= Huber score
{c}
{title2=$\psi_k(u)$}
The Huber score clips a residual at a fixed threshold:
$$
\psi_k(u)=\max(-k,\min(u,k)).
$$
For $k>0$, the <Huber score> is the <derivative> of the <Huber loss>:
$$
\psi_k(u)=\rho_k'(u)=\begin{cases}-k,&u<-k,\\u,&-k\leq u\leq k,\\k,&u>k.\end{cases}
$$
It is a continuous <odd function> and a <monotone function>, with <Lipschitz continuity> and $|\psi_k(u)|\leq k$. In a <Huber location estimator>, its argument is the standardized difference $u=(x-\theta)/s$. Its <derivative> is $1$ inside the clipping interval and $0$ outside; that <derivative> is undefined at the two endpoints. The score itself remains continuous there. This estimating score need not be the <score function> of a separately specified <statistical model>, nor is it already a normalized <influence function>.
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