# Moment-Based Parameter Estimation for the \(\Gamma\)-Parameterized TWDP Model
## 4 AsV and CRB  
To further assess the performance of proposed estimators, corresponding asymptotic variances for \(K\) and \(\Gamma\), \(AsV_{K}\) and \(AsV_{\Gamma}\), are calculated from (20) and (21), as [4]:  
\[AsV_{K}=g_{K}\text{CvM} \tag{22}\]  
\[AsV_{\Gamma}=g_{\Gamma}\text{CvM} \tag{23}\]  
where:  
\[g_{K}=\left[\frac{\partial\hat{K}}{\partial\hat{\mu}_{2}},\frac{\partial\hat{ K}}{\partial\hat{\mu}_{4}},\frac{\partial\hat{K}}{\partial\hat{\mu}_{6}} \right]_{\hat{\mu}_{2}=\mu_{2},\hat{\mu}_{4}=\mu_{4},\hat{\mu}_{6}=\mu_{6}} \tag{24}\]  
\[g_{\Gamma}=\left[\frac{\partial\hat{\Gamma}}{\partial\hat{\mu}_{2}},\frac{ \partial\hat{\Gamma}}{\partial\hat{\mu}_{4}},\frac{\partial\hat{\Gamma}}{ \partial\hat{\mu}_{6}}\right]_{\hat{\mu}_{2}=\mu_{2},\hat{\mu}_{4}=\mu_{4}, \hat{\mu}_{6}=\mu_{6}} \tag{25}\]  
and CvM is a covariance matrix with elements:  
\[\left[\text{CvM}\right]_{ij}=\frac{1}{N}\big{(}\mu_{2i+2j}-\mu_{2i}\mu_{2j} \big{)},\,\text{for}\,\,i,j=1,2,3 \tag{26}\]  
In addition, Cramer-Rao lower bounds are numerically calculated as [4]:  
\[CRB_{K}=[I(\theta)^{-1}]_{11} \tag{27}\]  
\[CRB_{\Gamma}=[I(\theta)^{-1}]_{22} \tag{28}\]  
where the elements of Fisher Information Matrix \(I(\theta)\), \([I(\theta)]_{ij}\), are determined as:  
\[[I(\theta)]_{ij}=N\mathbb{E}\Bigg{[}\frac{\partial\text{Inf}(r)}{\partial\theta _{i}}\frac{\partial\text{Inf}(r)}{\partial\theta_{j}}\Bigg{]},\,\text{for}\,\,i,j =1,2,3 \tag{29}\]  
and where \(N\) is the number of observations, \(f(r)\) is the closed-form TWDP envelope PDF given by [14] [Equation (7)], and \(\theta=[K,\Gamma,\Omega]\)  
Thereby, following the approach presented in [4,7], instead of using CRB and AsV, the square root of CRB and AsV normalized to \(N\) and the true value of estimating parameter are used in estimator performance assessment. Thus, to assess the estimation error, the sqrt-normalized CRB and AsV of \(\hat{K}\), \(\sqrt{CRB_{K}N/K^{2}}\) and \(\sqrt{AsV_{K}N/K^{2}}\),  
Figure 3: \(\hat{\Gamma}_{mean}\) (with absolute error bars) vs. \(\Gamma\) for **(a)**\(K=1\), **(b)**\(K=3\), **(c)**\(K=10\), and **(d)**\(K=30\). The solid line shows a linear regression fit to the data. The unit slope dashed line is illustrated as a benchmark. and sqrt-normalized CRB and AsV of \(\hat{\Gamma}\), \(\sqrt{CRB_{\Gamma}N/\Gamma^{2}}\) and \(\sqrt{AsV_{\Gamma}N/\Gamma^{2}}\), are plotted in Figure 5.  
From Figure 4, it can be observed that the estimation error of \(K\) increases with the decrease of parameter \(K\). Thereby, when the power of specular components \(V_{1}^{2}+V_{2}^{2}\) is small with respect to the power of diffuse components \(2\sigma^{2}\) (i.e., when the TWDP channel becomes Rayleigh-like), the error in estimation of \(K\) is very large. However, as the value of parameter \(K\) increases, i.e., as \(V_{1}^{2}+V_{2}^{2}\) overrides \(2\sigma^{2}\), the estimation of parameter \(K\) becomes very accurate. Figure 4 also shows that the error in estimation of \(K\) grows with the reduction of \(\Gamma\), indicating that it becomes harder to accurately estimate \(K\) as the specular component with the magnitude \(V_{2}\) becomes less significant with respect to one with the magnitude \(V_{1}\). From Figure 4, it can also be observed that the values of sqrt-normalized \(AsV_{K}\) are remarkably close to the sqrt-normalized \(CRB_{K}\) for the entire considered range of parameters \(K\) and \(\Gamma\), indicating almost asymptotic efficiency of the proposed estimator of a parameter \(K\). Figure 5 shows that the estimation error of parameter \(\Gamma\) behaves similarly as the estimation error of \(\hat{K}\), with respect to \(K\) and \(\Gamma\). Hence, the estimation of \(\Gamma\) deteriorates with the reduction of \(K\), i.e., as the power of diffuse components becomes more significant with respect to \(V_{1}^{2}+V_{2}^{2}\). The estimation error of \(\hat{\Gamma}\) is large for small values of \(\Gamma\), indicating that it is hard to estimate the values of \(\Gamma\) when \(V_{2}\) is insignificant with respect to \(V_{1}\).