## 4 Performance Analysis of Closed-Form Estimator
### The Simulation for Estimator Accuracy Analysis  
Due to the probability density function (PDF) of the \(\kappa\)-\(\mu\) fading channel model being controlled by the model parameters \(\kappa\) and \(\mu\), existing research has not yet been able to explain the conversion relationship between these two parameters. Consequently, it is not possible to evaluate the accuracy of the estimation algorithm using the deviation distance \(\left\|(k,\mu)-(\hat{k},\mu)\right\|_{2}\) between the theoretical and estimated values. Therefore, this paper employs the error \(\left\|f_{PDF}(k,\mu)-f_{PDF}(\hat{k},\hat{\mu})\right\|_{2}\) between the theoretical PDF curve and the estimated PDF curve, as well as the root mean square error (RMSE), to evaluate the estimation accuracy of the algorithm. The RMSE reflects the degree to which a set of data deviates from the true data. As the RMSE approaches 0, it indicates a smaller deviation between the theoretical PDF curve and the estimated PDF curve, resulting in higher estimation accuracy. The mathematical expression for RMSE is given as \(\sqrt{\sum_{1}^{n}\left(f_{PDF}(k,\mu,r_{i})-f_{PDF}(\hat{k},\hat{\mu},r_{i}) \right)^{2}/n}\), where \(r_{i}\sim r_{n}\) represents the minimum and maximum values of the envelope. In the experiments of this paper, the gradient \(\Delta r=r_{i+1}-r_{i}\) is 0.01. In the equation, \(f_{PDF}(k,\mu,r_{i})\) is the theoretical probability (assumed to be \(P_{i}\)) when the envelope is equal to \(r_{i}\), and \(f_{PDF}(\hat{k},\hat{\mu},r_{i})\) is the estimated probability (assumed to be \(\hat{P}_{i}\)) when the envelope is equal to \(r_{i}\). The calculation method of RMSE can be expanded as \(\left[\left(\left(P_{1}-\hat{P}_{1}\right)^{2}+\left(P_{2}-\hat{P}_{2}\right) ^{2}+\cdots+\left(P_{i}-\hat{P}_{i}\right)^{2}+\cdots+\left(P_{n}-\hat{P}_{n} \right)^{2}\right)/n\right]\) 0.5. Simulation experiment samples were generated using the Monte Carlo method, producing channel data that follows the \(\kappa\)-\(\mu\) distribution. In each estimation experiment, four estimators employed the same set of generated channel data, with a sample size of 10,000. Four simulation experiments were conducted with the following parameter settings: \(\kappa=2\) and \(\mu=1\); \(\kappa=4\) and \(\mu=1\); \(\kappa=4\) and \(\mu=2\); and \(\kappa=6\) and \(\mu=2\). The estimated PDF curves were plotted based on the estimation results and compared with the theoretical PDF curves. The comparison and error plots are illustrated in Figure 2. The estimated values and root mean square errors were rounded to the 12 th decimal place and recorded in Table 4.  
In Figure 2, the left column of the plots demonstrates the comparison between the PDF curves calculated from the estimated \(k\) and \(\mu\) parameters and the theoretical PDF curves. The right column of plots shows the errors between the four estimated PDF curves and the theoretical PDF curves. In Table 3, RMSE\({}_{pdf}\) refers to the root mean square error between the PDF curve and the theoretical PDF curve, while the last column displays the ranking of estimation accuracy for each estimator in each experiment. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Estimator Name** & \multicolumn{2}{c}{**Number of Samples 10,000**} & \multirow{2}{*}{**Ranking**} \\  & **Estimated 1 Time** & **Estimated 50 Time** & \\ \hline \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\)[24] & 0.001372 & 0.017895 & 1 st \\ \hline \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.001481 & 0.018135 & 2 nd \\ \hline \(\hat{k}\)-\(\hat{\mu}_{\text{A}}\) & 0.004415 & 0.048401 & 4 th \\ \hline \(\hat{k}\)-\(\hat{\mu}_{\text{B}}\) & 0.002582 & 0.028465 & 3 rd \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Total runtime of the estimator characteristics table.