mean_if {scoringfunctions} | R Documentation |
Mean identification function
Description
The function mean_if computes the mean identification function , when y
materialises and x
is the predictive mean.
The mean identification function is defined in Table 9 in Gneiting (2011).
Usage
mean_if(x, y)
Arguments
x |
Predictive mean (prediction). It can be a vector of length |
y |
Realisation (true value) of process. It can be a vector of length
|
Details
The mean identification function is defined by:
V(x, y) := x - y
Domain of function:
x \in \mathbb{R}
y \in \mathbb{R}
Range of function:
V(x, y) \in \mathbb{R}
Value
Vector of values of the mean identification function.
Note
The mean functional is the mean \textnormal{E}_F[Y]
of the probability
distribution F
of y
(Gneiting 2011).
The mean identification function is a strict \mathbb{F}
-identification
function for the mean 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 mean identification function.
df <- data.frame(
y = rep(x = 0, times = 3),
x = c(-2, 0, 2)
)
df$mean_if <- mean_if(x = df$x, y = df$y)