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