

$B=-\begin{bmatrix}E[\frac{\partial}{\partial\theta}\psi_{\theta}(X)]&E[\frac{\partial}{\partial\sigma}\psi_{\theta}(X)]&E[\frac{\partial}{\partial\varepsilon}\psi_{\theta}(X)]\\\
&E[\frac{\partial}{\partial\sigma}\psi_{\sigma}(X)]&E[\frac{\partial}{\partial\varepsilon}\psi_{\sigma}(X)]\\\
&&E[\frac{\partial}{\partial\varepsilon}\psi_{\varepsilon}(X)]\end{bmatrix}$

The elements of matrix $B$ are

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E[\frac{\partial}{\partial\theta}\psi_{\theta}(X)]$
$\displaystyle=$
$\displaystyle\frac{1}{(1+\varepsilon)^{3}\sigma^{2}2\sqrt{\pi}}\cdot\gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{1}{(1-\varepsilon)^{3}\sigma^{2}2\sqrt{\pi}}\cdot\gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\frac{\partial}{\partial\sigma}\psi_{\theta}(X)]$
$\displaystyle=$
$\displaystyle\frac{c_{1}}{\sigma^{2}(1+\varepsilon)^{2}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})-\frac{\sqrt{2}}{\sigma^{2}(1+\varepsilon)\sqrt{\pi}}\cdot\gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle+\frac{\sqrt{2}}{\sigma^{2}(1-\varepsilon)\sqrt{\pi}}\gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{c_{2}}{\sigma^{2}(1-\varepsilon)^{2}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\frac{\partial}{\partial\varepsilon}\psi_{\theta}(X)]$
$\displaystyle=$
$\displaystyle\frac{3c_{1}}{\sigma(1+\varepsilon)^{3}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})-\frac{2\sqrt{2}}{\sigma(1+\varepsilon)^{2}\sqrt{\pi}}\gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle-\frac{2\sqrt{2}}{\sigma(1-\varepsilon)^{2}\sqrt{\pi}}\gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})-\frac{3c_{2}}{\sigma(1-\varepsilon)^{3}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\frac{\partial}{\partial\sigma}\psi_{\sigma}(X)]$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{2}c_{1}}{\sigma^{2}\sqrt{\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{3(1+\varepsilon)}{\sigma^{2}\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle+\frac{3(1-\varepsilon)}{\sigma^{2}\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})+\frac{\sqrt{2}c_{2}}{\sigma^{2}\sqrt{\pi}}\cdot\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\frac{\partial}{\partial\varepsilon}\psi_{\sigma}(X)]$
$\displaystyle=$
$\displaystyle\frac{-3c_{1}}{\sigma(1+\varepsilon)\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{4}{\sigma\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle-\frac{4}{\sigma\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})-\frac{3c_{2}}{\sigma(1-\varepsilon)\sqrt{2\pi}}\cdot\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}),$
$\displaystyle E[\frac{\partial}{\partial\varepsilon}\psi_{\varepsilon}(X)]$
$\displaystyle=$
$\displaystyle\frac{-6\sqrt{2}c_{1}}{(1+\varepsilon)^{2}\sqrt{\pi}}\cdot\Gamma(1,\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})-\frac{3c_{1}^{2}}{(1+\varepsilon)^{3}2\sqrt{\pi}}\cdot\Gamma(\frac{1}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})$
$\displaystyle+\frac{10}{(1+\varepsilon)\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{1}^{2}}{2(1+\varepsilon)^{2}})+\frac{10}{(1-\varepsilon)\sqrt{\pi}}\cdot\gamma(\frac{3}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})$
$\displaystyle+\frac{6\sqrt{2}c_{2}}{(1-\varepsilon)^{2}\sqrt{\pi}}\Gamma(1,\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}})-\frac{3c_{2}^{2}}{(1-\varepsilon)^{3}2\sqrt{\pi}}\Gamma(\frac{1}{2},\frac{c_{2}^{2}}{2(1-\varepsilon)^{2}}).$

Table 1: Asymptotic variance of estimators | $Var(\hat{\tau})/n$ | $n=30$ | $n=50$ | $n=100$ | $n=150$
---|---|---|---|---|---
| $Var(\hat{\theta})/n$ | $0.190253$ | $0.114152$ | $0.057076$ | $0.038051$
$\varepsilon=-0.2$ | $Var(\hat{\sigma})/n$ | $0.018747$ | $0.011248$ | $0.005624$ | $0.003749$
$c_{1}=-1.1,c_{2}=3.7$ | $Var(\hat{\varepsilon})/n$ | $0.021061$ | $0.012637$ | $0.006318$ | $0.004212$
| $Var(\hat{\theta})/n$ | $0.059406$ | $0.035644$ | $0.017822$ | $0.011881$
$\varepsilon=-0.5$ | $Var(\hat{\sigma})/n$ | $0.022944$ | $0.013767$ | $0.006883$ | $0.004589$
$c_{1}=-0.7,c_{2}=5.0$ | $Var(\hat{\varepsilon})/n$ | $0.016147$ | $0.009688$ | $0.004844$ | $0.003229$
| $Var(\hat{\theta})/n$ | $0.010032$ | $0.006019$ | $0.003010$ | $0.002006$
$\varepsilon=-0.8$ | $Var(\hat{\sigma})/n$ | $0.035191$ | $0.021114$ | $0.010557$ | $0.007038$
$c_{1}=-0.1,c_{2}=6.4$ | $Var(\hat{\varepsilon})/n$ | $0.023486$ | $0.014091$ | $0.007046$ | $0.004697$

###  4.1 Simulation Study for Estimators of Location, Scale and Skewness
Parameters

To test the performance of asymmetric M-estimators, the contaminated version
of ESN distribution will be considered as follow:

$0.90ESN(\theta=0,\sigma=1,\varepsilon=\varepsilon_{0})+0.10ESL(\theta=0,\sigma=1,\varepsilon=\varepsilon_{0})$

$1000$ runs are performed. The sample sizes of each run are $30,50,100$ and
$150$. The relative efficiencies of estimators are also computed.

(4.17) $\displaystyle RE_{ESH}(\hat{\tau})$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{{MSE}_{ESH}(\hat{\tau})}{{MSE}_{ESN}(\hat{\tau})}\bigg{)}100$
(4.18) $\displaystyle RE_{ESH}(\hat{\tau})$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{{MSE}_{ESH}(\hat{\tau})}{{MSE}_{ESL}(\hat{\tau})}\bigg{)}100$
(4.19) $\displaystyle RE_{ESH}(\hat{\tau})$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{{MSE}_{ESH}(\hat{\tau})}{{MSE}_{ESt}(\hat{\tau})}\bigg{)}100$
(4.20) $\displaystyle RE_{ESH}(\hat{\tau})$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{{MSE}_{ESH}(\hat{\tau})}{{MSE}_{H}(\hat{\tau})}\bigg{)}100$

MSE is mean squared error obtained from simulation. ESH is epsilon-skew Huber
M-estimator, ESN, ESL and ESt epsilon-skew normal, Laplace and $t$
distributions, respectively. H is symmetric Huber M-estimator. Three degrees
of asymmetry are considered to test the asymmetric M-estimators. We make a
comparison betwen the symmetric Huber M-estimator and asymmetric Huber
M-estimator. In tables, maximum likelihood estimators of location, scale and
skewness parameters of ESN, ESL and epsilon-skew t (ESt) considered by [4]
distributions are given. The comparison of them with asymmetric M-estimator is
also considered. Tables 2-4 shows that asymmetric M-estimator (ESH)
outperforms generally than the maximum likelihood and M-estimators when the
data set has outliers. The inital points of $\theta$, $\sigma$ and
$\varepsilon$ to start the algorithm are median(x),
$MAD(x)=median(|x_{i}-median(x)|)$ and $0$, respectively. Here,
$x=\\{x_{1},x_{2},...,x_{n}\\}$. Three degress of asymmetry are considered.

Table 2: Asymmetric M (ESH), ML and M Estimators
($\varepsilon=-0.2$):$c_{1}=-1.10,c_{2}=3.70$, $k=1.4$

|  |  | $n=30$ |  |  |  | $n=50$ |  |
---|---|---|---|---|---|---|---|---|---
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$\theta$ | $0.0$ | $-0.0357$ | $0.1645$ | $0.1658$ | $100$ | $-0.0340$ | $0.0944$ | $0.0956$ | $100$
$\sigma$ | $1.0$ | $1.1180$ | $0.0633$ | $0.0773$ | $100$ | $1.1566$ | $0.0348$ | $0.0594$ | $100$
$\varepsilon$ | $-0.2$ | $-0.1603$ | $0.0200$ | $0.0216$ | $100$ | $-0.1804$ | $0.0153$ | $0.0160$ | $100$
$ESN$
$\theta$ | $0.0$ | $0.3057$ | $0.0911$ | $0.1846$ | $90$ | $0.3209$ | $0.0181$ | $0.1211$ | $79$
$\sigma$ | $1.0$ | $1.2466$ | $0.1105$ | $0.1714$ | $45$ | $1.2999$ | $0.0696$ | $0.1595$ | $37$
$\varepsilon$ | $-0.2$ | $-0.1462$ | $0.0569$ | $0.0598$ | $36$ | $-0.2200$ | $0.0454$ | $0.0455$ | $35$
$ESL$
$\theta$ | $0.0$ | $0.1736$ | $0.1014$ | $0.1315$ | $126$ | $0.1809$ | $0.0565$ | $0.0892$ | $107$
$\sigma$ | $1.0$ | $0.6316$ | $0.0137$ | $0.1494$ | $52$ | $0.6499$ | $0.0074$ | $0.1300$ | $46$
$\varepsilon$ | $-0.2$ | $-0.1306$ | $0.0308$ | $0.0356$ | $61$ | $-0.1263$ | $0.0172$ | $0.0226$ | $71$
$ESt$
$\theta$ | $0.0$ | $0.0096$ | $0.2177$ | $0.2178$ | $76$ | $0.0257$ | $0.1335$ | $0.1341$ | $71$
$\sigma$ | $1.0$ | $0.6850$ | $0.0145$ | $0.1137$ | $68$ | $0.7136$ | $0.0081$ | $0.0902$ | $66$
$\varepsilon$ | $-0.2$ | $-0.2156$ | $0.0873$ | $0.0876$ | $25$ | $-0.2007$ | $0.0510$ | $0.0510$ | $31$
$Huber$ $M$
$\theta$ | $0.0$ | $0.3342$ | $0.0714$ | $0.1831$ | $91$ | $0.3640$ | $0.0388$ | $0.1714$ | $56$
$\sigma$ | $1.0$ | $1.1378$ | $0.0425$ | $0.0615$ | $126$ | $1.0848$ | $0.0222$ | $0.0294$ | $202$
|  |  | $n=100$ |  |  |  | $n=150$ |  |
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$\theta$ | $0.0$ | $-0.0080$ | $0.0580$ | $0.0581$ | $100$ | $0.0176$ | $0.0364$ | $0.0367$ | $100$
$\sigma$ | $1.0$ | $1.1555$ | $0.0197$ | $0.0439$ | $100$ | $1.1702$ | $0.0106$ | $0.0396$ | $100$
$\varepsilon$ | $-0.2$ | $-0.1708$ | $0.0049$ | $0.0057$ | $100$ | $-0.1891$ | $0.0011$ | $0.0012$ | $100$
$ESN$
$\theta$ | $0.0$ | $0.1596$ | $0.0616$ | $0.0871$ | $67$ | $0.1904$ | $0.0213$ | $0.0575$ | $64$
$\sigma$ | $1.0$ | $1.2967$ | $0.0421$ | $0.1331$ | $33$ | $1.3220$ | $0.0274$ | $0.1311$ | $30$
$\varepsilon$ | $-0.2$ | $-0.1810$ | $0.0230$ | $0.0234$ | $24$ | $-0.2346$ | $0.0094$ | $0.0106$ | $11$
$ESL$
$\theta$ | $0.0$ | $0.1711$ | $0.0395$ | $0.0688$ | $84$ | $0.1566$ | $0.0223$ | $0.0468$ | $78$
$\sigma$ | $1.0$ | $0.6492$ | $0.0043$ | $0.1274$ | $34$ | $0.6531$ | $0.0024$ | $0.1228$ | $32$
$\varepsilon$ | $-0.2$ | $-0.1352$ | $0.0093$ | $0.0135$ | $42$ | $-0.1321$ | $0.0066$ | $0.0112$ | $11$
$ESt$
$\theta$ | $0.0$ | $-0.0206$ | $0.0741$ | $0.0746$ | $78$ | $0.0004$ | $0.0446$ | $0.0446$ | $82$
$\sigma$ | $1.0$ | $0.7239$ | $0.0041$ | $0.0804$ | $55$ | $0.7160$ | $0.0030$ | $0.0800$ | $49$
$\varepsilon$ | $-0.2$ | $-0.2270$ | $0.0252$ | $0.0259$ | $22$ | $-0.2103$ | $0.0156$ | $0.0157$ | $8$
$Huber$ $M$
$\theta$ | $0.0$ | $0.3298$ | $0.0168$ | $0.1256$ | $46$ | $0.3208$ | $0.0112$ | $0.1141$ | $32$
$\sigma$ | $1.0$ | $1.0469$ | $0.0184$ | $0.0206$ | $213$ | $1.0640$ | $0.0095$ | $0.0136$ | $291$

Table 3: Asymmetric M (ESH), ML and M Estimators
($\varepsilon=-0.5$):$c_{1}=-0.70,c_{2}=5.00$, $k=1.4$

|  |  | $n=30$ |  |  |  | $n=50$ |  |
---|---|---|---|---|---|---|---|---|---
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$\theta$ | $0.0$ | $-0.0828$ | $0.1534$ | $0.1602$ | $100$ | $-0.0741$ | $0.1083$ | $0.1138$ | $100$
$\sigma$ | $1.0$ | $1.0458$ | $0.0546$ | $0.0567$ | $100$ | $1.0274$ | $0.0369$ | $0.0377$ | $100$
$\varepsilon$ | $-0.5$ | $-0.4123$ | $0.0294$ | $0.0371$ | $100$ | $-0.4415$ | $0.0137$ | $0.0171$ | $100$
$ESN$
$\theta$ | $0.0$ | $0.4399$ | $0.1043$ | $0.2979$ | $54$ | $0.3843$ | $0.0774$ | $0.2251$ | $50$
$\sigma$ | $1.0$ | $1.4055$ | $0.2091$ | $0.3735$ | $15$ | $1.4629$ | $0.1720$ | $0.3863$ | $10$
$\varepsilon$ | $-0.5$ | $-0.3794$ | $0.0520$ | $0.0665$ | $56$ | $-0.4288$ | $0.0341$ | $0.0392$ | $44$
$ESL$
$\theta$ | $0.0$ | $0.4512$ | $0.1137$ | $0.3172$ | $51$ | $0.3933$ | $0.0694$ | $0.2241$ | $51$
$\sigma$ | $1.0$ | $0.6787$ | $0.0200$ | $0.1233$ | $46$ | $0.6933$ | $0.0118$ | $0.1059$ | $36$
$\varepsilon$ | $-0.5$ | $-0.2724$ | $0.0353$ | $0.0871$ | $43$ | $-0.3064$ | $0.0207$ | $0.0582$ | $29$
$ESt$
$\theta$ | $0.0$ | $0.1200$ | $0.2063$ | $0.2207$ | $73$ | $0.0367$ | $0.1215$ | $0.1228$ | $93$
$\sigma$ | $1.0$ | $0.6934$ | $0.0145$ | $0.1085$ | $52$ | $0.7172$ | $0.0068$ | $0.0868$ | $43$
$\varepsilon$ | $-0.5$ | $-0.4559$ | $0.0777$ | $0.0797$ | $47$ | $-0.4947$ | $0.0443$ | $0.0443$ | $39$
$Huber$ $M$
$\theta$ | $0.0$ | $1.0894$ | $0.1644$ | $1.3512$ | $12$ | $1.0201$ | $0.0895$ | $1.1299$ | $10$
$\sigma$ | $1.0$ | $1.2912$ | $0.1274$ | $0.2122$ | $27$ | $1.3318$ | $0.0893$ | $0.1994$ | $19$
|  |  | $n=100$ |  |  |  | $n=150$ |  |
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$\theta$ | $0.0$ | $-0.0824$ | $0.0504$ | $0.0572$ | $100$ | $-0.0692$ | $0.0458$ | $0.0506$ | $100$
$\sigma$ | $1.0$ | $0.9851$ | $0.0187$ | $0.0189$ | $100$ | $0.9636$ | $0.0106$ | $0.0119$ | $100$
$\varepsilon$ | $-0.5$ | $-0.4894$ | $0.0047$ | $0.0049$ | $100$ | $-0.4958$ | $0.0028$ | $0.0028$ | $100$
$ESN$
$\theta$ | $0.0$ | $0.3546$ | $0.0472$ | $0.1730$ | $33$ | $0.3134$ | $0.0359$ | $0.1341$ | $38$
$\sigma$ | $1.0$ | $1.4866$ | $0.0814$ | $0.3181$ | $6$ | $1.5389$ | $0.0716$ | $0.3619$ | $3$
$\varepsilon$ | $-0.5$ | $-0.4603$ | $0.0191$ | $0.0207$ | $24$ | $-0.4924$ | $0.0109$ | $0.0110$ | $26$
$ESL$
$\theta$ | $0.0$ | $0.3927$ | $0.0350$ | $0.1892$ | $30$ | $0.3832$ | $0.0271$ | $0.1739$ | $29$
$\sigma$ | $1.0$ | $0.6942$ | $0.0059$ | $0.0994$ | $19$ | $0.7056$ | $0.0045$ | $0.0912$ | $13$
$\varepsilon$ | $-0.5$ | $-0.3091$ | $0.0109$ | $0.0474$ | $10$ | $-0.3166$ | $0.0071$ | $0.0408$ | $7$
$ESt$
$\theta$ | $0.0$ | $-0.0568$ | $0.0594$ | $0.0626$ | $91$ | $-0.0796$ | $0.0348$ | $0.0509$ | $99$
$\sigma$ | $1.0$ | $0.7204$ | $0.0046$ | $0.0828$ | $23$ | $0.7276$ | $0.0032$ | $0.0774$ | $15$
$\varepsilon$ | $-0.5$ | $-0.5554$ | $0.0218$ | $0.0248$ | $20$ | $-0.5638$ | $0.0121$ | $0.0162$ | $17$
$Huber$ $M$
$\theta$ | $0.0$ | $1.0369$ | $0.0646$ | $1.1398$ | $5$ | $1.0252$ | $0.0588$ | $1.1099$ | $5$
$\sigma$ | $1.0$ | $1.4452$ | $0.0589$ | $0.2571$ | $7$ | $1.2744$ | $0.0780$ | $0.1533$ | $8$

Table 4: Asymmetric M (ESH), ML and M Estimators
($\varepsilon=-0.8$):$c_{1}=-0.10,c_{2}=6.40$, $k=1.4$

|  |  | $n=30$ |  |  |  | $n=50$ |  |
---|---|---|---|---|---|---|---|---|---
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$\theta$ | $0.0$ | $0.0689$ | $0.0886$ | $0.0934$ | $100$ | $-0.0262$ | $0.0377$ | $0.0383$ | $100$
$\sigma$ | $1.0$ | $1.0331$ | $0.0899$ | $0.0910$ | $100$ | $1.0235$ | $0.0578$ | $0.0584$ | $100$
$\varepsilon$ | $-0.8$ | $-0.7178$ | $0.0227$ | $0.0294$ | $100$ | $-0.7189$ | $0.0105$ | $0.0171$ | $100$
$ESN$
$\theta$ | $0.0$ | $0.4769$ | $0.1218$ | $0.3492$ | $27$ | $0.3285$ | $0.0700$ | $0.1779$ | $22$
$\sigma$ | $1.0$ | $1.7538$ | $0.4441$ | $1.0123$ | $9$ | $1.7556$ | $0.2643$ | $0.8352$ | $7$
$\varepsilon$ | $-0.8$ | $-0.6577$ | $0.0311$ | $0.0513$ | $57$ | $-0.7121$ | $0.0173$ | $0.0250$ | $68$
$ESL$
$\theta$ | $0.0$ | $0.7885$ | $0.1531$ | $0.7748$ | $12$ | $0.7100$ | $0.0721$ | $0.5762$ | $7$
$\sigma$ | $1.0$ | $0.8315$ | $0.0406$ | $0.0690$ | $132$ | $0.8008$ | $0.0194$ | $0.0591$ | $99$
$\varepsilon$ | $-0.8$ | $-0.4075$ | $0.0331$ | $0.1872$ | $16$ | $-0.4351$ | $0.0181$ | $0.1513$ | $11$
$ESt$
$\theta$ | $0.0$ | $0.1809$ | $0.1671$ | $0.1999$ | $47$ | $0.1678$ | $0.0788$ | $0.1069$ | $36$
$\sigma$ | $1.0$ | $0.7402$ | $0.0156$ | $0.0830$ | $110$ | $0.7365$ | $0.0101$ | $0.0795$ | $73$
$\varepsilon$ | $-0.8$ | $-0.7586$ | $0.0469$ | $0.0486$ | $61$ | $-0.7262$ | $0.0264$ | $0.0319$ | $53$
$Huber$ $M$
$\theta$ | $0.0$ | $1.8830$ | $8.1574$ | $11.7032$ | $1$ | $1.8797$ | $6.1185$ | $9.6521$ | $0$
$\sigma$ | $1.0$ | $1.3302$ | $0.8120$ | $0.9209$ | $10$ | $1.9495$ | $0.7885$ | $1.6899$ | $4$
|  |  | $n=100$ |  |  |  | $n=150$ |  |
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$\theta$ | $0.0$ | $-0.0811$ | $0.0166$ | $0.0232$ | $100$ | $-0.0959$ | $0.0060$ | $0.0152$ | $100$
$\sigma$ | $1.0$ | $0.9754$ | $0.0208$ | $0.0214$ | $100$ | $0.9858$ | $0.0171$ | $0.0173$ | $100$
$\varepsilon$ | $-0.8$ | $-0.7595$ | $0.0038$ | $0.0065$ | $100$ | $-0.7639$ | $0.0028$ | $0.0041$ | $100$
$ESN$
$\theta$ | $0.0$ | $0.2700$ | $0.0371$ | $0.1100$ | $21$ | $0.2194$ | $0.0244$ | $0.0726$ | $21$
$\sigma$ | $1.0$ | $1.7370$ | $0.1247$ | $0.6679$ | $3$ | $1.8112$ | $0.1056$ | $0.7637$ | $2$
$\varepsilon$ | $-0.8$ | $-0.7397$ | $0.0089$ | $0.0125$ | $52$ | $-0.7679$ | $0.0060$ | $0.0070$ | $58$
$ESL$
$\theta$ | $0.0$ | $0.6761$ | $0.0425$ | $0.4996$ | $5$ | $0.6810$ | $0.0281$ | $0.4918$ | $3$
$\sigma$ | $1.0$ | $0.7981$ | $0.0086$ | $0.0494$ | $43$ | $0.8164$ | $0.0062$ | $0.0399$ | $43$
$\varepsilon$ | $-0.8$ | $-0.4410$ | $0.0097$ | $0.1386$ | $5$ | $-0.4415$ | $0.0066$ | $0.1351$ | $3$
$ESt$
$\theta$ | $0.0$ | $0.0500$ | $0.0433$ | $0.0458$ | $51$ | $0.0323$ | $0.0223$ | $0.0234$ | $65$
$\sigma$ | $1.0$ | $0.7362$ | $0.0041$ | $0.0737$ | $29$ | $0.7427$ | $0.0033$ | $0.0695$ | $25$
$\varepsilon$ | $-0.8$ | $-0.7899$ | $0.0141$ | $0.0142$ | $45$ | $-0.8009$ | $0.0076$ | $0.0076$ | $54$
$Huber$ $M$
$\theta$ | $0.0$ | $2.0201$ | $1.8066$ | $5.8874$ | $0$ | $2.1522$ | $1.1568$ | $5.7889$ | $0$
$\sigma$ | $1.0$ | $1.8971$ | $0.9565$ | $1.7602$ | $1$ | $2.1260$ | $0.6387$ | $1.9067$ | $1$

###  4.2 Real Data Application on Estimations of Location, Scale and Skewness
Parameters

The real data sets are considerd to show the performance of the asymmetric
M-estimation (ESH). As an indicator of the goodness of fitness, $AIC$ (Akaike
information criterion) ve $BIC$ (Bayesian information criterion) are
considered. They are defined as the following forms:

(4.21) $\displaystyle AIC$ $\displaystyle=$ $\displaystyle
2k-2log(L(\hat{\tau};x_{1},x_{2},...,x_{n}))$ (4.22) $\displaystyle BIC$
$\displaystyle=$
$\displaystyle-2log(L(\hat{\tau};x_{1},x_{2},...,x_{n}))+klog(n)$

Example 1: The data set in the website
http://discover.nci.nih.gov/datasetsNature2000.jsp is analyized by [3] and
[22]. In this study, the asymmetric M-estimator (ESH), ESN, ESL, ESt, N
(Normal) and Huber M-estimation (H) distributions are used to analyze the data
set. The tuning constants of asymmetric Huber M-estimators are
$c_{1}=-0.1,c_{2}=0.3$, the tuning constant of Huber M-estimators is $k=0.2$
and the parameter $\nu$ that is a tuning constant for the ESt distribution is
$\nu=5$. Table 5 gives the estimates of parameters, log(L), AIC and BIC
values.

Table 5: Example $1$: Estimates of parameters, $logL$, AIC ve BIC

| ESH | ESN | ESL | ESt | N | H
---|---|---|---|---|---|---
$\hat{\theta}$ | 0.0386(0.0815) | 0.1240(0.1677) | 0.2480 | 0.2157(0.1170) | 0.2838(0.0047) | 0.0332(0.1606)
$\hat{\sigma}$ | 0.1195(0.0491) | 0.5139(0.0469) | 0.3033 | 0.3778(0.0329) | 0.5330(0.0023) | 0.1355(0.0875)
$\hat{\varepsilon}$ | -0.2049(0.2551) | -0.1839(0.1884) | -0.0452 | -0.0373(0.1788) | - | -
$logL$ | 27.1501 | -15.7852 | 8.2659 | -45.3664 | -17.8779 | 24.1149
AIC | -48.3002 | 37.5703 | -10.5318 | 96.7328 | 39.7559 | -44.2297
BIC | -42.0171 | 43.8534 | -4.2487 | 103.0159 | 43.9446 | -40.0410

Example 2: The tuning constants of asymmetric Huber M-estimators are
$c_{1}=-0.25,c_{2}=0.1$, the tuning constant of Huber M-estimators is $k=0.25$
and the parameter $\nu$ that is a tuning constant for the ESt distribution is
$\nu=5$. Table 6 gives the estimates of parameters, log(L), AIC and BIC
values. The tuning constants were tired until the smallest values of AIC and
BIC are gotten for these two examples.

Table 6: Example $2$: Estimates of parameters, $logL$, AIC ve BIC

| ESH | ESN | ESL | ESt | N | H
---|---|---|---|---|---|---
$\hat{\theta}$ | 0.1130(0.1287) | 0.2371(0.4414) | -0.4931 | -0.1179(0.3899) | -0.8721(0.0505) | -0.2550(0.3717)
$\hat{\sigma}$ | 0.2260(0.0928) | 1.5598(0.1423) | 0.9042 | 1.3172(0.4006) | 1.7419(0.0252) | 0.3837(0.1983)
$\hat{\varepsilon}$ | 0.3144(0.2286) | 0.5231(0.1633) | 0.1345 | 0.2950(0.1709) | - | -
$logL$ | -20.5648 | -82.3323 | -57.3653 | -113.2674 | -88.9332 | -48.8577
AIC | 47.1297 | 170.6645 | 120.7305 | 232.5347 | 181.8664 | 101.7154
BIC | 53.4127 | 176.9476 | 127.0135 | 238.8178 | 186.0551 | 105.9041

##  5 Regression Application on Asymmetric $M-$Estimation

The regression model is considered as an application of asymmetric
$M-$estimation.

(5.1)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}y_{i}=\mathbf{x}_{i}^{T}\mathbf{b}+u_{i},~{}~{}~{}~{}i=1,2,...,n$

where $y$ is dependent variable. $\mathbf{x}$ is explanatory variable.
$b=(b_{0},b_{1},...,b_{p-1})$ is a vector of parameters. $u$ is error terms.
We will get the asymmetric $M-$estimators.

###  5.1 Asymmetric $M-$estimation and its Estimators

The regression model in equation (5.1) is considered. Suppose that the error
terms $u$ are asymmetrically distributed. Then, it can be possible to model
the asymmetry in data set via the skewness parameter $\varepsilon$. To
estimate the parameters $\mathbf{b}$, $\sigma$ and $\varepsilon$, the
following function $Q$ will be considered.

$Q(\mathbf{b},\sigma,\varepsilon)=\sum_{i=1}^{n}\rho_{ESH}\bigg{(}\frac{y_{i}-\mathbf{x}_{i}^{T}\mathbf{b}}{\sigma(1-sign(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})\varepsilon)}\bigg{)}+nlog(\sigma)\\\
+\sum_{i=1}^{n}log(1-sign(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})\varepsilon)$

The minimization of function $Q$ with respect to parameters will give the
estimators of parameters.

(5.2) $\frac{\partial
Q}{\partial\mathbf{b}}=\sum_{i=1}^{n}\psi_{\mathbf{b}}\bigg{(}\frac{y_{i}-\mathbf{x}_{i}^{T}\mathbf{b}}{\sigma(1-sign(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})\varepsilon)}\bigg{)}\frac{\mathbf{x}_{i}}{\sigma(1-sign(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})\varepsilon)}=0$

let $r_{i}$ be
$\frac{u_{i}}{\sigma(1-sign(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})\varepsilon)}$.
Then, $w(r_{i})=\psi_{\mathbf{b}}(r_{i})/r_{i}$ is defined in the robustness.
Then, asymmetric $M-$estimator is

(5.3)
$\hat{\mathbf{b}}=[\sum_{i=1}^{n}\mathbf{x}_{i}\frac{w_{i}}{(\hat{\sigma}(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon}))^{2}}\mathbf{x}_{i}^{T}]^{-1}\sum_{i=1}^{n}\mathbf{x}_{i}\frac{w_{i}}{(\hat{\sigma}(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon}))^{2}}y_{i}$

where
$w_{i}=w(\frac{y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}}}{\hat{\sigma}(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon})})$
is weight function.

To get the estimator for the parameter $\sigma$, we will get the derivative
with respect to $\sigma$

(5.4) $\frac{\partial
Q}{\partial\sigma}=\frac{-n}{\sigma}+\sum_{i=1}^{n}\psi_{\sigma}\bigg{(}\frac{y_{i}-\mathbf{x}_{i}^{T}\mathbf{b}}{\sigma(1-sign(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})\varepsilon)}\bigg{)}\frac{y_{i}-\mathbf{x}_{i}^{T}\mathbf{b}}{(1-sign(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})\varepsilon)\sigma^{2}}=0$

$r_{i}=\frac{u_{i}}{\sigma(1-sign(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})\varepsilon)}$.
Then, $w(r_{i})=\psi_{\sigma}(r_{i})/r_{i}$ is weight function. The asymmetric
$M-$estimator of scale parameter is as follow:

(5.5)
$\hat{\sigma}^{2}=\frac{1}{n}\sum_{i=1}^{n}w_{i}\frac{(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})^{2}}{(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon})^{2}}$

where
$w_{i}=w(\frac{y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}}}{\hat{\sigma}(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon})})$.
The derivative of function $Q(\mathbf{b},\sigma,\varepsilon)$ with respect to
parameter $\varepsilon$ is taken

(5.6) $\frac{\partial
Q}{\partial\varepsilon}=\frac{sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})}{1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\varepsilon}-\frac{1}{\sigma}\sum_{i=1}^{n}\psi_{\varepsilon}(r_{i})\frac{(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})}{(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\varepsilon)^{2}}=0$

$r_{i}=\frac{u_{i}}{\sigma(1-sign(y_{i}-\mathbf{x}_{i}^{T}\mathbf{b})\varepsilon)}$.
$w(r_{i})=\psi_{\varepsilon}(r_{i})/r_{i}$. The asymmetric $M-$estimator of
skewness parameter is as follow:

(5.7) $\displaystyle\hat{\varepsilon}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\bigg{[}\frac{sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})}{(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon})^{2}}-w_{i}\frac{(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})^{2}sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})}{\hat{\sigma}^{2}(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon})^{3}}\bigg{]}$
$\displaystyle/$
$\displaystyle\sum_{i=1}^{n}\frac{1}{(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon})^{2}}$

$w_{i}=w(\frac{y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}}}{\hat{\sigma}(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon})})$.
The weight function for the asymmetric $M-$estimators is as follow

(5.8)
$w(r_{i})=\left\\{\begin{array}[]{ll}\frac{c_{1}}{(1+\varepsilon)^{2}r_{i}},&(-\infty,c_{1});\\\
\frac{1}{(1+\varepsilon)^{2}},&[c_{1},0);\\\
\frac{1}{(1-\varepsilon)^{2}},&[0,c_{2}];\\\
\frac{c_{2}}{(1-\varepsilon)^{2}r_{i}},&(c_{2},\infty).\end{array}\right.$

where
$r_{i}=\frac{y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}}}{\hat{\sigma}(1-sign(y_{i}-\mathbf{x}_{i}^{T}\hat{\mathbf{b}})\hat{\varepsilon})}$.
The computation steps are similar to the previous case, because the estimates
of parameters location, scale and skewness are considered. Thus, we omitted
the steps for the regression case.

###  5.2 Simulation Study for Estimators of Regression, Scale and Skewness
Parameters

In the simulation setting, the simulation plan is same with the estimations of
location, scale and skewness parameters. The true regression model is as
follow:

(5.9)
$y_{i}=3x_{0i}+5x_{1i}+x_{2i}-4x_{3i}+2x_{4i}-2x_{5i}+u_{i},~{}~{}~{}~{}i=1,2,...,n$

The error terms $u$ are distributed asymmetrically. The explanatory variables
are $x_{0},x_{1},x_{2},...,x_{5}$. The inital points of
$\mathbf{b}=(b_{0},b_{1},b_{2},b_{3},b_{4},b_{5})$, $\sigma$ and $\varepsilon$
to start the algorithm are the vector of $(0,0,0,0,0,0)$, $MAD$ and $0$,
respectively. Three degress of asymmetry are considered. In tables, the
results shows that the asymmetric M-estimators outperforms than the maximum
likelihood estimators of regression, scale and skewness parameters of ESN, ESL
and ESt distributions generally.

Table 7: Asymmetric M (ESH) and ML Estimators ($\varepsilon=-0.2$):
$c_{1}=-1.10,c_{2}=5.20$

|  |  | $n=30$ |  |  |  | $n=50$ |  |
---|---|---|---|---|---|---|---|---|---
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$b_{0}$ | $3.0$ | $2.9675$ | $0.1484$ | $0.1495$ | $100$ | $2.8862$ | $0.0841$ | $0.0929$ | $100$
$b_{1}$ | $5.0$ | $4.9565$ | $0.0344$ | $0.0362$ | $100$ | $5.0044$ | $0.0253$ | $0.0253$ | $100$
$b_{2}$ | $1.0$ | $1.0536$ | $0.0618$ | $0.0647$ | $100$ | $0.9990$ | $0.0230$ | $0.0230$ | $100$
$b_{3}$ | $-4.0$ | $-3.9775$ | $0.0351$ | $0.0356$ | $100$ | $-3.9779$ | $0.0260$ | $0.0265$ | $100$
$b_{4}$ | $2.0$ | $1.9788$ | $0.0458$ | $0.0463$ | $100$ | $2.0315$ | $0.0329$ | $0.0339$ | $100$
$b_{5}$ | $-2.0$ | $-1.9260$ | $0.0457$ | $0.0511$ | $100$ | $-1.9911$ | $0.0236$ | $0.0237$ | $100$
$\sigma$ | $1.0$ | $1.0267$ | $0.0135$ | $0.0142$ | $100$ | $1.0364$ | $0.0098$ | $0.0112$ | $100$
$\varepsilon$ | $-0.2$ | $-0.1685$ | $0.0141$ | $0.0151$ | $100$ | $-0.2003$ | $0.0124$ | $0.0124$ | $100$
$ESN$
$b_{0}$ | $3.0$ | $3.3478$ | $0.0591$ | $0.1801$ | $83$ | $3.3773$ | $0.0219$ | $0.1642$ | $57$
$b_{1}$ | $5.0$ | $5.0192$ | $0.0807$ | $0.0811$ | $45$ | $5.0420$ | $0.0517$ | $0.0535$ | $47$
$b_{2}$ | $1.0$ | $1.0348$ | $0.0929$ | $0.0941$ | $69$ | $1.0235$ | $0.0344$ | $0.0350$ | $66$
$b_{3}$ | $-4.0$ | $-3.9830$ | $0.0779$ | $0.0781$ | $46$ | $-3.9705$ | $0.0473$ | $0.0482$ | $55$
$b_{4}$ | $2.0$ | $1.9189$ | $0.1256$ | $0.1322$ | $35$ | $2.0516$ | $0.0344$ | $0.0371$ | $91$
$b_{5}$ | $-2.0$ | $-1.9313$ | $0.0697$ | $0.0745$ | $69$ | $-1.9883$ | $0.0339$ | $0.0340$ | $70$
$\sigma$ | $1.0$ | $1.1339$ | $0.0783$ | $0.0963$ | $15$ | $1.2276$ | $0.1005$ | $0.1523$ | $7$
$\varepsilon$ | $-0.2$ | $-0.0354$ | $0.0068$ | $0.0339$ | $44$ | $-0.0810$ | $0.0089$ | $0.0231$ | $54$
$ESL$
$b_{0}$ | $3.0$ | $3.1078$ | $0.1612$ | $0.1728$ | $87$ | $3.1346$ | $0.0803$ | $0.0998$ | $93$
$b_{1}$ | $5.0$ | $4.7403$ | $0.1761$ | $0.2435$ | $15$ | $4.6403$ | $0.1550$ | $0.2844$ | $8$
$b_{2}$ | $1.0$ | $0.8516$ | $0.0902$ | $0.1122$ | $58$ | $0.9095$ | $0.0978$ | $0.1060$ | $22$
$b_{3}$ | $-4.0$ | $-3.7824$ | $0.2043$ | $0.2517$ | $14$ | $-3.7205$ | $0.1581$ | $0.2363$ | $11$
$b_{4}$ | $2.0$ | $1.7326$ | $0.1880$ | $0.2595$ | $18$ | $1.8225$ | $0.0721$ | $0.1071$ | $32$
$b_{5}$ | $-2.0$ | $-1.8504$ | $0.1347$ | $0.1571$ | $33$ | $-1.8451$ | $0.0905$ | $0.1145$ | $21$
$\sigma$ | $1.0$ | $0.6916$ | $0.0384$ | $0.1335$ | $11$ | $0.7090$ | $0.0209$ | $0.1056$ | $11$
$\varepsilon$ | $-0.2$ | $-0.1064$ | $0.0194$ | $0.0282$ | $53$ | $-0.1351$ | $0.0122$ | $0.0164$ | $76$
$ESt$
$b_{0}$ | $3.0$ | $3.1056$ | $0.3187$ | $0.3298$ | $45$ | $2.9574$ | $0.2092$ | $0.2111$ | $44$
$b_{1}$ | $5.0$ | $4.9588$ | $0.0862$ | $0.0879$ | $41$ | $5.0119$ | $0.0388$ | $0.0390$ | $65$
$b_{2}$ | $1.0$ | $1.0232$ | $0.1097$ | $0.1103$ | $59$ | $1.0038$ | $0.0334$ | $0.0334$ | $69$
$b_{3}$ | $-4.0$ | $-4.0213$ | $0.0663$ | $0.0668$ | $53$ | $-3.9777$ | $0.0398$ | $0.0403$ | $66$
$b_{4}$ | $2.0$ | $1.9617$ | $0.0981$ | $0.0996$ | $46$ | $1.9986$ | $0.0354$ | $0.0354$ | $96$
$b_{5}$ | $-2.0$ | $-1.9408$ | $0.0581$ | $0.0616$ | $83$ | $-1.9755$ | $0.0394$ | $0.0400$ | $59$
$\sigma$ | $1.0$ | $0.7318$ | $0.2703$ | $0.3422$ | $4$ | $0.6817$ | $0.0086$ | $0.1100$ | $10$
$\varepsilon$ | $-0.2$ | $-0.1841$ | $0.1730$ | $0.1733$ | $9$ | $-0.2764$ | $0.1022$ | $0.1080$ | $12$

Table 8: Asymmetric M (ESH) and ML Estimators ($\varepsilon=-0.2$):
$c_{1}=-1.10,c_{2}=5.20$

|  |  | $n=100$ |  |  |  | $n=150$ |  |
---|---|---|---|---|---|---|---|---|---
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$b_{0}$ | $3.0$ | $2.7998$ | $0.0428$ | $0.0828$ | $100$ | $2.8049$ | $0.0311$ | $0.0692$ | $100$
$b_{1}$ | $5.0$ | $5.0042$ | $0.0099$ | $0.0099$ | $100$ | $4.9814$ | $0.0094$ | $0.0098$ | $100$
$b_{2}$ | $1.0$ | $0.9832$ | $0.0066$ | $0.0069$ | $100$ | $0.9984$ | $0.0068$ | $0.0068$ | $100$
$b_{3}$ | $-4.0$ | $-4.0094$ | $0.0144$ | $0.0144$ | $100$ | $-3.9802$ | $0.0065$ | $0.0069$ | $100$
$b_{4}$ | $2.0$ | $2.0029$ | $0.0081$ | $0.0081$ | $100$ | $2.0063$ | $0.0064$ | $0.0064$ | $100$
$b_{5}$ | $-2.0$ | $-1.9821$ | $0.0092$ | $0.0095$ | $100$ | $-1.9935$ | $0.0079$ | $0.0079$ | $100$
$\sigma$ | $1.0$ | $1.0473$ | $0.0044$ | $0.0066$ | $100$ | $1.0447$ | $0.0025$ | $0.0045$ | $100$
$\varepsilon$ | $-0.2$ | $-0.2070$ | $0.0044$ | $0.0044$ | $100$ | $-0.2157$ | $0.0039$ | $0.0041$ | $100$
$ESN$
$b_{0}$ | $3.0$ | $3.3573$ | $0.0224$ | $0.1501$ | $55$ | $3.3669$ | $0.0093$ | $0.1439$ | $48$
$b_{1}$ | $5.0$ | $5.0270$ | $0.0187$ | $0.0194$ | $51$ | $4.9870$ | $0.0181$ | $0.0183$ | $53$
$b_{2}$ | $1.0$ | $0.9920$ | $0.0137$ | $0.0138$ | $50$ | $1.0008$ | $0.0096$ | $0.0096$ | $71$
$b_{3}$ | $-4.0$ | $-4.0387$ | $0.0267$ | $0.0282$ | $51$ | $-3.9871$ | $0.0094$ | $0.0095$ | $72$
$b_{4}$ | $2.0$ | $2.0043$ | $0.0215$ | $0.0215$ | $38$ | $2.0051$ | $0.0114$ | $0.0114$ | $56$
$b_{5}$ | $-2.0$ | $-1.9846$ | $0.0155$ | $0.0157$ | $60$ | $-1.9977$ | $0.0131$ | $0.0131$ | $60$
$\sigma$ | $1.0$ | $1.2661$ | $0.0389$ | $0.1097$ | $6$ | $1.2943$ | $0.0254$ | $0.1120$ | $4$
$\varepsilon$ | $-0.2$ | $-0.0951$ | $0.0051$ | $0.0161$ | $27$ | $-0.0980$ | $0.0046$ | $0.0150$ | $27$
$ESL$
$b_{0}$ | $3.0$ | $3.0491$ | $0.0971$ | $0.0996$ | $83$ | $3.0047$ | $0.0839$ | $0.0840$ | $82$
$b_{1}$ | $5.0$ | $4.6588$ | $0.1551$ | $0.2815$ | $3$ | $4.6069$ | $0.1201$ | $0.2746$ | $4$
$b_{2}$ | $1.0$ | $0.8519$ | $0.0355$ | $0.0679$ | $10$ | $0.9336$ | $0.0409$ | $0.0453$ | $15$
$b_{3}$ | $-4.0$ | $-3.6718$ | $0.0844$ | $0.1921$ | $8$ | $-3.6635$ | $0.0846$ | $0.1908$ | $4$
$b_{4}$ | $2.0$ | $1.8236$ | $0.0774$ | $0.1065$ | $8$ | $1.8640$ | $0.0835$ | $0.1020$ | $6$
$b_{5}$ | $-2.0$ | $-1.8170$ | $0.0808$ | $0.1143$ | $8$ | $-1.8302$ | $0.0596$ | $0.0884$ | $9$
$\sigma$ | $1.0$ | $0.7443$ | $0.0227$ | $0.0881$ | $8$ | $0.7855$ | $0.0212$ | $0.0673$ | $7$
$\varepsilon$ | $-0.2$ | $-0.1175$ | $0.0079$ | $0.0147$ | $30$ | $-0.1789$ | $0.0130$ | $0.0135$ | $31$
$ESt$
$b_{0}$ | $3.0$ | $2.9470$ | $0.0871$ | $0.0982$ | $84$ | $2.8777$ | $0.0823$ | $0.0973$ | $71$
$b_{1}$ | $5.0$ | $4.9989$ | $0.0134$ | $0.0134$ | $74$ | $4.9699$ | $0.0128$ | $0.0130$ | $75$
$b_{2}$ | $1.0$ | $0.9859$ | $0.0109$ | $0.0111$ | $62$ | $1.0080$ | $0.0092$ | $0.0093$ | $73$
$b_{3}$ | $-4.0$ | $-4.0174$ | $0.0184$ | $0.0187$ | $77$ | $-3.9771$ | $0.0076$ | $0.0081$ | $85$
$b_{4}$ | $2.0$ | $2.0119$ | $0.0162$ | $0.0163$ | $50$ | $2.0060$ | $0.0068$ | $0.0069$ | $93$
$b_{5}$ | $-2.0$ | $-1.9831$ | $0.0142$ | $0.0145$ | $66$ | $-2.0023$ | $0.0127$ | $0.0127$ | $63$
$\sigma$ | $1.0$ | $0.7386$ | $0.0042$ | $0.0726$ | $9$ | $0.7685$ | $0.0030$ | $0.0566$ | $8$
$\varepsilon$ | $-0.2$ | $-0.2604$ | $0.0497$ | $0.0513$ | $9$ | $-0.2935$ | $0.0332$ | $0.0419$ | $10$

Table 9: DAsymmetric M (ESH) and ML Estimators ($\varepsilon=-0.5$):
$c_{1}=-0.30,c_{2}=5.30$

|  |  | $n=30$ |  |  |  | $n=50$ |  |
---|---|---|---|---|---|---|---|---|---
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$b_{0}$ | $3.0$ | $3.0629$ | $0.1043$ | $0.1083$ | $100$ | $2.9531$ | $0.0506$ | $0.0528$ | $100$
$b_{1}$ | $5.0$ | $5.0174$ | $0.0496$ | $0.0499$ | $100$ | $5.0219$ | $0.0193$ | $0.0197$ | $100$
$b_{2}$ | $1.0$ | $1.0027$ | $0.0419$ | $0.0419$ | $100$ | $1.0106$ | $0.0163$ | $0.0164$ | $100$
$b_{3}$ | $-4.0$ | $-4.0016$ | $0.0511$ | $0.0511$ | $100$ | $-4.0097$ | $0.0266$ | $0.0267$ | $100$
$b_{4}$ | $2.0$ | $2.0080$ | $0.0472$ | $0.0473$ | $100$ | $2.0126$ | $0.0122$ | $0.0123$ | $100$
$b_{5}$ | $-2.0$ | $-1.9778$ | $0.0446$ | $0.0451$ | $100$ | $-2.0076$ | $0.0171$ | $0.0172$ | $100$
$\sigma$ | $1.0$ | $1.0301$ | $0.1086$ | $0.1095$ | $100$ | $0.9721$ | $0.0100$ | $0.0108$ | $100$
$\varepsilon$ | $-0.5$ | $-0.5097$ | $0.0278$ | $0.0279$ | $100$ | $-0.5230$ | $0.0089$ | $0.0094$ | $100$
$ESN$
$b_{0}$ | $3.0$ | $3.9324$ | $0.0643$ | $0.9338$ | $12$ | $3.9961$ | $0.0563$ | $1.0485$ | $5$
$b_{1}$ | $5.0$ | $5.0071$ | $0.0898$ | $0.0899$ | $56$ | $5.0391$ | $0.0630$ | $0.0645$ | $31$
$b_{2}$ | $1.0$ | $1.0402$ | $0.0853$ | $0.0869$ | $48$ | $1.0118$ | $0.0603$ | $0.0605$ | $27$
$b_{3}$ | $-4.0$ | $-4.0389$ | $0.0957$ | $0.0972$ | $53$ | $-4.0706$ | $0.0783$ | $0.0833$ | $32$
$b_{4}$ | $2.0$ | $2.0174$ | $0.0877$ | $0.0880$ | $54$ | $2.0291$ | $0.0631$ | $0.0639$ | $19$
$b_{5}$ | $-2.0$ | $-2.0281$ | $0.1258$ | $0.1266$ | $36$ | $-2.0227$ | $0.0709$ | $0.0714$ | $24$
$\sigma$ | $1.0$ | $1.4340$ | $0.3739$ | $0.5623$ | $19$ | $1.5196$ | $0.2371$ | $0.5071$ | $2$
$\varepsilon$ | $-0.5$ | $-0.1376$ | $0.0126$ | $0.1439$ | $19$ | $-0.1717$ | $0.0098$ | $0.1176$ | $8$
$ESL$
$b_{0}$ | $3.0$ | $4.3295$ | $0.1289$ | $0.5785$ | $19$ | $3.6835$ | $0.1095$ | $0.5767$ | $9$
$b_{1}$ | $5.0$ | $4.7333$ | $0.1948$ | $0.2659$ | $19$ | $4.7566$ | $0.1906$ | $0.2499$ | $8$
$b_{2}$ | $1.0$ | $0.9364$ | $0.1432$ | $0.1473$ | $28$ | $0.8366$ | $0.0845$ | $0.1112$ | $15$
$b_{3}$ | $-4.0$ | $-3.7703$ | $0.2556$ | $0.3084$ | $17$ | $-3.7676$ | $0.1708$ | $0.2248$ | $12$
$b_{4}$ | $2.0$ | $1.9062$ | $0.1776$ | $0.1864$ | $25$ | $1.9135$ | $0.1227$ | $0.1302$ | $9$
$b_{5}$ | $-2.0$ | $-1.9351$ | $0.1649$ | $0.1691$ | $27$ | $-1.8726$ | $0.1472$ | $0.1634$ | $11$
$\sigma$ | $1.0$ | $0.7930$ | $0.0520$ | $0.0949$ | $115$ | $0.7978$ | $0.0513$ | $0.0922$ | $12$
$\varepsilon$ | $-0.5$ | $-0.1640$ | $0.0205$ | $0.1334$ | $21$ | $-0.1658$ | $0.0114$ | $0.1231$ | $8$
$ESt$
$b_{0}$ | $3.0$ | $3.0856$ | $0.1987$ | $0.2060$ | $53$ | $3.0557$ | $0.1434$ | $0.1465$ | $36$
$b_{1}$ | $5.0$ | $4.9587$ | $0.0886$ | $0.0903$ | $55$ | $5.0196$ | $0.0340$ | $0.0344$ | $57$
$b_{2}$ | $1.0$ | $0.9923$ | $0.0832$ | $0.0832$ | $50$ | $1.0168$ | $0.0378$ | $0.0381$ | $43$
$b_{3}$ | $-4.0$ | $-4.0140$ | $0.1452$ | $0.1454$ | $35$ | $-4.0211$ | $0.0310$ | $0.0315$ | $85$
$b_{4}$ | $2.0$ | $1.9574$ | $0.1279$ | $0.1297$ | $36$ | $2.0144$ | $0.0359$ | $0.0361$ | $34$
$b_{5}$ | $-2.0$ | $-2.0362$ | $0.0874$ | $0.0887$ | $51$ | $-1.9977$ | $0.0250$ | $0.0250$ | $69$
$\sigma$ | $1.0$ | $0.6531$ | $0.0376$ | $0.1580$ | $69$ | $0.6898$ | $0.0104$ | $0.1066$ | $10$
$\varepsilon$ | $-0.5$ | $-0.5929$ | $0.0841$ | $0.0927$ | $30$ | $-0.5810$ | $0.0680$ | $0.0746$ | $13$

Table 10: Asymmetric M (ESH) and ML Estimators ($\varepsilon=-0.5$):
$c_{1}=-0.30,c_{2}=5.30$

|  |  | $n=100$ |  |  |  | $n=150$ |  |
---|---|---|---|---|---|---|---|---|---
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$b_{0}$ | $3.0$ | $2.8901$ | $0.0155$ | $0.0285$ | $100$ | $2.8595$ | $0.0083$ | $0.0280$ | $100$
$b_{1}$ | $5.0$ | $5.0040$ | $0.0039$ | $0.0039$ | $100$ | $4.9915$ | $0.0033$ | $0.0034$ | $100$
$b_{2}$ | $1.0$ | $1.0027$ | $0.0053$ | $0.0053$ | $100$ | $0.9958$ | $0.0041$ | $0.0041$ | $100$
$b_{3}$ | $-4.0$ | $-3.9976$ | $0.0037$ | $0.0037$ | $100$ | $-3.9993$ | $0.0029$ | $0.0029$ | $100$
$b_{4}$ | $2.0$ | $1.9966$ | $0.0025$ | $0.0025$ | $100$ | $2.0003$ | $0.0014$ | $0.0014$ | $100$
$b_{5}$ | $-2.0$ | $-2.0032$ | $0.0036$ | $0.0036$ | $100$ | $-2.0024$ | $0.0024$ | $0.0024$ | $100$
$\sigma$ | $1.0$ | $0.9736$ | $0.0038$ | $0.0045$ | $100$ | $0.9815$ | $0.0028$ | $0.0032$ | $100$
$\varepsilon$ | $-0.5$ | $-0.5164$ | $0.0027$ | $0.0030$ | $100$ | $-0.5072$ | $0.0016$ | $0.0016$ | $100$
$ESN$
$b_{0}$ | $3.0$ | $3.9450$ | $0.0240$ | $0.9171$ | $3$ | $3.9202$ | $0.0151$ | $0.8619$ | $3$
$b_{1}$ | $5.0$ | $5.0383$ | $0.0316$ | $0.0331$ | $12$ | $5.0192$ | $0.0206$ | $0.0210$ | $16$
$b_{2}$ | $1.0$ | $0.9864$ | $0.0287$ | $0.0289$ | $18$ | $1.0054$ | $0.0171$ | $0.0171$ | $24$
$b_{3}$ | $-4.0$ | $-4.0314$ | $0.0328$ | $0.0338$ | $11$ | $-4.0368$ | $0.0171$ | $0.0184$ | $16$
$b_{4}$ | $2.0$ | $1.9962$ | $0.0234$ | $0.0234$ | $11$ | $2.0151$ | $0.0148$ | $0.0150$ | $9$
$b_{5}$ | $-2.0$ | $-2.0215$ | $0.0220$ | $0.0225$ | $16$ | $-2.0180$ | $0.0107$ | $0.0111$ | $22$
$\sigma$ | $1.0$ | $1.5933$ | $0.1231$ | $0.4751$ | $1$ | $1.6149$ | $0.0927$ | $0.4807$ | $1$
$\varepsilon$ | $-0.5$ | $-0.2053$ | $0.0074$ | $0.0942$ | $3$ | $-0.2249$ | $0.0066$ | $0.0823$ | $2$
$ESL$
$b_{0}$ | $3.0$ | $3.5221$ | $0.1011$ | $0.3737$ | $8$ | $3.4696$ | $0.0808$ | $0.3014$ | $9$
$b_{1}$ | $5.0$ | $4.5857$ | $0.3390$ | $0.5106$ | $1$ | $4.5264$ | $0.1515$ | $0.3757$ | $1$
$b_{2}$ | $1.0$ | $0.8813$ | $0.0979$ | $0.1120$ | $5$ | $0.9225$ | $0.0603$ | $0.0663$ | $6$
$b_{3}$ | $-4.0$ | $-3.6655$ | $0.1793$ | $0.2912$ | $1$ | $-3.6455$ | $0.1292$ | $0.2549$ | $1$
$b_{4}$ | $2.0$ | $1.8401$ | $0.0709$ | $0.0965$ | $3$ | $1.8302$ | $0.0605$ | $0.0893$ | $2$
$b_{5}$ | $-2.0$ | $-1.8937$ | $0.1010$ | $0.1119$ | $3$ | $-1.8087$ | $0.0618$ | $0.1009$ | $2$
$\sigma$ | $1.0$ | $0.8658$ | $0.1095$ | $0.1275$ | $4$ | $0.8737$ | $0.0444$ | $0.0603$ | $5$
$\varepsilon$ | $-0.5$ | $-0.2029$ | $0.0118$ | $0.1000$ | $3$ | $-0.2169$ | $0.0107$ | $0.0909$ | $2$
$ESt$
$b_{0}$ | $3.0$ | $2.9438$ | $0.0577$ | $0.0608$ | $47$ | $2.9242$ | $0.0342$ | $0.0400$ | $70$
$b_{1}$ | $5.0$ | $4.9897$ | $0.0192$ | $0.0192$ | $20$ | $4.9879$ | $0.0099$ | $0.0100$ | $34$
$b_{2}$ | $1.0$ | $0.9879$ | $0.0157$ | $0.0158$ | $33$ | $0.9934$ | $0.0075$ | $0.0075$ | $55$
$b_{3}$ | $-4.0$ | $-3.9983$ | $0.0127$ | $0.0127$ | $30$ | $-4.0147$ | $0.0091$ | $0.0093$ | $31$
$b_{4}$ | $2.0$ | $1.9915$ | $0.0091$ | $0.0092$ | $27$ | $2.0051$ | $0.0053$ | $0.0053$ | $26$
$b_{5}$ | $-2.0$ | $-2.0110$ | $0.0085$ | $0.0086$ | $41$ | $-1.9994$ | $0.0074$ | $0.0074$ | $32$
$\sigma$ | $1.0$ | $0.7626$ | $0.0052$ | $0.0615$ | $7$ | $0.7793$ | $0.0039$ | $0.0526$ | $6$
$\varepsilon$ | $-0.5$ | $-0.5857$ | $0.0205$ | $0.0279$ | $11$ | $-0.5775$ | $0.0113$ | $0.0173$ | $9$

Table 11: Asymmetric M (ESH) and ML Estimators ($\varepsilon=-0.8$):
$c_{1}=-0.01,c_{2}=6.20$

|  |  | $n=30$ |  |  |  | $n=50$ |  |
---|---|---|---|---|---|---|---|---|---
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$b_{0}$ | $3.0$ | $3.4040$ | $0.2139$ | $0.3771$ | $100$ | $3.1880$ | $0.1209$ | $0.1562$ | $100$
$b_{1}$ | $5.0$ | $4.9738$ | $0.1024$ | $0.1031$ | $100$ | $5.0228$ | $0.0768$ | $0.0773$ | $100$
$b_{2}$ | $1.0$ | $1.0093$ | $0.1496$ | $0.1496$ | $100$ | $1.0040$ | $0.0468$ | $0.0468$ | $100$
$b_{3}$ | $-4.0$ | $-4.0286$ | $0.1756$ | $0.1764$ | $100$ | $-3.9875$ | $0.0475$ | $0.0476$ | $100$
$b_{4}$ | $2.0$ | $1.8981$ | $0.1859$ | $0.1963$ | $100$ | $2.0424$ | $0.0417$ | $0.0435$ | $100$
$b_{5}$ | $-2.0$ | $-1.9979$ | $0.1245$ | $0.1245$ | $100$ | $-1.9866$ | $0.0307$ | $0.0308$ | $100$
$\sigma$ | $1.0$ | $1.1666$ | $0.3282$ | $0.3559$ | $100$ | $1.0377$ | $0.1754$ | $0.1768$ | $100$
$\varepsilon$ | $-0.8$ | $-0.7395$ | $0.0463$ | $0.0500$ | $100$ | $-0.8319$ | $0.0223$ | $0.0233$ | $100$
$ESN$
$b_{0}$ | $3.0$ | $4.6000$ | $0.2060$ | $2.7660$ | $14$ | $4.5671$ | $0.0794$ | $2.5350$ | $6$
$b_{1}$ | $5.0$ | $5.0640$ | $0.2286$ | $0.2327$ | $44$ | $5.0713$ | $0.1417$ | $0.1468$ | $53$
$b_{2}$ | $1.0$ | $1.0032$ | $0.1768$ | $0.1769$ | $85$ | $1.0373$ | $0.1224$ | $0.1238$ | $38$
$b_{3}$ | $-4.0$ | $-4.0691$ | $0.2639$ | $0.2687$ | $66$ | $-4.0339$ | $0.0845$ | $0.0857$ | $56$
$b_{4}$ | $2.0$ | $1.9688$ | $0.2828$ | $0.2837$ | $69$ | $2.0409$ | $0.1182$ | $0.1199$ | $36$
$b_{5}$ | $-2.0$ | $-2.0679$ | $0.2640$ | $0.2686$ | $46$ | $-2.0494$ | $0.0867$ | $0.0892$ | $35$
$\sigma$ | $1.0$ | $2.0024$ | $0.8743$ | $1.8791$ | $19$ | $2.1712$ | $0.7601$ | $2.1317$ | $8$
$\varepsilon$ | $-0.8$ | $-0.1903$ | $0.0095$ | $0.3812$ | $13$ | $-0.2500$ | $0.0109$ | $0.3134$ | $7$
$ESL$
$b_{0}$ | $3.0$ | $4.1359$ | $0.2345$ | $1.5248$ | $25$ | $4.0596$ | $0.1143$ | $1.2372$ | $13$
$b_{1}$ | $5.0$ | $4.7404$ | $0.3929$ | $0.4602$ | $22$ | $4.7188$ | $0.1857$ | $0.2647$ | $29$
$b_{2}$ | $1.0$ | $1.0128$ | $0.1411$ | $0.1412$ | $106$ | $0.9760$ | $0.1342$ | $0.1348$ | $35$
$b_{3}$ | $-4.0$ | $-3.7965$ | $0.3274$ | $0.3688$ | $48$ | $-3.6758$ | $0.1485$ | $0.2536$ | $19$
$b_{4}$ | $2.0$ | $1.8841$ | $0.2825$ | $0.2960$ | $66$ | $1.8651$ | $0.1080$ | $0.1262$ | $35$
$b_{5}$ | $-2.0$ | $-1.9028$ | $0.2800$ | $0.2894$ | $43$ | $-1.9026$ | $0.1127$ | $0.1222$ | $25$
$\sigma$ | $1.0$ | $0.9353$ | $0.1219$ | $0.1260$ | $282$ | $0.9547$ | $0.0549$ | $0.0570$ | $310$
$\varepsilon$ | $-0.8$ | $-0.1981$ | $0.0150$ | $0.3773$ | $13$ | $-0.2408$ | $0.0118$ | $0.3245$ | $7$
$ESt$
$b_{0}$ | $3.0$ | $3.2988$ | $0.5966$ | $0.6859$ | $55$ | $3.2092$ | $0.1182$ | $0.1619$ | $96$
$b_{1}$ | $5.0$ | $4.8167$ | $0.5956$ | $0.6292$ | $17$ | $4.9793$ | $0.0493$ | $0.0497$ | $155$
$b_{2}$ | $1.0$ | $0.9902$ | $0.4074$ | $0.4075$ | $37$ | $0.9896$ | $0.0538$ | $0.0539$ | $87$
$b_{3}$ | $-4.0$ | $-3.7916$ | $0.6577$ | $0.7012$ | $25$ | $-3.9680$ | $0.0785$ | $0.0795$ | $60$
$b_{4}$ | $2.0$ | $1.8722$ | $0.2551$ | $0.2714$ | $72$ | $2.0209$ | $0.0895$ | $0.0899$ | $48$
$b_{5}$ | $-2.0$ | $-1.8194$ | $0.4842$ | $0.5168$ | $24$ | $-2.0189$ | $0.0477$ | $0.0480$ | $64$
$\sigma$ | $1.0$ | $0.6930$ | $0.0436$ | $0.1379$ | $258$ | $0.7431$ | $0.0278$ | $0.0938$ | $189$
$\varepsilon$ | $-0.8$ | $-0.6726$ | $0.0726$ | $0.0889$ | $56$ | $-0.7778$ | $0.0458$ | $0.0463$ | $50$

Table 12: Asymmetric M (ESH) and ML Estimators ($\varepsilon=-0.8$):
$c_{1}=-0.01,c_{2}=6.20$

|  |  | $n=100$ |  |  |  | $n=150$ |  |
---|---|---|---|---|---|---|---|---|---
| $\tau$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$ | $\hat{\tau}$ | ${Var}(\hat{\tau})$ | ${MSE}(\hat{\tau})$ | $RE$
$ESH$
$b_{0}$ | $3.0$ | $3.0883$ | $0.0105$ | $0.0183$ | $100$ | $2.9272$ | $0.0092$ | $0.0145$ | $100$
$b_{1}$ | $5.0$ | $5.0099$ | $0.0081$ | $0.0082$ | $100$ | $5.0020$ | $0.0012$ | $0.0012$ | $100$
$b_{2}$ | $1.0$ | $0.9911$ | $0.0115$ | $0.0116$ | $100$ | $1.0028$ | $0.0013$ | $0.0014$ | $100$
$b_{3}$ | $-4.0$ | $-4.0027$ | $0.0060$ | $0.0061$ | $100$ | $-4.0009$ | $0.0022$ | $0.0022$ | $100$
$b_{4}$ | $2.0$ | $1.9883$ | $0.0063$ | $0.0064$ | $100$ | $1.9970$ | $0.0023$ | $0.0023$ | $100$
$b_{5}$ | $-2.0$ | $-1.9932$ | $0.0047$ | $0.0047$ | $100$ | $-1.9987$ | $0.0016$ | $0.0016$ | $100$
$\sigma$ | $1.0$ | $0.9697$ | $0.0077$ | $0.0086$ | $100$ | $0.9668$ | $0.0034$ | $0.0045$ | $100$
$\varepsilon$ | $-0.8$ | $-0.7452$ | $0.0042$ | $0.0072$ | $100$ | $-0.7639$ | $0.0045$ | $0.0058$ | $100$
$ESN$
$b_{0}$ | $3.0$ | $3.5584$ | $0.3346$ | $0.5631$ | $3$ | $3.5236$ | $0.2240$ | $0.4760$ | $3$
$b_{1}$ | $5.0$ | $5.0601$ | $0.0545$ | $0.0581$ | $14$ | $5.0768$ | $0.0218$ | $0.0277$ | $4$
$b_{2}$ | $1.0$ | $1.0179$ | $0.0397$ | $0.0400$ | $29$ | $1.0039$ | $0.0320$ | $0.0321$ | $4$
$b_{3}$ | $-4.0$ | $-4.0586$ | $0.0418$ | $0.0452$ | $13$ | $-4.0589$ | $0.0317$ | $0.0352$ | $6$
$b_{4}$ | $2.0$ | $2.0107$ | $0.0409$ | $0.0410$ | $16$ | $2.0366$ | $0.0191$ | $0.0205$ | $11$
$b_{5}$ | $-2.0$ | $-2.0415$ | $0.0511$ | $0.0529$ | $9$ | $-2.0498$ | $0.0295$ | $0.0320$ | $5$
$\sigma$ | $1.0$ | $2.1763$ | $0.3331$ | $1.7168$ | $1$ | $2.1919$ | $0.2047$ | $1.6253$ | $0$
$\varepsilon$ | $-0.8$ | $-0.2917$ | $0.0068$ | $0.2652$ | $2$ | $-0.3090$ | $0.0052$ | $0.2462$ | $2$
$ESL$
$b_{0}$ | $3.0$ | $3.9831$ | $0.1062$ | $1.0727$ | $2$ | $3.9723$ | $0.1267$ | $1.0721$ | $1$
$b_{1}$ | $5.0$ | $4.6342$ | $0.1360$ | $0.2698$ | $3$ | $4.6800$ | $0.1799$ | $0.2563$ | $1$
$b_{2}$ | $1.0$ | $0.9216$ | $0.0817$ | $0.0878$ | $13$ | $0.9271$ | $0.0729$ | $0.0782$ | $2$
$b_{3}$ | $-4.0$ | $-3.6813$ | $0.1384$ | $0.2400$ | $3$ | $-3.6560$ | $0.1091$ | $0.2274$ | $1$
$b_{4}$ | $2.0$ | $1.8291$ | $0.0965$ | $0.1257$ | $5$ | $1.8276$ | $0.0551$ | $0.0848$ | $3$
$b_{5}$ | $-2.0$ | $-1.8124$ | $0.0778$ | $0.1130$ | $4$ | $-1.8239$ | $0.0724$ | $0.1035$ | $2$
$\sigma$ | $1.0$ | $0.9879$ | $0.0429$ | $0.0430$ | $20$ | $0.9977$ | $0.0327$ | $0.0327$ | $14$
$\varepsilon$ | $-0.8$ | $-0.2682$ | $0.0125$ | $0.2953$ | $2$ | $-0.2683$ | $0.0116$ | $0.2942$ | $2$
$ESt$
$b_{0}$ | $3.0$ | $3.0865$ | $0.0394$ | $0.0469$ | $39$ | $3.0720$ | $0.0270$ | $0.0322$ | $45$
$b_{1}$ | $5.0$ | $5.0086$ | $0.0081$ | $0.0082$ | $100$ | $5.0022$ | $0.0046$ | $0.0046$ | $27$
$b_{2}$ | $1.0$ | $0.9993$ | $0.0108$ | $0.0108$ | $107$ | $0.9977$ | $0.0031$ | $0.0031$ | $43$
$b_{3}$ | $-4.0$ | $-4.0001$ | $0.0063$ | $0.0063$ | $97$ | $-3.9974$ | $0.0054$ | $0.0054$ | $40$
$b_{4}$ | $2.0$ | $1.9980$ | $0.0082$ | $0.0082$ | $78$ | $1.9895$ | $0.0058$ | $0.0059$ | $39$
$b_{5}$ | $-2.0$ | $-1.9989$ | $0.0073$ | $0.0073$ | $64$ | $-2.0069$ | $0.0046$ | $0.0046$ | $35$
$\sigma$ | $1.0$ | $0.7816$ | $0.0055$ | $0.0532$ | $16$ | $0.7993$ | $0.0037$ | $0.0440$ | $10$
$\varepsilon$ | $-0.8$ | $-0.8105$ | $0.0146$ | $0.0147$ | $49$ | $-0.7969$ | $0.0082$ | $0.0082$ | $71$

###  5.3 Real Data Application on Estimations of Regression, Scale and
Skewness Parameters

Example 3: The Martin Marietta data set was analyezed by [6], [5, 8, 9, 3, 4]
and [1]. The estimates of the regression parameters were obitaned by these
studies. They assume that th The data set consists of 60 monthly observations
from January 1982 to December 1986. [6] introduce a linear regression model
$y=b_{0}+b_{1}CRSP+u$ where $y$ is the excess rate of the Martin Marietta
company, $x=$ CRSP is an index of the excess rate of return for the New York
market, and $u$ is an error term. The tuning constants of asymmetric Huber
M-estimators are $c_{1}=-0.015,c_{2}=0.03$, the tuning constant of Huber
M-estimators is $k=0.03$ and the parameter $\nu$ that is a tuning constant for
the ESt distribution is $\nu=1.5$. Table 13 gives the estimates of parameters,
log(L), AIC and BIC values.

Table 13: Example $3$: Estimates of parameters, $logL$, AIC ve BIC

| ESH | ESN | ESL | ESt | N | H
---|---|---|---|---|---|---
$\hat{b}_{0}$ | -0.0047 | -0.0177 | -0.0099 | -0.0092 | 0.0011 | -0.0016
$\hat{b}_{1}$ | 0.3607 | 1.5118 | 0.6846 | 1.0009 | 1.8025 | 0.4241
$\hat{\sigma}$ | 0.1240 | 0.1105 | 0.0546 | 0.0633 | 0.1210 | 0.1373
$\hat{\varepsilon}$ | -0.1116 | -0.2581 | -0.1093 | -0.0459 | - | -
$logL$ | 70.1209 | 58.8884 | 63.7627 | 66.1233 | 53.9302 | 63.0189
AIC | -134.2418 | -111.7768 | -121.5255 | -126.2466 | -101.8604 | -120.0377
BIC | -127.9588 | -105.4938 | -115.2424 | -119.9636 | -95.5774 | -113.7547

Example 4: The same regression model considered in the previous example $3$ is
again taken. This data set can get the PET package. It is called as ”la”. The
dependent variable ”y=la$PET510” and the explanatory variable ”x=la$bflow” are
standardized. ”bflow” is a variable representing the blood flow. The variable
”PET510” is measured via the PET (Positron Emission Tomography) machines. The
detailed discussion can be found in the package named as gamlss.nl. The [20,
15, 23] studies also analyzed this data set. The tuning constants of
asymmetric Huber M-estimators are $c_{1}=-1,c_{2}=0.7$, the tuning constant of
Huber M-estimators is $k=1$ and the parameter $\nu$ that is a tuning constant
for the ESt distribution is $\nu=2$. If a value is bigger than the
$Q_{3}+1.5IQR=1.6686$, the added value with $y$ direction is considered to be
an outlier. The maximum value of explanatory variable is $5.4276$, the added
value is $5$. Here, $Q_{3}$ is the third quantile. $IQR$ is an interquartile
range. The sample size is $n=251$. After adding one outlier, the sample size
is $n=252$. Table 14 and 15 give the estimates of parameters, log(L), AIC and
BIC values.

Table 14: Example $3$: Estimates of parameters, $logL$, AIC ve BIC

| ESH | ESN | ESL | ESt | N | H
---|---|---|---|---|---|---
$\hat{b}_{0}$ | -0.0773 | -0.0841 | -0.0280 | 0.0118 | -0.0000 | -0.0229
$\hat{b}_{1}$ | 0.7299 | 0.8094 | 0.4586 | 0.6881 | 0.8046 | 0.7688
$\hat{\sigma}$ | 0.6013 | 0.9618 | 0.4908 | 0.5486 | 0.9980 | 0.6629
$\hat{\varepsilon}$ | -0.0320 | -0.1599 | 0.0000 | 0.0813 | - | -
$logL$ | -176.1575 | -265.8107 | -258.1030 | -223.6345 | -274.4067 | -191.9880
AIC | 360.3149 | 539.6215 | 524.2060 | 455.2690 | 554.8134 | 389.9761
BIC | 374.4167 | 553.7233 | 538.3078 | 469.3708 | 565.3897 | 400.5524

Table 15: Example $4$(Added outlier): Estimates of parameters, $logL$, AIC ve
BIC

| ESH | ESN | ESL | ESt | N | H
---|---|---|---|---|---|---
$\hat{b}_{0}$ | -0.0876 | -0.0850 | -0.0316 | 0.0103 | 0.0288 | -0.0175
$\hat{b}_{1}$ | 0.7440 | 0.9262 | 0.4746 | 0.6930 | 0.9491 | 0.7959
$\hat{\sigma}$ | 0.6305 | 1.1451 | 0.5220 | 0.5538 | 1.2504 | 0.6943
$\hat{\varepsilon}$ | -0.0528 | -0.2532 | -0.0323 | 0.0734 | - | -
$logL$ | -189.6646 | -314.7171 | -273.7550 | -231.9491 | -334.5094 | -207.4125
AIC | 387.3292 | 637.4342 | 555.5100 | 471.8983 | 675.0188 | 420.8250
BIC | 401.4469 | 651.5519 | 569.6277 | 486.0160 | 685.6070 | 431.4133

##  6 Conclusions

Asymmetric Huber M-estimators are suggested by using the $-log(f)$ as an
objective function in robustness. Asymmetric M-estimators have a skewness
parameter to model the potential skewness in data set. The asymptotic
properties of asymmetric M-estimators are examined. Firstly, the existence and
uniqueness of the proposed objective function with respect to parameters are
examined. After that, the asymptotic normality of estimators can be shown via
the well known Taylor expansion of the proposed asymmetric M-function and the
multivariate Slutsky’s lemma is used. For the proposed estimators, the
asymptotic variance-covariance matrix is provided. The influence function as a
local robustness property of estimators was provided. The breakdown point as
an indicator of global robustness of location estimator is shown to be $1/2$
after providing the assumptions of the robustness criteria of breakdown. An
application on regression was also considered. Real data examples for both
cases were provided. The results show the competence of our proposed
estimators when there is a potential asymmetry in data set. The asymptotic
properties of regression case will be considered as a comprehensive study. The
asymmetric M-estimates in R software will be added. The asymmetric forms of
Welsch, Hampel functions will be suggested to model the asymmetry in data set.

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