# Moment-Based Parameter Estimation for the \(\Gamma\)-Parameterized TWDP Model
## 4 AsV and CRB
For moderate values of \(\Gamma\), \(\hat{\Gamma}\) given by (21) starts to provide pretty accurate results, especially for large values of \(K\). However, as \(\Gamma\) approaches one, estimation of \(\Gamma\) becomes more inaccurate. In these conditions, the magnitudes of specular components, \(V_{1}\) and \(V_{2}\), become  
similar. Considering the aforementioned and the fact that the phase difference between these components is uniform, the probability of destructive superposition of specular components becomes more likely, making their overall magnitude often insignificant. Thus, as \(\Gamma\) approaches one, it gets harder to accurately determine the value of \(V_{2}/V_{1}\), especially when the power of diffuse components is large with respect to \(V_{1}^{2}+V_{2}^{2}\). When it comes to the observation of the proximity of the proposed \(\Gamma\) estimator to its CRB, it can be concluded from Figure 5 that the values of the sqrt-normalized \(AsV_{\Gamma}\) are remarkably close to the sqrt-normalized \(CRB_{\Gamma}\) for \(K\geq 2\) in the entire range of \(\Gamma\), making the proposed estimator asymptotically efficient for the considered values of \(K\). Accordingly, except for providing estimation errors significantly close to the corresponding CRBs, moment-based estimators (20) and (21) provide accurate estimates obtained from a relatively small number of samples, which can be clearly observed from Figures 4 and 5. For example, if we assume that the sufficient accuracy of estimation process is 20%, it can be achieved with \(N=10^{4}\) samples for \(\Gamma\in[0.3,0.9]\) and \(K\geq 3\) (obtained by multiplying estimation errors \(\sqrt{AsV_{K}N/K^{2}}\) and \(\sqrt{AsV_{\Gamma}N/\Gamma^{2}}\) by \(1/\sqrt{N}\)). If necessary, the estimation accuracy in the determined region can be increased, or the region itself can be further expanded, by involving more samples within the estimation process (e.g., by employing \(N=10^{6}\) samples, relative estimation error in the considered region of parameters \(K\) and \(\Gamma\) could be reduced to 2%, or the estimation error of 20% could be achieved for the wider range of \(K\) and \(\Gamma\), i.e., \(K\geq 3\) and \(\Gamma\in[0.16,0.99]\)). In this way, the procedures used to create Figures 4 and 5 can be used to determine the number of samples needed to obtain desired estimation accuracy within the desired range of parameters \(K\) and \(\Gamma\). Comparison of Moment-Based Estimators for the \(\Delta\)- and \(\Gamma\)-Parameterized TWDP Model  
In order to observe qualitative differences between \(\Gamma\)- and \(\Delta\)-based parameterization and to gain more precise insight into the relation between estimation errors for considered parameterizations, \(AsV_{\Delta}\) and \(AsV_{\Gamma}\) are normalized to the same parameter ratio \(V_{2}/V_{1}\) and presented in \[AsV_{\Gamma}=g_{\Gamma}\text{CvM} \tag{23}\]0 This enables us to compare absolute values of AsV for \(\Delta\) and \(\Gamma\) and to discover differences in their estimation errors for considered ratios between \(V_{1}\) and \(V_{2}\). Figure 6 shows that, for \(0\leq V_{2}/V_{1}\leq 0.5\) (which corresponds to \(0\leq\Delta\leq 0.8\)), estimation error of parameter \(\Gamma\) is up to two times smaller than the error obtained for \(\Delta\) parameter estimation. On the other side, for \(0.5<V_{2}/V_{1}<0.8\) (i.e., \(0.8<\Delta<0.96\)) and \(K\geq 3\), there is no significant difference in accuracy of \(\hat{\Delta}\) and \(\hat{\Gamma}\). Finally, for \(V_{2}/V_{1}\in[0.8,1]\), error in estimation of \(\Gamma\) starts to increase with the increment of \(V_{2}/V_{1}\), thus being in line with the model's physical mechanisms. On the contrary, in the considered region, \(\Delta\)-based parameterization provides false accurate results, obtained by also considering values of \(\hat{\Delta}\) greater than one in calculation of \(AsV_{\Delta}\).