# 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

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

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