info prev up next book cdrom email home

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 $z=0$, 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.




© 1996-9 Eric W. Weisstein
1999-05-26