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.

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