## 4 Performance Analysis of Closed-Form Estimator
### The Simulation for Estimator Accuracy Analysis
Throughout numerous experiments conducted during the research process, we discovered that the \(k\), \(\mu\), and RMSE\({}_{pdf}\) values for estimators \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) and \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) were nearly identical, with differences only appearing beyond the seventh decimal place. Consequently, we consider them to have the same estimation accuracy. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number** & **Factor** & \(\mathbf{k}\) & \(\mathbf{\mu}\) & \(\mathbf{RMSE}_{pdf}\) & **Top** & **Figure** \\ \hline \multirow{4}{*}{1} & \multicolumn{3}{c}{Truth value} & 2 & 1 & / & / \\ \cline{2-7}  & \multirow{4}{*}{Estimation Method} & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\)[(24)] & 7.345803258921 & 0.403866253717 & 0.028617213697 & 3 rd \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 1.836935521234 & 1.056679155891 & 0.004732242111 & 1 st \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 1.836935521235 & 1.056679155891 & 0.004732242111 & 1 st \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.071707062788 & 1.811409799105 & 0.053510658991 & 4 th \\ \hline \multirow{4}{*}{2} & \multicolumn{3}{c}{Truth value} & 4 & 1 & / & / \\ \cline{2-7}  & \multirow{4}{*}{Estimation Method} & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\)[(24)] & 1.557613414953 & 1.783643777602 & 0.016863852210 & 3 rd \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 13.638564206393 & 0.370451235968 & 0.009715164656 & 1 st \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 13.638564206103 & 0.370451235975 & 0.009715164656 & 1 st \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.09509435442 & 2.786154089865 & 0.048716935761 & 4 th \\ \hline \multirow{4}{*}{3} & \multicolumn{3}{c}{Truth value} & 4 & 2 & / & / \\ \cline{2-7}  & \multirow{4}{*}{Estimation Method} & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\)[(24)] & 0.581701867497 & 4.950940693780 & 0.026659232023 & 3 rd \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 2.623205011929 & 2.715157714236 & 0.010815350647 & 1 st \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 2.623205011909 & 2.715157714249 & 0.010815350647 & 1 st \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.079660408693 & 5.675180409116 & 0.037623713820 & 4 th \\ \hline \multirow{4}{*}{4} & \multicolumn{3}{c}{Truth value} & 6 & 2 & / & / \\ \cline{2-7}  & \multirow{4}{*}{Estimation Method} & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\)[(24)] & 0.881337139779 & 5.787975718336 & 0.019774889712 & 3 rd \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 3.255105553371 & 3.070458936537 & 0.008496533664 & 1 st \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 3.255105553489 & 3.070458936463 & 0.008496533664 & 1 st \\ \cline{3-7}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.083446313862 & 7.358476845620 & 0.034970287599 & 4 th \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance comparison table of estimators for \(k\)-\(\mu\) fading channels when \(\mu=1\). Figure 2: \(Cont.\)  
Meanwhile, in multiple experiments, Estimator \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) frequently exhibits instances of negative or complex values for \(k\) and \(\mu\), which we consider to be incorrect estimations. The cause of this phenomenon is the lack of stability in Estimator \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\), and there are three factors contributing to its instability:  
Firstly, in Equation (6) \(\hat{k}_{yacoub}\), the denominator of the term \(2E[\alpha^{4}]^{2}-E[\alpha^{4}]-E[\alpha^{6}]\) often becomes negative, resulting in complex-valued estimates. Secondly, Estimator \(\hat{k}_{yacoub}\) violates two principles of numerical computation that should be adhered to.