lqquantile_rs: Realised L_q-quantile score

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Realised L_q-quantile score

Description

The function lqquantile_rs computes the realised L_q-quantile score at a specific level p and parameter q, when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Realised L_q-quantile score is a realised score corresponding to the L_q-quantile scoring function lqquantile_sf.

Usage

lqquantile_rs(x, y, p, q)

Arguments

x	Prediction. It can be a vector of length n (must have the same length as \textbf{\textit{y}}).
y	Realisation (true value) of process. It can be a vector of length n (must have the same length as \textbf{\textit{x}}).
p	It can be a scalar.
q	It can be a scalar.

Details

The realised L_q-quantile score is defined by:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, p, q) := (1/n) \sum_{i = 1}^{n} L(x_i, y_i, p, q)

where

\textbf{\textit{x}} = (x_1, ..., x_n)^\mathsf{T}

\textbf{\textit{y}} = (y_1, ..., y_n)^\mathsf{T}

and

L(x, y, p, q) := |\textbf{1} \lbrace x \geq y \rbrace - p| |x - y|^q

Domain of function:

\textbf{\textit{x}} \in \mathbb{R}^n

\textbf{\textit{y}} \in \mathbb{R}^n

0 < p < 1

q > 1

Range of function:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, p, q) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n, p \in (0, 1), q > 1

Value

Value of the realised L_q-quantile score.

Note

For details on the L_q-quantile scoring function, see lqquantile_sf.

The concept of realised (average) scores is defined by Gneiting (2011) and Fissler and Ziegel (2019).

The realised L_q-quantile score is the realised (average) score corresponding to the L_q-quantile scoring function.

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166–1211. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/19-EJS1552")}.

Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746–762. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/jasa.2011.r10138")}.

See Also

lqquantile_sf

Examples

# Compute the realised Lq-quantile score.

set.seed(12345)

p <- 0.7

q <- 2

x <- qnorm(p = p, mean = 0, sd = 1)

y <- rnorm(n = 100, mean = 0, sd = 1)

print(lqquantile_rs(x = x, y = y, p = p, q = q))

print(lqquantile_rs(x = rep(x = x, times = 100), y = y, p = p, q = q))

scoringfunctions documentation built on Aug. 30, 2026, 5:07 p.m.