The Exponential-Centred Skew-Normal Distribution
Round 1
Reviewer 1 Report
In this paper, the authors present a new distribution which combines a skew normal distribution and an exponential distribution for the purpose of fitting skew data. Corresponding statistical properties such as moments and MLEs are derived. The centered version is also proposed to modify the drawback of Fisher information matrix being singular. Real data are provided to illustrate the estimation process. In general, the proposed results are some interesting. I would recommend after some minor changes such as sentence structure and grammar mistakes.
Author Response
Thanks for your comments. We'll have our manuscript proofread by a professional editor.
Reviewer 2 Report
This work introduces a new family of continuous probability distributions that generalizes, using Azzalini's skew normal distribution \citetext{\citealp{azzalini_1985}} and the classical exponential one, the family proposed by Hohle. The new family is suitable for modeling data with positive and negative bias. The authors denote this new family as the Exponential-Centred Skew-Normal (ECSN) distribution.
As it is written at the moment, the work does not believe that it is publishable in this journal unless the authors try to attend to the suggestions that I highlight below.
\section*{Comments}
The proposed methodology is very simple, although sometimes the expressions obtained are not very attractive.
The motivation of the work is not really convincing and it would be desirable for the authors to provide a more real one. For example, the family initially proposed by Azzalini and the one proposed by Hohl seem (the first one undoubtedly has it) to be connected with the modeling of the technical efficiency of an industry or market, for example. See, for instance \cite{kumbhakarandlovell_2000} and \cite{greene_1980a} and \cite{greene_1980b}. The authors could try to connect in some way the proposed modeling with the problem of an economic nature that seems to underlie it.
If the normal equations provided in the work are not used (I suppose the authors will have directly maximized the logarithm function of the likelihood.) They should be incorporated, as well as the Fisher information matrix, in an appendix.
The list of references provided does not seem exhaustive enough. See for example the list of papers appearing in the special issue "Symmetric and Asymmetric Distributions: Theoretical Developments and Applications (I \& II)"
Comments for author File: 
Comments.pdf
Author Response
=== we type the reviewer's comments and our responses (the new changes in the pdf are shown in blue font)
This work introduces a new family of continuous probability distributions that generalizes, using Azzalini's skew normal distribution \citetext{\citealp{azzalini_1985}} and the classical exponential one, the family proposed by Hohle. The new family is suitable for modeling data with positive and negative bias. The authors denote this new family as the Exponential-Centred Skew-Normal (ECSN) distribution.
As it is written at the moment, the work does not believe that it is publishable in this journal unless the authors try to attend to the suggestions that I highlight below.
\section*{Comments}
The proposed methodology is very simple, although sometimes the expressions obtained are not very attractive.
The motivation of the work is not really convincing and it would be desirable for the authors to provide a more real one. For example, the family initially proposed by Azzalini and the one proposed by Hohl seem (the first one undoubtedly has it) to be connected with the modeling of the technical efficiency of an industry or market, for example. See, for instance \cite{kumbhakarandlovell_2000} and \cite{greene_1980a} and \cite{greene_1980b}. The authors could try to connect in some way the proposed modeling with the problem of an economic nature that seems to underlie it.
R/ we’ve added at the outset some sentences and recent references that motivate and substantiate our proposed distribution. Note, though, that our distribution isn’t related to an economic problem but to a type of data occurring in neurosciences and experimental psychology. Note that the Azzalini 1985 is already cited in the version submitted to the journal.
If the normal equations provided in the work are not used (I suppose the authors will have directly maximized the logarithm function of the likelihood.) They should be incorporated, as well as the Fisher information matrix, in an appendix.
R/ All those equations are now placed in the appendix
The list of references provided does not seem exhaustive enough. See for example the list of papers appearing in the special issue "Symmetric and Asymmetric Distributions: Theoretical Developments and Applications (I \& II)"
R/ we now cite a couple of articles from those two special issues and that we believe have a place within our work. Also, in the discussion we’ve added a footnote that points the keen reader to those two special issues.
Round 2
Reviewer 2 Report
In my opinion, I think that the authors of this work have not responded well enough to the suggestions I made in the first report on the work.
\begin{itemize}
\item Where do the references now appear? What are the added references? I only see a few question marks
\item I continue to insist that the methodology initially proposed by Azzalini, as well as the subsequent ones, are related to the problem of estimating the technical efficiency of an industry. See in this regard the text \cite{azzalini_2014}. See also \cite{kumbhakarandlovell_2000} and \cite{greene_1980a} and \cite{greene_1980b}
Here, I present some ideas.
The classical stochastic frontier model with normal and
exponential assumptions is described by the following stochastic
representation: $(i)$ $v_i\sim \,\mbox{iid}\,N(0,\sigma_{\nu}^2)$;
$(ii)$ $u_i\sim\,\mbox{iid}$ exponential with
parameter $\sigma_u>0$; and $(iii)$ $u_i$ and
$v_i$ are distributed independently of each other
and of the regressors. The probability density functions of $v_i$ and $u_i$ are as
follows:
\begin{eqnarray*}
f_{\sigma_{\nu}}(\nu) =
\frac{1}{\sigma_{\nu}\sqrt{2\pi}}\,e^{-\frac{\nu^2}{2\sigma_{\nu}^2}},\quad
f_{\sigma_u}(u) = \frac{1}{\sigma_u}\,e^{-\frac{u}{\sigma_u}},
\end{eqnarray*}
where $-\infty<\nu<\infty$, $\sigma_{\nu}>0$, $u>0$ and
$\sigma_u>0$.
In this case for $\varepsilon=\nu-u$, we have that,
\begin{eqnarray}
f_{\sigma_u,\sigma_{\nu}}(\varepsilon)=
\frac{1}{\sigma_u}\,\Phi\left(-\frac{\varepsilon}
{\sigma_{\nu}}-\frac{\sigma_{\nu}}{\sigma_u}\right)\,\exp\left\{\frac{\varepsilon}{\sigma_u}+\frac{\sigma_{\nu}^2}{2\sigma_u^2}\right\},\label{marginal0}
\end{eqnarray}
\begin{eqnarray}
f(u|\varepsilon)=\frac{1}{\sqrt{2\pi}\,\sigma_{\nu}\,\Phi(\widetilde\mu/\sigma_{\nu})}\,\exp\left\{-\frac{1}{2\sigma_{\nu}^2}(u-\widetilde\mu)^2
\right\},\label{halfnormal}
\end{eqnarray}
where $\widetilde\mu=-\varepsilon-\sigma_{\nu}^2/\sigma_u$.
The marginal $f(\varepsilon)$ is asymmetrically distributed with
given by $ E(\varepsilon) = -\sigma_u $ and the variance by
$var(\varepsilon)= \sigma_u^2+\sigma_{\nu}^2$. On the other hand,
$u|\varepsilon=\varepsilon$ follows a half--normal distribution,
$N^{+}(\widetilde\mu,\sigma_{\nu}^2)$, with mean
\begin{eqnarray*}
E(u|\varepsilon)=\widetilde
\mu+\sigma_{\nu}\frac{\phi(-\widetilde\mu/\sigma_{\nu})}{\Phi(-\widetilde\mu/\sigma_{\nu})}=\sigma_{\nu}\left(\frac{\phi(A)}{\Phi(-A)}-A\right),
\end{eqnarray*}
and where $A=-\widetilde\mu/\sigma_{\nu}$.
The technical efficiency is computed as follows:
\begin{eqnarray*}
TE_i &=& E(e^{-u}|\varepsilon_i)= \int_0^{\infty} e^{-u}f(u|\varepsilon_i)\,du\\
&=& \frac{\Phi\left(-\frac{\varepsilon_i}{\sigma_{\nu}}-\frac{(1+\sigma_u)\sigma_{\nu}}{\sigma_u}\right)}
{\Phi\left(-\frac{\varepsilon_i}{\sigma_{\nu}}-\frac{\sigma_{\nu}}{\sigma_u}\right)}\exp\left[\varepsilon_i+\frac{(2+\sigma_u)
\sigma_{\nu}^2}{2\sigma_u}\right],\quad i=1,2,\dots,n.
\end{eqnarray*}
Comments for author File: Comments.pdf
Author Response
Please see the attchment for revised version.
Author Response File: Author Response.pdf
