## 4 A Novel Loss Function: The Weighted Q-Symmetric Entropy Loss (WQEL) Function
### The Definition of the WQEL Function  
The WQEL function is derived from the QEL function and is defined as  
\[L_{\text{WQEL}}\big{(}\hat{\theta},\theta\big{)}=\frac{1}{\theta^{z}}\left[ \left(\frac{\hat{\theta}}{\theta}\right)^{q}+\left(\frac{\hat{\theta}}{\theta }\right)^{-q}-2\right],z\in R,q\
eq 0, \tag{33}\]  
where \(\frac{1}{\theta^{z}}\) denotes the weighted function imposed on the expression of the QEL function. Obviously, the WQEL function is a generalized form of the QEL function, and it is asymmetric. When \(z=0\), the WQEL devolves into the QEL. For symmetric loss functions, such as the QEL function, overestimates and underestimates have the same effects on its function value. Thus, the estimates obtained from the QEL are inaccurate in cases where there is a bias toward overestimated or underestimated values. In contrast to the symmetry of the QEL function, the new WQEL function is asymmetric, which means that it is more practical and flexible than the QEL function. Specifically, when \(z=0\), the WQEL function degrades into the QEL function, the estimator of which has no preference for overestimation or underestimation. When \(z>0\), the estimator under the WQEL function is more biased toward the case of \(\frac{\hat{\theta}}{\theta}<1\), and vice versa for the case of \(\frac{\hat{\theta}}{\theta}>1\). Figure 4 provides a graphical representation of the WQEL function with the given parameter values \(\theta=1,2\) and the chosen values of \(q\) and \(z\). From Figure 4, we can conclude that the WQEL function is strictly convex and non-negative. Moreover, regardless of the values of the parameters \(q\) and \(z\), as the deviation of \(\frac{\hat{\theta}}{\hat{\theta}}\) from 1 increases, the value of the WQEL function grows larger, fulfilling the requirements for a loss function. However, when \(\theta=1\), the WQEL is equivalent to the QEL function, which means that the value of \(z\) has no effect on the value of the WQEL function. Additionally, increasing the value of \(q\) or decreasing the value of \(z\) leads to a larger function value. Under the WQEL function \(L_{WQEL}\big{(}\hat{\theta},\theta\big{)}=\frac{1}{\mu^{p}}\left[\left(\hat{ \hat{\theta}}\right)^{q}+\left(\hat{\hat{\theta}}\right)^{-q}-2\right],z\in R,q \
eq 0\), the Bayesian estimate \(\hat{\theta}\) of \(\theta\) to be estimated is derived as  
\[\hat{\theta}_{WQEL}=\left[\frac{E\big{(}\theta^{q-z}|\underline{X}\big{)}}{E \big{(}\theta^{-q-z}|\underline{X}\big{)}}\right]^{\frac{1}{2q}}. \tag{34}\]  
The Bayesian estimation \(\hat{\theta}\) for \(\theta\) is the estimate used to minimize the posterior risk, which is expressed as  
\[Risk_{L}\big{(}\hat{\theta},\theta\big{)} =E\big{(}L_{WQEL}\big{(}\hat{\theta},\theta\big{)}\big{)}\] \[=\int_{\forall\theta}\frac{1}{\theta^{z}}\left[\left(\frac{\hat{ \theta}}{\theta}\right)^{q}+\left(\frac{\hat{\theta}}{\hat{\theta}}\right)^{-q }-2\right]\cdot\pi(\theta|\underline{X})d\theta\] \[=\int_{\forall\theta}\hat{\theta}^{q-q-z}\pi(\theta|\underline{X })d\theta+\int_{\forall\theta}\hat{\theta}^{q-q}\theta^{q-z}\pi(\theta| \underline{X})d\theta-2\int_{\forall\theta}\frac{1}{\theta^{z}}\pi(\theta| \underline{X})d\theta\] \[=\hat{\theta}^{q}E\big{(}\theta^{-q-z}|\underline{X}\big{)}+ \hat{\theta}^{-q}E\big{(}\theta^{q-z}|\underline{X}\big{)}-2E\big{(}\theta^{ -z}|\underline{X}\big{)},\]  
where \(\pi(\theta|\underline{X})\) refers to the given posterior distribution. Considering that the parameter estimate under the WQEL function aims to minimize the posterior risk, we find  
\[\frac{\partial Risk_{L}}{\partial\hat{\theta}}=q\hat{\theta}^{q-1}E\big{(} \theta^{-q-z}|\underline{X}\big{)}-q\hat{\theta}^{-q-1}E\big{(}\theta^{q-z}| \underline{X}\big{)}.\]  
Figure 4: The plots of the WQEL function. It is worth clarifying that \(Risk_{L}(\hat{\theta},\theta)\) is a convex function; thus, we can straightforwardly obtain the minima of the posterior risk by setting \(\frac{\partial Risk_{L}}{\partial\theta}=0\).