scoringfunctions (version 1.1)

expectile_if: 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)

Value

Vector of values of the expectile identification function.

Arguments

x

Predictive expectile (prediction) at level \(p\). It can be a vector of length \(n\) (must have the same length as \(y\)).

y

Realisation (true value) of process. It can be a vector of length \(n\) (must have the same length as \(x\)).

p

It can be a vector of length \(n\) (must have the same length as \(y\)).

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:

$$V(x, y, p) \in \mathbb{R}$$

References

Dimitriadis T, Fissler T, Ziegel JF (2024) Osband's principle for identification functions. Statistical Papers 65:1125--1132. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1214/16-AOS1439").

Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746--762. tools:::Rd_expr_doi("10.1198/jasa.2011.r10138").

Newey WK, Powell JL (1987) Asymmetric least squares estimation and testing. Econometrica 55(4):819--847. tools:::Rd_expr_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)