Tukey's Biweight and Redescending Psi Functions

Fit a mean or a regression line to a batch of returns that includes a couple of fat-finger prints, and ordinary least squares will bend itself out of shape trying to accommodate them — because in OLS every point's pull on the fit grows without limit the further it sits from the line. Tukey's biweight fixes this with one idea: let an observation's pull increase at first as it moves away from the fit, like OLS does, but then have that pull fall back to zero once the observation is far enough out. A point in another county gets no vote at all.

An analogy: a committee that stops listening to shouting

Imagine a committee vote where quiet disagreement is weighed normally, louder disagreement is weighed a bit more, but once someone starts shouting incoherently, the room simply tunes them out — their volume stops translating into influence. That's the shape of the biweight's influence: rises with distance from consensus, peaks, then collapses to nothing for the truly extreme voices. OLS, by contrast, is a committee that gives the loudest shouter the most weight, no matter how unreasonable they are.

The math, one symbol at a time

For a residual $u$ (how far a point's observed value sits from the fitted value), scaled by a measure of typical spread $s$, define $r = u/s$. The psi function $\psi(r)$ describes how much "pull" a residual of size $r$ exerts on the fit. Tukey's biweight psi function is

$$\psi(r) = \begin{cases} r\left(1 - (r/c)^2\right)^2 & |r| \le c \\ 0 & |r| > c \end{cases}$$

where $c$ is a tuning constant (commonly $c \approx 4.685$). In plain English: within $c$ scaled units, a residual pulls the fit toward itself, rising then tapering; beyond $c$ its pull is exactly zero — "redescending," unlike methods where influence flattens but never vanishes, or OLS where it grows forever. The weight $w(r) = \psi(r)/r$ starts near 1 for small residuals and hits exactly 0 beyond $c$.

Worked example 1: three residuals, three treatments

Suppose $s = 1$ (so $r$ equals the residual itself) and $c = 4.685$. Consider residuals of $0.50, $3.00, and $9.00 in a P&L reconciliation.

• $r = 0.5$: well inside $c$, so $\psi(0.5) \approx 0.5(1-0.011)^2 \approx 0.489$ — almost full pull, close to OLS.
• $r = 3.0$: $(3/4.685)^2 \approx 0.410$, so $\psi(3) \approx 3(0.590)^2 \approx 3(0.348) \approx 1.045$ — pull has grown but is already curving down relative to a linear OLS pull of 3.
• $r = 9.0$: exceeds $c = 4.685$, so $\psi(9) = 0$ — this point is completely excluded from the fit, as if it weren't in the sample at all.

Worked example 2: comparing a mean estimate

Five daily P&L figures: $10, $12, $9, $11, and $200 (a data error). The sample mean is $(10+12+9+11+200)/5 = 48.4$ — dominated by the one bad print. A biweight-fitted location estimate, computed by iterating (see Iteratively Reweighted Least Squares), converges to $10.5, because the $200 print's residual quickly earns weight zero, leaving only the four sensible values. The estimate auto-detects and drops the bad point without anyone flagging it in advance.

What this means in practice

Biweight-based fitting is used wherever a fit must not be hijacked by a handful of bad prints, fat-finger trades, or genuine but rare regime breaks — cleaning return series before estimating covariances, robustifying factor-model regressions, or building P&L reconciliation checks that should flag rather than absorb data errors. It's typically implemented via iteratively reweighted least squares, and its variance is best assessed with the sandwich variance estimator rather than the OLS formula.