## 3 The Bias in the Nonlinear Least-Squares Solution  
The nonlinear least-squares estimator \(\mathbf{\hat{X}}\) can be written as a function of the random observation vector \(\mathbf{L}\):  
\[\mathbf{\hat{X}}=M(\mathbf{L}) \tag{6}\]  
Generally speaking, there is no closed expressions available for the nonlinear observation model, therefore the nonlinear least-squares solution cannot be given explicitly. A Taylor expansion for the estimator is given as:  
\[\mathbf{\hat{X}}=\mathbf{\overline{X}}+\partial M(\mathbf{\overline{L}})( \mathbf{L}-\mathbf{\overline{L}})+\frac{1}{2}\partial^{2}M(\mathbf{\overline{ L}})(\mathbf{L}-\mathbf{\overline{L}})^{2}+\frac{1}{6}\partial^{2}M( \mathbf{\overline{L}})(\mathbf{L}-\mathbf{\overline{L}})^{3} \tag{7}\]  
If the terms of the third order and higher are ignored, take the expectation of the Equation (7), then the bias in the nonlinear least-squares parameter estimator can be expressed as [22]:  
\[b_{\mathbf{\hat{X}}}=E[\mathbf{\hat{X}}-\mathbf{\overline{X}}]=E[\mathbf{ \hat{X}}]-\mathbf{\overline{X}}\cong\partial M(\mathbf{\overline{L}})b_{ \mathbf{\overline{L}}}+\frac{1}{2}\Big{\{}\sigma_{0}^{2}\mathrm{tr}\left[ \partial^{2}M(\mathbf{\overline{L}})\mathbf{Q_{\overline{L}}}\right]+\partial^ {2}M(\mathbf{\overline{L}})b_{\mathbf{\overline{L}}}^{2}\Big{\}}\Big{\}} \tag{8}\]where \(b_{\widetilde{\mathbf{L}}}=E[\mathbf{L}]-\widetilde{\mathbf{L}}\), \(\mathbf{Q}_{\widetilde{\mathbf{L}}}=\mathbf{P}^{-1}\) represents the variance matrix of the observation vector. We can see that the bias comes from non-zero higher order terms. Assuming that the observations are normally distributed, the least-squares estimator is equivalent to the maximum likelihood estimator, then \(b_{\widetilde{\mathbf{L}}}=0\), and \(\widetilde{\mathbf{L}}\) will be an unbiased estimator of \(\mathbf{L}\), and Equation (8) can be simplified to:  
\[b_{\widetilde{\mathbf{X}}}\cong\frac{1}{2}\sigma_{0}^{2}\text{tr}\big{[} \partial^{2}M(\widetilde{\mathbf{L}})\mathbf{Q}_{\widetilde{\mathbf{L}}}\big{]} \tag{9}\]  
For the nonlinear cases, \(\partial^{2}M(\widetilde{\mathbf{L}})\) is non-zero, and therefore, the least-squares estimator is biased. Based on the orthogonality condition, we can get the first and second order partial derivatives of the function \(M(\bullet)\), then the bias \(b_{\widehat{\mathbf{X}}}\) is finally estimated [24]:  
\[b_{\widehat{\mathbf{X}}}\cong\mathbf{Q}_{\widetilde{\mathbf{X}}}\partial d^{ T}(\widetilde{\mathbf{X}})\mathbf{Q}_{\widetilde{\mathbf{L}}}^{-1}b \tag{10}\]  
where \(b{=-\frac{1}{2}\sigma_{0}^{2}\text{tr}\big{[}\partial^{2}d(\widetilde{ \mathbf{X}})\mathbf{Q}_{\widetilde{\mathbf{X}}}\big{]}}\) is the amount of nonlinearity. According to [PERSON]'s theory, the distance function is obtained as follows [21]:  
\[d(\widehat{\mathbf{X}})\cong d(\widetilde{\mathbf{X}})+\mathbf{J}(\widetilde{ \mathbf{X}})(\widehat{\mathbf{X}}-\widetilde{\mathbf{X}})+\frac{1}{2}( \widehat{\mathbf{X}}-\widetilde{\mathbf{X}})^{\text{T}}\mathbf{G}(\widetilde {\mathbf{X}})(\widehat{\mathbf{X}}-\widetilde{\mathbf{X}}) \tag{11}\]  
We can see that \(b\) is actually the expected difference between the linear and quadratic approximation of the distance function \(d(\widehat{\mathbf{X}})\), and it can be expressed as [27]:  
\[ \begin{array}{rl}\hat{b}\cong b&=-\frac{1}{2}E[(\widehat{\mathbf{X}}- \widetilde{\mathbf{X}})^{\text{T}}\mathbf{G}(\widetilde{\mathbf{X}})( \widehat{\mathbf{X}}-\widetilde{\mathbf{X}})]\\ &=\frac{\text{I}^{\text{T}}(\widetilde{\mathbf{X}})E[(\widehat{\mathbf{X}}- \widetilde{\mathbf{X}})^{\text{T}}(\widehat{\mathbf{X}}-\widetilde{\mathbf{X }})](\widetilde{\mathbf{X}}-E[(\widehat{\mathbf{X}}-\widetilde{\mathbf{X}})^{ \text{T}}(\widehat{\mathbf{X}}-\widetilde{\mathbf{X}})]}{2d(\widetilde{ \mathbf{X}})}\\ &=\frac{\sigma_{0}^{2}\big{[}\text{J}^{\text{T}}(\widetilde{\mathbf{X}}) \mathbf{Q}_{\widetilde{\mathbf{X}}}\big{]}^{\text{T}}(\widetilde{\mathbf{X}}) -\text{tr}(\mathbf{Q}_{\widetilde{\mathbf{X}}})\big{]}}{2d(\widetilde{ \mathbf{X}})}\end{array}