## 6 MSE of Direct Position Determination Method in Presence of Array Model Errors
### Second-Order Perturbation Analysis on the Cost Function
\begin{array}{l}\Pr\{|<\widetilde{\mathbf{p}}>_{1}|\leq\Delta_{1}\,,\,|<\widetilde{ \mathbf{p}}>_{2}|\leq\Delta_{2}\,,\,\cdots\,,\,|<\widetilde{\mathbf{p}}>_{L}|\leq\Delta _{L}\}\\ =\int_{-\Delta_{L}}^{\Delta_{L}}\cdots\int_{-\Delta_{2}}^{\Delta_{2}}\int_{- \Delta_{1}}^{\Delta_{1}}\left(2\pi\right)^{-L/2}\cdot|\det[\mathbf{P}]|^{-1/2}\cdot \exp\{-\mathbf{z}^{T}\mathbf{p}^{-1}\mathbf{z}/2\}\,\mathrm{d}z_{1}\mathrm{d}z_{2}\cdots \mathrm{d}z_{L}\,.\end{array}  \tag{59}\]  
It is apparent from (59) that the first SP can be approximately obtained via numerical integration over a cube in high dimensional Euclidean space. However, the high-dimensional numerical integration is not attractive from a computational viewpoint. If possible, it is preferable to get an explicit formula. Obviously, this is a non-trivial task and we only consider two-dimensional (2-D) localization scenarios (i.e., \(L=2\)) for simplicity of mathematical analysis. First, an explicit formula with which to evaluate the joint probability of the Gaussian distribution is formally concluded in a proposition as below. **Proposition 2**. : _Consider two joint Gaussian random variables \(z_{1}\) and \(z_{2}\). The mean and variance of \(z_{1}\) are \(m_{1}\) and \(v_{11}\), respectively. The mean and variance of \(z_{2}\) are \(m_{2}\) and \(v_{22}\), respectively. In addition, the covariance of the two random variables is \(v_{12}\). It follows that_  
\[\Pr\biggl{\{}z_{1}\leq\alpha_{1}\,,\,z_{2}\leq\alpha_{2}\}=\Gamma_{0}[\alpha_{ 10}/\sqrt{v_{11}}]\cdot\Gamma_{0}[(\alpha_{2}-\mathrm{E}[\overline{z}_{2}]) \,/\sqrt{\mathrm{var}[\overline{z}_{2}]}]\,, \tag{60}\]  
where  
\[\left\{ \begin{array}{l}\mathrm{E}[\overline{z}_{2}]=m_{2}-\frac{v_{12}\cdot \exp\{-\frac{4v_{10}^{2}/(2v_{11})}{\sqrt{2\pi v_{11}\cdot\Gamma_{0}}[\alpha_{1 0}/\sqrt{v_{11}}]}}}{\mathrm{var}[\overline{z}_{2}]=v_{22}-\frac{v_{12}^{2}}{ \sqrt{2\pi v_{11}\cdot\Gamma_{0}}[\alpha_{10}/\sqrt{v_{11}}]}}\cdot\left(\frac {\alpha_{10}\cdot\exp\{-\alpha_{10}^{2}/(2v_{11})\}}{\sqrt{v_{11}}}+\frac{ \exp\{-\alpha_{10}^{2}/v_{11}\}}{\sqrt{2\pi\cdot\Gamma_{0}}[\alpha_{10}/\sqrt {v_{11}}]}\right)\end{array} \right. \tag{61}\]  
with \(\alpha_{10}=\alpha_{1}-m_{1}\) and \(\Gamma_{0}[x]=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}\cdot\exp\{-t^{2}/2\} \cdot\mathrm{d}t\). Appendix E shows the proof of Proposition 2, which is along the lines of incomplete conditional moments theory presented in [46]. Note that Proposition 2 plays a significant role in the subsequent derivation process. When \(L=2\), it can be verified by algebraic manipulation that  
\[ \begin{array}{l}\Pr\{|<\widetilde{\mathbf{p}}>_{1}|\leq\Delta_{1}\,,\,|< \widetilde{\mathbf{p}}>_{2}|\leq\Delta_{2}\}\\ =\Pr\{-\Delta_{1}\leq\widetilde{\mathbf{p}}>_{1}\leq\Delta_{1}\}-\Pr\{<\widetilde {\mathbf{p}}>_{2}\geq\Delta_{2}\}-\Pr\{<\widetilde{\mathbf{p}}>_{2}\leq-\Delta_{2}\}+ \Pr\{-<\widetilde{\mathbf{p}}>_{1}\leq-\Delta_{1}\,,\,-<\widetilde{\mathbf{p}}>_{2}\leq -\Delta_{2}\}\\ +\Pr\{<\widetilde{\mathbf{p}}>_{1}\leq-\Delta_{1}\,,\,-<\widetilde{\mathbf{p}}>_{2} \leq-\Delta_{2}\}+\Pr\{-<\widetilde{\mathbf{p}}>_{1}\leq-\Delta_{1}\,,\,< \widetilde{\mathbf{p}}>_{2}\leq-\Delta_{2}\}+\Pr\{<\widetilde{\mathbf{p}}>_{1}\leq- \Delta_{1}\,,\,<\widetilde{\mathbf{p}}>_{2}\leq-\Delta_{2}\}\,.\end{array}  \tag{62}\]  
The proof of (62) is shown in Appendix F. Applying the result in Proposition 2 and the definition of \(\Gamma_{0}[x]\), we have  
\[ \begin{array}{l}\Pr\{|<\widetilde{\mathbf{p}}>_{1}|\leq\Delta_{1}\,,\,|< \widetilde{\mathbf{p}}>_{2}|\leq\Delta_{2}\}\\ =\Gamma_{0}[\Delta_{1}/\sqrt{<\mathbf{P}>_{11}}]-\Gamma_{0}[-\Delta_{1}/\sqrt{<\mathbf{ P}>_{11}}]-2\cdot\Gamma_{0}[-\Delta_{2}/\sqrt{<\mathbf{P}>_{22}}]\\ +2\cdot\Gamma_{0}[-\Delta_{1}/\sqrt{<\mathbf{P}>_{11}}]\cdot\left(\Gamma_{0}[(- \Delta_{2}+\kappa_{1})/\sqrt{\kappa_{2}}]+\Gamma_{0}[(-\Delta_{2}-\kappa_{1})/ \sqrt{\kappa_{2}}]\right),\end{array}  \tag{63}\]  
where  
\[\left\{