\noindent According to Tabels \ref{4-tb7}-\ref{4-tb12}, we can get the following conclusions, which will correspond with Corollary \ref{4-cl7} (d)-(f). Note that $e^{exp}_{\alpha,\beta}/\rho(X)$ is denoted as the 2-parameter expectile when $X\sim Exp(100)$ and $e^{pareto}_{\alpha,\beta}/\rho(X)$ is denoted as the 2-parameter expectile when $X\sim Pareto(3,200)$.
\begin{enumerate}
\item[(i)] Table \ref{4-tb7}-\ref{4-tb9} and Tabel \ref{4-tb10}-\ref{4-tb12} illustrates that the 2-parameter expectiles increase as $\alpha$ increases for any fixed $\beta\geq \alpha$, $\rho$ and $X$. For example, $$e^{exp}_{0.8,0.9}/E[X]=166.33 \leq 204.01=e^{exp}_{0.9,0.9}/E[X]$$ and $$e^{pareto}_{0.7,0.9}/ VaR_{0.95}(X)=157.07 \leq 189.65=e^{pareto}_{0.8,0.9}/VaR_{0.95}(X).$$ …show more content…
For instance, in the third column of Table \ref{4-tb10},
$$e^{pareto}_{0.9,0.92}/VaR_{0.95}(X)=256.88\leq 263.19=e^{pareto}_{0.9,0.98}/VaR_{0.95}(X).$$
\item[(iii)] By comparing the values in the second column and the third column of Tabels \ref{4-tb7}-\ref{4-tb12}, we find that the 2-parameter expectiles with $\rho(X)=VaR_{0.95}(X)$ are always less than those with $\rho(X)=E[X]$ for any fixed $\alpha \leq \beta$ and $X$ since $VaR_{0.95}(X)\geq E[X]$ for $X\sim Exp(100)$ or $X\sim Pareto(3,200)$. Note that $\rho(X)$ plays as a benchmark to determine the threshold for larger value of $X$. If the benchmark increases and other assumptions in the objective function are fixed, the minimizer in our model might decrease.
\item[(iv)] Moreover, $e^{exp}_{\alpha,\beta}/\rho(X) \leq e^{pareto}_{\alpha,\beta}/\rho(X)$ is ture for any fixed $\alpha \leq \beta$ when $\rho(X)=VaR_{0.95}(X)\ or\ E[X]$ by checking the corresponding value for the 2-parameter expectiles. For example,
$$e^{exp}_{0.9,0.96}/E[X] =207.73\leq
\begin{enumerate}
\item[(i)] Table \ref{4-tb7}-\ref{4-tb9} and Tabel \ref{4-tb10}-\ref{4-tb12} illustrates that the 2-parameter expectiles increase as $\alpha$ increases for any fixed $\beta\geq \alpha$, $\rho$ and $X$. For example, $$e^{exp}_{0.8,0.9}/E[X]=166.33 \leq 204.01=e^{exp}_{0.9,0.9}/E[X]$$ and $$e^{pareto}_{0.7,0.9}/ VaR_{0.95}(X)=157.07 \leq 189.65=e^{pareto}_{0.8,0.9}/VaR_{0.95}(X).$$ …show more content…
For instance, in the third column of Table \ref{4-tb10},
$$e^{pareto}_{0.9,0.92}/VaR_{0.95}(X)=256.88\leq 263.19=e^{pareto}_{0.9,0.98}/VaR_{0.95}(X).$$
\item[(iii)] By comparing the values in the second column and the third column of Tabels \ref{4-tb7}-\ref{4-tb12}, we find that the 2-parameter expectiles with $\rho(X)=VaR_{0.95}(X)$ are always less than those with $\rho(X)=E[X]$ for any fixed $\alpha \leq \beta$ and $X$ since $VaR_{0.95}(X)\geq E[X]$ for $X\sim Exp(100)$ or $X\sim Pareto(3,200)$. Note that $\rho(X)$ plays as a benchmark to determine the threshold for larger value of $X$. If the benchmark increases and other assumptions in the objective function are fixed, the minimizer in our model might decrease.
\item[(iv)] Moreover, $e^{exp}_{\alpha,\beta}/\rho(X) \leq e^{pareto}_{\alpha,\beta}/\rho(X)$ is ture for any fixed $\alpha \leq \beta$ when $\rho(X)=VaR_{0.95}(X)\ or\ E[X]$ by checking the corresponding value for the 2-parameter expectiles. For example,
$$e^{exp}_{0.9,0.96}/E[X] =207.73\leq