The Tukey loss function

The Tukey loss function, also known as Tukey’s biweight function, is a loss function that is used in robust statistics. Tukey’s loss is similar to Huber loss in that it demonstrates quadratic behavior near the origin. However, it is even more insensitive to outliers because the loss incurred by large residuals is constant, rather than scaling linearly as it would for the Huber loss.

The loss function is defined by the formula

$latex begin{aligned} ell (r) = begin{cases} frac{c^2}{6} left(1 – left[ 1 – left( frac{r}{c}right)^2 right]^3 right) &text{if } |r| leq c, frac{c^2}{6} &text{otherwise}. end{cases} end{aligned}$

In the above, I use $latex r$ as the argument to the function to represent “residual”, while $latex c$ is a positive parameter that the user has to choose. A common choice of this parameter is $latex c = 4.685$: Reference 1 notes that this value results…