# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 3 Results and Discussion
### Method 1
#### 3.1.4 Investigation of Parameter Solution Spaces of the WJ Distribution
Figure 3c summarizes a series of curves under the conditions of \(\hat{\sigma}_{\text{II}}=1\) and a set of \(\mu_{WJ}\in(0.5,\,1,\,2,\,5)\). When \(\mu_{WJ}\in(0.5,\,1)\), the curves have comparable trending as in Figure 3a, shifting to the left as \(\mu_{WJ}\) increases. With \(\mu_{WJ}\) rising from two to five, the available solution range of \(\beta\) diminishes. Generally, the curves become closer for the larger \(\alpha\) and the effect of \(\mu_{WJ}\) on both \(\alpha\) and \(\beta\) is weakened. Figure 3d is the case of \(\hat{\sigma}_{\text{II}}=1\) and \(\mu_{WJ}\in(-5,\,-2,-1,-0.5)\). Of interest are the curves developing connected branches in the solution space and manifesting the shape of a boomerang [43; 44]. As \(\mu_{WJ}\) becomes smaller, the curves move to the right, and the branches become closer. In Figure 3e, branching similarly ensues in the curves of \(\alpha\) vs. \(\beta\) under the condition of \(\hat{\sigma}_{\text{I2}}=1\) and a chosen set of \(\mu_{WJ}\in(0.1,\,0.2,\,0.5,\,1)\), but it is a disconnected bifurcation instead. However, the scenery of branching varies significantly with smaller values of \(\mu_{WJ}\). When \(\mu_{WJ}\in(0.1,\,0.2)\), one branch is observable in the graph, but the other branch is far away (out of the graph). As \(\mu_{WJ}\) increases to (0.5, 1), the two separate parts of the curves gradually approach each other. The upper branch looks like the shape of an arc or a boomerang, while the lower branch is distinguishly analogous to the case of \(\hat{\sigma}_{\text{I2}}=1\) and \(\mu_{WJ}=0\) in Figure 3b, accompanied by the gradual rise of the peak value with the increase in \(\mu_{WJ}\). In parallel, the right parts in the lower branches of the curves in the series nearly overlap, implying an insensitive contribution of \(\mu_{WJ}\) to the change in the value of \(\alpha\) and \(\beta\) in the specific range. In light of Equation (8), Figure 3f exposes the solution space when \(\hat{\sigma}_{\text{I2}}=1\) and \(\mu_{WJ}\in(-1,-0.5,-0.2,-0.1)\). The shapes of the curves behave like that of \(\mu_{WJ}=0\), featuring lower peak values as \(\mu_{WJ}\) decreases. By the same token, we work on the behavior of the estimated mean \(\hat{\mu}_{\rm I}\) as a function of \(\alpha\) and \(\beta\) for a reference value of \(\mu_{W\!J}\). Substituting the expression  
\[\
u=\mu_{W\!J}+\frac{1}{\beta}\Psi\!\left(\frac{\alpha}{\beta}\right)\]  
into Equation (9) yields  
\[\hat{\mu}_{\rm I}=\frac{\beta e^{\beta[\mu_{W\!J}+\frac{1}{\beta}\Psi\!\left( \frac{\alpha}{\beta}\right)]}-\alpha}{\beta^{2}e^{\beta[\mu_{W\!J}+\frac{1}{ \beta}\Psi\!\left(\frac{\alpha}{\beta}\right)]}}.\]  
Figure 4a illustrates the 3D plot of \(\hat{\mu}_{\rm I}\) as a function of \(\alpha\) and \(\beta\) when \(\mu_{W\!J}=0\). It is apparent that for a fixed \(\beta\), \(\hat{\mu}_{\rm I}\) increases with \(\alpha\) and the growth gradually slows down, with  
the maximal value across the surface in the approach of 0. Figure 4b presents the pattern for the case of \(\mu_{WJ}=3\), which is comparable to Figure 4a and has the maximum value near three within the displayed range of \(\alpha\).