## 3 Methods
### Complex Parameter Wald Test
#### 3.2.1 Complex Wald Test for General Cases  
Observe that \(p_{\mathbf{x}}(\tilde{\mathbf{x}};\tilde{\boldsymbol{\theta}})\) is a real function of \(\tilde{\boldsymbol{\theta}}\), depending on \(\tilde{\boldsymbol{\theta}}\) and \(\tilde{\boldsymbol{\theta}}^{*}\), where \(*\) denotes complex conjugate. Denote \(p_{\mathbf{x}}(\tilde{\mathbf{x}};\tilde{\boldsymbol{\theta}})\) as \(p_{\mathbf{x}}(\tilde{\mathbf{x}};\tilde{\boldsymbol{\Theta}})\equiv p_{ \mathbf{x}}(\tilde{\mathbf{x}};\tilde{\boldsymbol{\theta}},\tilde{\boldsymbol {\theta}}^{*})\), where \(\tilde{\boldsymbol{\Theta}}=\left[\tilde{\boldsymbol{\theta}}^{T}\ \tilde{\boldsymbol{\theta}}^{H}\right]^{T}\), \(\tilde{\boldsymbol{\Theta}}\in\mathbb{C}^{2p\times 1}\), and \((\cdot)^{H}\) represents Hermitian. Also, the complex partial derivatives of a real scalar function \(g(\tilde{z},\tilde{z}^{*})\equiv f(x,y)\) are  
\[\frac{\partial g(\tilde{z},\tilde{z}^{*})}{\partial\tilde{z}}=\frac{1}{2}( \frac{\partial f}{\partial x}-j\frac{\partial f}{\partial y}) \tag{27}\]  
and  
\[\frac{\partial g(\tilde{z},\tilde{z}^{*})}{\partial\tilde{z}^{*}}=\frac{1}{2}( \frac{\partial f}{\partial x}+j\frac{\partial f}{\partial y}), \tag{28}\]  
where \(\tilde{z}=x+jy\). And, for a real function \(g\) of complex vectors \(\tilde{\mathbf{z}}\) and \(\tilde{\mathbf{z}}^{*}\). \[\left[\frac{\partial g(\tilde{\mathbf{z}},\tilde{\mathbf{z}}^{*})}{\partial \tilde{\mathbf{z}}}\right]_{i}=\frac{\partial g(\tilde{\mathbf{z}},\tilde{ \mathbf{z}}^{*})}{\partial\tilde{\mathbf{z}}_{i}} \tag{29}\]  
\[\left[\frac{\partial g(\tilde{\mathbf{z}},\tilde{\mathbf{z}}^{*})}{\partial \tilde{\mathbf{z}}^{*}}\right]_{i}=\frac{\partial g(\tilde{\mathbf{z}},\tilde{ \mathbf{z}}^{*})}{\partial\tilde{\mathbf{z}}_{i}^{*}}. \tag{30}\]  
With these definitions, we have  
**Theorem 1**. : _Complex Wald Test_  
\[T_{\tilde{W}}(\tilde{\mathbf{x}})=(\tilde{\mathbf{\Theta}}-\tilde{\mathbf{\Theta} }_{0})^{H}\tilde{\mathbf{I}}(\tilde{\mathbf{\Theta}})(\tilde{\mathbf{\Theta}}- \tilde{\mathbf{\Theta}}_{0})\big{|}_{\tilde{\mathbf{\Theta}}=\tilde{\mathbf{ \Theta}}_{1}}=T_{W}(\mathbf{x}), \tag{31}\]  
_where_  
\[\tilde{\mathbf{I}}(\tilde{\mathbf{\Theta}})=E\Bigg{(}\frac{\partial\ln p_{ \mathbf{x}}(\tilde{\mathbf{x}};\tilde{\mathbf{\Theta}})}{\partial\tilde{ \mathbf{\Theta}}^{*}}\frac{\partial\ln p_{\mathbf{x}}(\tilde{\mathbf{x}}; \tilde{\mathbf{\Theta}})}{\partial\tilde{\mathbf{\Theta}}^{*}}^{H}\Bigg{)}, \tag{32}\]  
\(T_{\tilde{W}}(\tilde{\mathbf{x}})\) _is a complex Wald test statistic, and \(T_{W}(\mathbf{x})\) is real Wald test statistic. Note that \(\hat{\tilde{\mathbf{\Theta}}}_{1}\) is the MLE of \(\tilde{\mathbf{\Theta}}\) under \(\mathcal{H}_{1}\), and \(\hat{\tilde{\mathbf{\Theta}}}_{1}\) is the MLE of \(\tilde{\mathbf{\theta}}\) under \(\mathcal{H}_{1}\); \(\tilde{\mathbf{\Theta}}_{0}=\left[ \begin{array}{c}\tilde{\mathbf{\theta}}_{ 0}\\ \tilde{\mathbf{\theta}}_{0}^{*}\end{array} \right]\), and \(\tilde{\tilde{\mathbf{\theta}}}_{0}\) is the true value of the unknown parameter under \(\mathcal{H}_{0}\). Note that \(\tilde{\mathbf{I}}(\tilde{\mathbf{\Theta}})\) can be used as a FIM for an unbiased estimation of \(\tilde{\mathbf{\Theta}}\). Also, \(\tilde{\mathbf{I}}(\tilde{\mathbf{\Theta}})\) is a \(2p\times 2p\) complex hermitian matrix. Hence, \(T_{\tilde{W}}(\tilde{\mathbf{x}})\) is real._  
Note that no assumption has been imposed on the form of \(\mathbf{I}(\xi)\). Proof. : Let  
\[\mathbf{T}=\left[ \begin{array}{cc}\frac{1}{2}\mathbf{I}_{p}&\frac{i}{2} \mathbf{I}_{p}\\ \frac{1}{2}\mathbf{I}_{p}&-\frac{i}{2}\mathbf{I}_{p}\end{array} \right] \tag{33}\]  
where \(\mathbf{I}_{p}\) is a \(p\times p\) identity matrix. Then,  
\[\tilde{\mathbf{I}}(\tilde{\mathbf{\Theta}})=\mathbf{T}\,\mathbf{I}(\xi)\, \mathbf{T}^{H} \tag{34}\]  
Hence, we have  
\[