Redescending M-estimator - Examples

Examples

1. Hampel's three-part M estimators have Ψ functions which are odd functions and defined for any x by:


\Psi(x)=
\begin{cases}
x, & 0\le |x| \le a \text{ (central segment)}\\
a\, \operatorname{sign}(x), & a\le |x| \le b \text{ (high and low flat segments)}\\
\frac{a(r-|x|)}{r-b}\,\operatorname{sign}(x),& b\le |x| \le r \text{ (end slopes)}\\
0,& r\le |x| \qquad\, \text{(left and right tails)}
\end{cases}

This function is plotted in the following figure for a=1.645, b=3 and r=6.5.

2. Tukey's biweight or bisquare M estimators have Ψ functions for any positive k, which defined by:

This function is plotted in the following figure for k=5.

3. Andrew's sine wave M estimator has the following Ψ function:

This function is plotted in the following figure.

Read more about this topic:  Redescending M-estimator

Famous quotes containing the word examples:

    In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.
    Michel de Montaigne (1533–1592)

    It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.
    —G.C. (Georg Christoph)

    Histories are more full of examples of the fidelity of dogs than of friends.
    Alexander Pope (1688–1744)