expectile_if: Expectile identification function

In scoringfunctions: A Collection of Loss Functions for Assessing Point Forecasts

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 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}

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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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. \Sexpr[results=rd]{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)

scoringfunctions documentation built on April 4, 2025, 12:28 a.m.