expectile_if {scoringfunctions} | R Documentation |
Expectile identification function
Description
The function expectile_if computes the expectile identification function at a
specific level p
, when y
materialises and x
is the predictive
expectile at level p
.
The expectile identification function is defined in Table 9 in Gneiting (2011).
Usage
expectile_if(x, y, p)
Arguments
x |
Predictive expectile (prediction) at level |
y |
Realisation (true value) of process. It can be a vector of length
|
p |
It can be a vector of length |
Details
The expectile identification function is defined by:
V(x, y, p) := 2 |\textbf{1} \lbrace x \geq y \rbrace - p| (x - y)
Domain of function:
x \in \mathbb{R}
y \in \mathbb{R}
0 < p < 1
Range of function:
V(x, y, p) \in \mathbb{R}
Value
Vector of values of the expectile identification function.
Note
For the definition of expectiles, see Newey and Powell (1987).
The expectile identification function is a strict
\mathbb{F}
-identification function for the p
-expectile functional
(Gneiting 2011; Fissler and Ziegel 2016; Dimitriadis et al. 2024).
\mathbb{F}
is the family of probability distributions F
for which
\textnormal{E}_F[Y]
exists and is finite (Gneiting 2011; Fissler and
Ziegel 2016; Dimitriadis et al. 2024).
References
Dimitriadis T, Fissler T, Ziegel JF (2024) Osband's principle for identification functions. Statistical Papers 65:1125–1132. doi:10.1007/s00362-023-01428-x.
Fissler T, Ziegel JF (2016) Higher order elicitability and Osband's principle. The Annals of Statistics 44(4):1680–1707. doi:10.1214/16-AOS1439.
Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746–762. doi:10.1198/jasa.2011.r10138.
Newey WK, Powell JL (1987) Asymmetric least squares estimation and testing. Econometrica 55(4):819–847. doi:10.2307/1911031.
Examples
# Compute the expectile identification function.
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)
)
df$expectile_if <- expectile_if(x = df$x, y = df$y, p = df$p)