tions in consideration of predictions errors. Actually, in some cases, it is best not to act as predicted because of prediction errors. For example, the paper [@Yamaguchi2018] formulates a method for minimizing the expected value of the procurement cost of electricity in two popular spot markets: [*day-ahead*]{} and [*intra-day*]{}, under the assumption that the expected value of the unit prices and the distributions of the prediction errors for the electricity demand traded in two markets are known. The paper showed that if the procurement is increased or decreased from the prediction, in some cases, the expected value of the procurement cost is reduced. (2) In recent years, prediction methods have been black boxed by the big data and machine learning (see, e.g., Ref. [@10.1145/3236009]). The day will soon come, when we must minimize the objective function by using predictions obtained by such black boxed methods. In our method, even if we do not know the prediction $\hat{y}$, we can determine the parameter $C$ if we know the prediction error distribution $f$ and asymmetric loss function $L$.

To obtain $y^{*}$, we derive $\operatorname{{E}}[\Pe(Z + c)]$ for any $c \in \mathbb{R}$. Let $\G(a, x)$ and $\g(a, x)$ be the upper and the lower incomplete gamma functions, respectively (see, e.g., Ref. [@doi:10.1142/0653]). The expected value and the variance of $\Pe(Z + c)$ are as follows:

\[lem:1.1\] For any $c \in \mathbb{R}$, we have $$\begin{aligned}
(1)\quad 
\operatorname{{E}}[\Pe(Z + c)] 
&= \frac{(k_{1} - k_{2}) c}{2} 
+ \frac{(k_{1} + k_{2}) \lvert c \rvert}{2 \G(a)} 
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) 
+ \frac{(k_{1} + k_{2}) b}{2 \G(a)} 
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right), \\
(2)\quad 
\operatorname{{V}}[\Pe(Z + c)] 
&= \frac{(k_{1} + k_{2})^{2} c^{2}}{4} 
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{2 \G(a)} 
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} b \lvert c \rve