M-estimators

M-estimators

Introduction

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

\(m_\theta (\theta )=-(x-\theta )^2\)

\(z_\theta (\theta )=x-\theta \)