hubermean_if {scoringfunctions} | R Documentation |
Huber mean identification function
Description
The function hubermean_if computes the Huber mean identification function with
parameter a
, when y
materialises and x
is the predictive Huber
mean.
The Huber mean identification function is defined by eq. (3.5) in Taggart (2022).
Usage
hubermean_if(x, y, a)
Arguments
x |
Predictive Huber mean (prediction). It can be a vector of length
|
y |
Realisation (true value) of process. It can be a vector of length
|
a |
It can be a vector of length |
Details
The Huber mean identification function is defined by:
V(x, y, a) := (1/2) \kappa_{a,a}(x - y)
where \kappa_{a,b}(t)
is the capping function defined by:
\kappa_{a,b}(t) := \max \lbrace \min \lbrace t,b \rbrace, -a \rbrace
Domain of function:
x \in \mathbb{R}
y \in \mathbb{R}
a > 0
Value
Vector of values of the Huber mean identification function.
Note
For the definition of Huber mean, see Taggart (2022).
The Huber mean identification function is a strict
\mathbb{F}
-identification function for the Huber mean functional (Taggart
2022).
\mathbb{F}
is the family of probability distributions F
for which
for which \textnormal{E}_F[Y]
exists and is finite (Taggart 2022).
References
Taggart RJ (2022) Point forecasting and forecast evaluation with generalized Huber loss. Electronic Journal of Statistics 16:201–231. doi:10.1214/21-EJS1957.
Examples
# Compute the Huber mean identification function.
df <- data.frame(
x = c(-3, -2, -1, 0, 1, 2, 3),
y = c(0, 0, 0, 0, 0, 0, 0),
a = c(2.7, 2.5, 0.6, 0.7, 0.9, 1.2, 5)
)
df$hubermean_if <- hubermean_if(x = df$x, y = df$y, a = df$a)
print(df)