# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
## 3 Results and Discussion
### Method II
#### 3.2.2 Numerical Assessment  
Following the tactic employed in Method I to appraise the effectiveness of Method II, we consider the case of the standard normal distribution and replace the fitted parameter values \(\alpha_{0}=12.52\), \(\beta_{0}=0.07915\) and \(\
u_{0}=63.96\) of the WJ distribution into Equations (18) and (19), respectively. It is immediately found that both \(\hat{\
u}_{\Pi 1}\) and \(\hat{\
u}_{\Pi 2}\) differ enormously from one, for instance, \(\hat{\
u}_{\Pi 1}\approx 12.63\), which is obviously much larger than one. As a consequence, the results obtained by means of the expansion in Method II may not appositely be discussed in the same way as in Method I. In view of the discrepancy, we observe Equation (18) which shows an inverse relation between \(\hat{\
u}_{\Pi 1}\) and \(\hat{\
u}_{\Pi 2}\) equal to one, it is convenient to obtain \(\beta=1\) and then substitute it into Equation (19) to obtain the computed result of \(\alpha\approx 1.15\). As a remark, the substitution of \(\alpha\) and \(\beta\) into Equation (20) results in \(\
u=0.15\) under the condition of \(\hat{\mu}_{\Pi}=0\). The corresponding curve of the WJ function (2) with the calculated parameters, labeled by \(\mathrm{WJ}_{\Pi}\), is shown in Figure 5a. It is transparent that the WJ function fits the Gaussian better near the peak, but the discrepancy grows bigger as \(x\) moves away from the origin. The errors of the curve relative to the standard normal distribution are presented as the scatter points \(\Delta_{\Pi}\) in Figure 5b, plainly larger than that of Method I. Nonetheless, the value of \(\sigma\) is to be determined when the Gaussian function is unknown; thus, we are not in the position to set a value to \(\beta\) by applying Equation (18). We propose an alternative estimation scheme. According to Equations (16) or (17), the parameter \(\
u\) appears only as \((x-v)\) in the exponential part and affects only \(\hat{\mu}_{\Pi}\). Taking the point into consideration, we could move forward by assuming \((x-v)\) as the independent variable (equivalent to the case of taking \(v=0\) which reflects the properties of some special distributions [12; 13; 14]). In actuality, fitting the standard normal distribution by the WJ function with \((x-v)\) as the independent variable produces the parameter set of \(\alpha=1.188\) and \(\beta=0.9522\). The associated curve is shown by the label of \(\mathrm{WJ}_{0}\) in Figure 5a and demonstrates better performance than that of \(\mathrm{WJ}_{\Pi}\). In Figure 5b, the scatters \(\Delta_{0}\) chart the errors of the fitted WJ function versus the standard normal distribution. Employing Equation (20), we directly obtain \(\hat{\mu}_{\Pi}=-0.26\), which surely reveals a systematic deviation due to selecting \((x-v)\) as the independent variable. We need to correct it when analyzing the truncated functions involving \(\hat{\mu}_{\Pi}\) and the magnitude of the correction is \(-\hat{\mu}_{\Pi}\) or \(\
u=0.26\). In Figure 5b, the symbols \(\Delta_{\Pi 1}\) and \(\Delta_{\Pi 2}\) represent the residual plots of \(\mathrm{N}(\hat{\mu}_{\Pi},\hat{\
u}_{\Pi 1})\) and \(\mathrm{N}(\hat{\mu}_{\Pi},\hat{\
u}_{\Pi 2})\) relative to \(\mathrm{N}(0,\ 1)\) after substituting the values of \(\alpha\), \(\beta\) and \(v\) into the equations for \(\hat{\mu}_{\Pi}\), \(\hat{\
u}_{\Pi 1}\) and \(\hat{\
u}_{\Pi 2}\). The errors are certainly larger when compared to Method I. The observation is further supported by calculating the related cumulative probability distributions, as presented in Figure 5c (where Gaussian, CaseII1, CaseII2, and \(\mathrm{WJ}_{0}\) denote the cumulative probability distribution functions of \(\mathrm{N}(0,\ 1)\), \(\mathrm{N}(\hat{\mu}_{\Pi},\hat{\
u}_{\Pi 1})\), \(\mathrm{N}(\hat{\mu}_{\Pi},\hat{\
u}_{\Pi 2})\) and the fitted WJ function, respectively). It is evident that the first three curves are close together as a result of the choice of the parameters, but the difference with the fitted WJ function is large.