## 4 MM Estimation Theory for Positioning  
Conventional statistical estimation schemes are based upon some assumptions about the observations. If the real observations deviate from the assumptions, estimation might deviate from the truth, such as LSE. The robust estimator tries to dynamically search a best fitting criterion to the real observations to reduce the effect of unusual data. Generally, a robust estimator minimizes the cost function \(J(\mathbf{r})\),  
\[J(\mathbf{r})=\sum\limits_{l=1}^{n}\mu(r_{l})=\sum\limits_{l=1}^{n}\mu(\rho_{ l}-\mathbf{H}_{l}^{T}X) \tag{16}\]  
\(\rho_{l}\) is the \(l\_th\) observation, \(X\) is the estimated unknown (here we consider the unknown term as generic), \(\mathrm{H}_{l}\) is the transformation vector for the \(l\_th\) observation, and \(r_{l}=\rho_{l}-\mathbf{H}_{l}^{T}X\) is the error residual for the \(l\_th\) observation. \(\mu(\mathbf{r})\) is the objective function that gives the contribution of each residual to the cost function \(J(\mathbf{r})\). For the least squares estimator \(\mu(\mathbf{r})=\mathbf{r}^{2}\), while in the robust estimation process, a scaled observation residual is minimized. The scaled estimation process minimizes the weight of out of the bound observations dynamically,  
\[\mathrm{min}\sum\limits_{l=1}^{n}\mu\left(\frac{r_{l}}{s}\right)=\mathrm{min} \sum\limits_{i=1}^{n}\mu\left(\frac{\rho_{l}-\mathbf{H}_{l}^{T}X}{s}\right) \tag{17}\]  
\(s\) is the scale factor. \(\overset{\sim}{\mathbf{r}}_{s}\) is the scaled residual \(\mathbf{r}/s\). Let if \(\psi\) be the derivative of the objective function for the unknowns \(X\), \(\psi=\left.\mu^{\prime}\right|_{X}\). Differentiating the objective function in Equation (17) and setting the derivative to 0,  
\[ \begin{array}{c}-\frac{1}{s}\sum\limits_{l=1}^{n}\psi\left(\frac{\rho_{l}- \mathbf{H}_{l}^{T}X}{s}\right)\mathbf{H}_{l}{}^{T}=0\\ \text{\emph{equivalent to}}\quad\sum\limits_{l=1}^{n}\psi\left(\frac{\rho_{l}- \mathbf{H}_{l}^{T}X}{s}\right)\mathbf{H}_{l}{}^{T}=0\end{array}  \tag{18}\]  
By introducing a weighting function \(w\left(\overset{\sim}{r}_{s,l}\right)\) to consider the effects of each scaled residual \(\overset{\sim}{r}_{s,l}\), Equation (18) can be modified as,  
\[\sum\limits_{l=1}^{n}w\left(\overset{\sim}{r}_{s,l}\right)\psi\left(\overset{ \sim}{r}_{s,l}\right)\mathbf{H}_{l}{}^{T}=0 \tag{19}\]  
Solution of Equation (19) is a weighted least-squares problem. However, the weights depend upon the scaled residuals, the scaled residuals depend upon the estimated coefficients, and the estimated coefficients depend upon the weights. Hence, it is an iterative process, iteratively reweighted least squares (IRWLS) [31; 32]. Three most popular objective functions and their corresponding weighting functions are shown in Table 1.  
\(\stackrel{{\rightarrow}}{{r}}\) is the observation residual and it can be either true or scaled. The \(\alpha\), for Huber and Bi-square, is a tuning constant with optimal value to attain more resistance against outliers (high breakdown point) as well as the convergence efficiency. \(\alpha\) is a tradeoff parameter in terms of robustness and efficiency. The table of \(\alpha\) is provided in Appendix A to represent the relation between efficiency, robustness and tuning constant. \(\alpha\) is selected such that to balance the performance of high breakdown point and efficiency. An outline of robust estimation procedure is defined in the following steps:  
1. Find an initial estimate, \(X_{0}\) with IRWLS (Appendix B) or subsampling method (Appendix C). In IRWLS (simplest subsampling) procedure, all observations are used in \(X_{0}\) estimation. However, in subsampling procedure, subsamples (subsets) of observations are selected each with size of \(n_{sub}\), number of observations in one subset. For each subsample, a solution is computed using IRWLS. Best candidate solution is selected as \(X_{0}\) and the initial residuals are estimated, \(r_{l}(X_{0})=\rho_{l}-\mathbf{H}_{l}{}^{T}X_{0}\) for each observation. Then we get a residual vector \(\mathbf{r}^{0}=[