# Robust Statistics for GNSS Positioning under Harsh Conditions: A Useful Tool?
## 2 Robust Statistics Principles
### Dictionary of Robust Statistics Terms  
In [45], [PERSON] described the main notations of robustness in analogy to the stability of a bridge: (i) the qualitative aspect: a small perturbation should induce small effects; (ii) the breakdown aspect: how big could a perturbation be before the bridge would fall apart; and (iii) the infinitesimal aspect: how is the structure altered under the effects of infinitesimal perturbations. This section covers the basic concepts of robust statistics. First, _qualitative robustness_ is described adopting [PERSON]'s definition [11]. In plain words, if a bounded change in the distribution of the observations is seen as a bounded change in the distribution of the estimates, then the claim is that the estimator is robust. More precisely, let \(\mathcal{X}=\{x_{1},\ldots,x_{n}\}\) be a set of i.i.d. observations from a distribution \(F\), and let \(T_{n}=T_{n}(\mathcal{X})\) be a sequence of estimates. Then, \(T_{n}\) is called _robust_ at \(F=F_{0}\) if the sequence of maps of distributions (\(\mathfrak{L}_{F}(T_{n})\) stands for the distribution of an estimator (or test statistic) \(T_{n}\) under \(F\)), \(\mathfrak{L}_{F}(T_{n})\) is equicontinuous at \(F_{0}\), that is, if we take a suitable distance \(d_{*}\), in the space of probability measures, and assume that for all \(\delta_{2}>0\) there exists a \(\delta_{1}>0\) such that,  
\[d_{*}(F_{0},F)\leq\delta_{1}\Rightarrow d_{*}\left(\mathfrak{L}_{F_{0}}(T_{n} ),\mathfrak{L}_{F}(T_{n})\right)\leq\delta_{2}\;. \tag{2}\]  
Another important concept is that of _breakdown point_\(\varepsilon^{*}\) of an estimator, defined as the smallest percentage of contamination that can cause the estimator to take on arbitrarily large aberrant value [11]. Later, the concept of breakdown point on finite sets was introduced in [46]. Consider any sample \(\mathcal{X}\) of \(n\) observations and any estimator \(T\triangleq T_{n}\). The corrupted sample \(\mathcal{X}^{\prime}\) is obtained via \(\varepsilon\)-replacement, for which a random subset of size \(m\) of the original \(\mathcal{X}\) samples is replaced by arbitrary values, with a contamination fraction of \(\varepsilon=m/n\). The maximum estimation bias due to \(\varepsilon\)-contaminated is defined as  
\[MC\left(\varepsilon;T,\mathcal{X}\right)=\sup\lvert T(\mathcal{X}^{\prime})-T (\mathcal{X})\rvert \tag{3}\]  
where the supremum is evaluated over all the set of \(\varepsilon\)-corrupted samples [46]. Thus, the breakdown point \(\varepsilon^{*}\) of an estimator \(T\) reads:  
\[\varepsilon^{*}\left(T,\mathcal{X}\right)=\inf\left\{\varepsilon:MC\left(T, \mathcal{X}\right)=\infty\right\} \tag{4}\]  
For an in-depth discussion on the breakdown point of the most relevant robust estimators, the reader might refer to [47]. The _influence function_\(1F\), first introduced by [PERSON] [48] under the name _influence curve_, has often been considered as the most useful heuristic tool of robust statistics [42; 44]. The \(IF\) measures the change of the estimator \(T\) at the distribution \(F\) when the sample contains a fraction \(\varepsilon\) of outliers, as  
\[IF\left(x,T(F),F\right)=\lim_{\varepsilon\to 0}\frac{T\left(\left(1- \varepsilon\right)F+\varepsilon\,\delta_{x}\right)-T(F)}{\varepsilon} \tag{5}\]  
where \(x\) is the position of the infinitesimal contamination and \(\delta_{x}\) is the point-mass probability at \(x\). \(IF\)'s main use is to assess the relative influence of individual observations toward the value of an estimate. If it is unbounded, an outlier might cause trouble. Robust estimators provide resiliency to outliers, but they do it at the price of some performance degradation under the nominal model, that is, when all observations are inliers. Such degradation is quantified via the so-called _loss-of-efficiency_ (LoE), defined as the performance ratio between a robust estimator and the optimal method under the nominal noise model.