Looking for some intuition regarding the MCD estimator The Minimum Covariance Determinant (MCD) estimator may be used to achieve robustness when estimating a covariance matrix. It looks for the subset of $h$ data points (out of $n > h$) whose covariance matrix has the smallest determinant and it computes the covariance of those $h$ points.
Why does minimising the determinant do the job? 
 A: Okay, i tried an intuitive overview.  
Suppose that $X\sim F_p(m,C)$, a known continuous distribution on $\mathbb{R}^p$ (often assumed elliptical but this can be generalized). So $X$ is a collection of $n$ draws from $F_p$.
I show it for the estimator of scatter, the proof for the estimator of location follows easily.
Denote $S(X)$ an estimator of $C$ computed from the uncontaminated data $X$, $S(X_{\epsilon})$ another estimator of $C$ computed on $X_{\epsilon}$, the contaminated data. $X_{\epsilon}$ is $X$ but for $\lfloor n\epsilon\rfloor$ of the observations which have been replaced by draws from $G_q$.


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*max-bias of an estimator $S$, for a rate of contamination $\epsilon$ of a dataset $X$: 


$\verb+bias+(\epsilon,S,X)=\displaystyle\underset{X_{\epsilon}}{\sup}||S(X)-S(X_{\epsilon})||_{\kappa}$
where $||.||_{\kappa}$ is the condition number metric.


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*breakdown point of an estimator $S$: 


$\epsilon^*(S,X)=\min\{\epsilon:\verb+bias+(\epsilon,S,X)=\infty\}$
The max bias is often hard to manipulate theoretically because it depends on the  distribution of $X_{\epsilon}$, whereas the definition of the breakdown point contains no probability distribution.


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*maximum breakdown point of an affine equivariant estimator $S$
For an AE estimator S and a dataset $X$ in general linear position in $\mathbb{R}^p$: 
$\epsilon^*(S,X)< \left\lfloor \frac{n-p-1}{2n}\right\rfloor$


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*breakdown point of an affine equivariant estimator $S$
For an AE estimator S and a dataset $X$ in general linear position in the expression of the breakdown point simplifies to 
$\epsilon^*(S,X)=\min\{\epsilon:\frac{\lambda_1(S(X_{\epsilon}))}{\lambda_p(S(X_{\epsilon}))}=\infty\}$ 
where $\lambda_p(C)$ ($\lambda_1(C)$) is the smallest (largest) eigen-value of a covariance matrix $C$.
There is then two types of breakdown. Implosion breakdown are caused by cases where $\lambda_p(S(X_{\epsilon}))=0$ (cases where $h$ observations lie on an affine hyper-plane) and explosion breakdown by cases where 
$\lambda_1(S(X_{\epsilon}))=\infty$. 
The exact fit property of the MCD protects it from implosion breakdowns. For explosion breakdowns, it is enough to show that the chosen MCD H-subset $H_*$


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*$H_*=\underset{H:|H|=h}{\Arg.\min.}\det(\underset{i\in H}{\text{cov}}(x_i))$


cannot belong to the set of $H$-subsets $H^+$: 


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*$H^+=\cup \{H:|H|=h, \lambda_1(\underset{i\in H}{\text{cov}}(x_i))=\infty\}$

