expectile_rs: Realised expectile score

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

Realised expectile score

Description

The function expectile_rs computes the realised expectile score at a specific level p when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Realised expectile score is a realised score corresponding to the expectile scoring function expectile_sf.

Usage

expectile_rs(x, y, p)

Arguments

x	Prediction. It can be a vector of length n (must have the same length as \textbf{\textit{y}}).
y	Realisation (true value) of process. It can be a vector of length n (must have the same length as \textbf{\textit{x}}).
p	It can be a vector of length n (must have the same length as \textbf{\textit{y}}) or a scalar value.

Details

The realized expectile score is defined by:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, p) := (1/n) \sum_{i = 1}^{n} L(x_i, y_i, p)

where

\textbf{\textit{x}} = (x_1, ..., x_n)^\mathsf{T}

\textbf{\textit{y}} = (y_1, ..., y_n)^\mathsf{T}

and

L(x, y, p) := |\textbf{1} \lbrace x \geq y \rbrace - p| (x - y)^2

Domain of function:

\textbf{\textit{x}} \in \mathbb{R}^n

\textbf{\textit{y}} \in \mathbb{R}^n

0 < p < 1

Range of function:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, p) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n, p \in (0, 1)

Value

Value of the realised expectile score.

Note

For details on the expectile scoring function, see expectile_sf.

The concept of realised (average) scores is defined by Gneiting (2011) and Fissler and Ziegel (2019).

The realised expectile score is the realised (average) score corresponding to the expectile scoring function.

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166–1211. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/19-EJS1552")}.

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")}.

Examples

# Compute the realised expectile score.

set.seed(12345)

x <- 0.5

y <- rnorm(n = 100, mean = 0, sd = 1)

print(expectile_rs(x = x, y = y, p = 0.7))

print(expectile_rs(x = rep(x = x, times = 100), y = y, p = 0.7))

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