The realised generalized Huber score is defined by:
$$S(\textbf{\textit{x}}, \textbf{\textit{y}}, p, a, b) := (1/n)
\sum_{i = 1}^{n} L(x_i, y_i, p, a, b)$$
where
$$\textbf{\textit{x}} = (x_1, ..., x_n)^\mathsf{T}$$
$$\textbf{\textit{y}} = (y_1, ..., y_n)^\mathsf{T}$$
and
$$
L(x, y, p, a, b) :=
|\textbf{1} \lbrace x \geq y \rbrace - p| f_{a, b}(x - y)
$$
where
$$f_{a, b}(t) := \kappa_{a, b}(t) (2 t - \kappa_{a, b}(t))$$
and \(\kappa_{a, b}(t)\) is the capping function defined by:
$$\kappa_{a, b}(t) := \max \lbrace \min \lbrace t, b \rbrace, -a \rbrace$$
Domain of function:
$$\textbf{\textit{x}} \in \mathbb{R}^n$$
$$\textbf{\textit{y}} \in \mathbb{R}^n$$
$$0 < p < 1$$
$$a > 0$$
$$b > 0$$
Range of function:
$$S(\textbf{\textit{x}}, \textbf{\textit{y}}, p, a, b) \geq 0,
\forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n,
p \in (0, 1), a, b > 0$$