# Mirror Image

## Robust estimators II

In this post I was complaining that I don’t know what breakdown point for redescending M-estimators is. Now I found out that upper bound for breakdown point of redescending of M-estimators was given by Mueller in 1995, for linear regression (that is statisticians word for simple estimation of p-dimensional hyperplane):
$\frac{1}{N}(\frac{N - \mathcal{N}(x) + 1)}{2})$
$N$ – number of measurements and $\mathcal{N}(x)$ is little tricky: it is a maximum number of measurement vectors X lying in the same p-dimensional hyperplane. If number of measurements N >> p that mean breakdown point is near 50% – You can have half measurement results completely out of the blue and estimator will still work.
That only work if the error present only in results of measurements, which is reasonable condition – in most cases we can move random error from x part to y part.
Now which M-estimators attain this upper bound?
The condition is “slow variation”(Mizera and Mueller 1999)
$\lim_{t\to \infty} \frac{\rho(t x)}{\rho(t)} = 1$
Mentioned in previous post Cauchy estimator is satisfy that condition:
$\rho(x) = -\ln(1 +(\frac{x}{\gamma})^2)$ and its derivative $\psi(x) = \frac{x}{\gamma^2+ x^2}$
In practice we always work with $\psi$, not $\rho$ so Cauchy estimator is easy to calculate.
Rule of the thumb: if you don’t know which robust estimator to use, use Cauchy: It’s fast(which is important in real time apps), its easy to understand, it’s differentiable, and it is as robust as possible (that is for redescending M-estimator)

19, April, 2011 -

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