lqquantile_sf: L_q-quantile scoring function
View source: R/lqquantile_sf.R
| lqquantile_sf | R Documentation |
L_q-quantile scoring function
Description
The function lqquantile_sf computes the L_q-quantile scoring function at a
specific level p, when y materialises and x is the predictive
L_q-quantile at level p.
The L_q-quantile scoring function is defined by Chen (1996).
Usage
lqquantile_sf(x, y, p, q)
Arguments
x |
Predictive |
y |
Realisation (true value) of process. It can be a vector of length
|
p |
It can be a vector of length |
q |
It can be a vector of length |
Details
The L_q-quantile scoring function is defined by:
S(x, y, p, q) := |\textbf{1} \lbrace x \geq y \rbrace - p| |x - y|^q
or equivalently,
S(x, y, p, q) := p |\max \lbrace -(x - y), 0 \rbrace|^q +
(1 - p) |\max \lbrace x - y, 0 \rbrace|^q
Domain of function:
x \in \mathbb{R}
y \in \mathbb{R}
0 < p < 1
q \geq 2
Range of function:
S(x, y, p, q) \geq 0, \forall x, y \in \mathbb{R}, p \in (0, 1),
q \geq 2
Value
Vector of L_q-quantile losses.
Note
For the definition of L_q-quantiles, see Chen (1996). In particular,
L_q-quantiles at level p are the solution of the equation
\textnormal{E}_F[V(x, Y, p, q)] = 0, where
V(x, y, p, q) := q (\textbf{1} \lbrace x \geq y \rbrace - p)
|x - y|^{q - 1}
The L_q-quantile scoring function is negatively oriented (i.e. the
smaller, the better).
The L_q-quantile scoring function is strictly \mathbb{F}-consistent
for the L_q-quantile functional at level p. \mathbb{F} is the
family of probability distributions F for which
\textnormal{E}_F[Y^q] exists and is finite (Chen 2016; Bellini 2014).
References
Bellini F, Klar B, Muller A, Gianin ER (2014) Generalized quantiles as risk measures. Insurance: Mathematics and Economics 54:41–48. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.insmatheco.2013.10.015")}.
Chen Z (1996) Conditional L_p-quantiles and their application to the
testing of symmetry in non-parametric regression.
Statistics and Probability Letters 29(2):107–115.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/0167-7152(95)00163-8")}.
Examples
# Compute the Lq-quantile scoring function at level p.
df <- data.frame(
y = rep(x = 0, times = 6),
x = c(2, 2, -2, -2, 0, 0),
p = rep(x = c(0.05, 0.95), times = 3),
q = c(2, 3, 2, 3, 2, 3)
)
df$lqquantile_penalty <- lqquantile_sf(x = df$x, y = df$y, p = df$p, q = df$q)
print(df)
