page setting out linear stuff to come
OLS, generalised linea rmodels etc are m-estimators, as are gmm
h3 on parametric
With maximum likelihood estimation we maximise a function.
We could choose other functions to maximise or minimise.
\(\sum_i f(x_i, \theta )\)
If \(f(x_i, \theta )\) is differentiable wrt to \(\theta\) this can be solved by finding the stationay point.
This is a \(\phi\) type.
Otherwise it is a \(\rho\) type.
page on influence funcitons there
Generalisation of MLE.
\(m_\theta =m_\theta (x, \theta )\)
Z-estimator is where this is met, through diff
\(\frac{\delta }{\delta \theta }m_\theta =z_\theta (\theta , x)=0\)
M-estimator for mean
\(m_\theta (\theta )=-(x-\theta )^2\)
\(z_\theta (\theta )=x-\theta\)