M-estimator

M- estimator ( by maximum likelihood like) are a class of estimation functions which can be regarded as a generalization of the maximum likelihood method. M- estimators are compared to other estimators such as the maximum likelihood estimators more robust to outliers.

This article deals with M- estimator to determine the location parameter.

Deriving by generalizing the maximum likelihood method

The principle of maximum likelihood estimators is based on the function

With an appropriate probability density function or a function of minimizing.

The idea behind M- estimators is to replace the function by a function which reacts less sensitive to outliers. Task, then, is the expression

As a function of minimizing or the equation

With

To solve.

Any solution of this equation is called the M- estimator.

Implicit definition

Be an arbitrary distribution function and a straight and monotonically increasing function equal to 0 then is defined as the solution of the equation

It must be noted that depending on the choice of and there may be either no, one or more solutions. In the case of a concrete sample is, the solution of

Called M- estimators.

Suitable functions

The following are the according

Standardized to achieve scale invariance. this represents a dispersion estimator is used for most of the MAP (median absolute deviation).

The weight functions in the following picture show the differences between the estimates: at Huber -k extreme observations have a low weight, the Hampel, Andrews and Tukey 's biweight wave estimator extreme observations is assigned the weight zero.

Robustness

With a suitable choice of (even, bounded and monotonically increasing ), M - estimator a break of

Numerical solution method

For many functions, no explicit solution can be stated, it must be computed numerically. As usual for the computation of zeros of problems has also here the Newton -Raphson method, and it results in the following iteration, again:

A suitable starting value of the median is usually used. This iteration converges very fast, usually two or three iterations is sufficient.

W - estimator

W - estimators are M- estimators provide very similar and normally the same results. The only difference lies in the solution of the minimization problem. W - estimators are usually used in the robust regression.

It is the weighting function

With

Introduced with the aid of the minimization problem can be rewritten as

Inserting the definition of multiply out, and change ultimately results about the fixed point equation

The iteration