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Tukey's Biweight


The function

z\left({1-{z^2\over c^2}}\right)^2 & for $\vert z\vert<c$\cr
0 & for $\vert z\vert>c$\cr}

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

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

and an inflection point at $z=0$, where

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


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.

© 1996-9 Eric W. Weisstein