Tukey's Biweight

$\begin{figure}\begin{center}\BoxedEPSF{TukeysBiweight.epsf}\end{center}\end{figure}$

The function

$\begin{displaymath} \psi(z)=\cases{ z\left({1-{z^2\over c^2}}\right)^2 & for $\vert z\vert<c$\cr 0 & for $\vert z\vert>c$\cr} \end{displaymath}$

sometimes used in Robust Estimation. It has a minimum at $z=-c/\sqrt{3}$ and a maximum at $z=c/\sqrt{3}$ , where

$\begin{displaymath} \psi'(z)=1-{3x^2\over c^2}=0, \end{displaymath}$

and an inflection point at

, where

$\begin{displaymath} \psi''(z)=-{6z\over c^2}=0. \end{displaymath}$

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 697, 1992.