The realised generalized piecewise linear power score (type 1) is defined by:
$$S(\textbf{\textit{x}}, \textbf{\textit{y}}, p, b) := (1/n)
\sum_{i = 1}^{n} L(x_i, y_i, p, b)$$
where
$$\textbf{\textit{x}} = (x_1, ..., x_n)^\mathsf{T}$$
$$\textbf{\textit{y}} = (y_1, ..., y_n)^\mathsf{T}$$
and
$$
L(x, y, p, b) :=
(1/b) (\textbf{1} \lbrace x \geq y \rbrace - p) (x^b - y^b)
$$
Domain of function:
$$\textbf{\textit{x}} > \textbf{0}$$
$$\textbf{\textit{y}} > \textbf{0}$$
$$0 < p < 1$$
$$b > 0$$
where
$$\textbf{0} = (0, ..., 0)^\mathsf{T}$$
is the zero vector of length \(n\) and the symbol \(>\) indicates pairwise
inequality.
Range of function:
$$S(\textbf{\textit{x}}, \textbf{\textit{y}}, p, b) \geq 0,
\forall \textbf{\textit{x}}, \textbf{\textit{y}} > \textbf{0}, p \in (0, 1),
b > 0$$