scoringfunctions (version 1.1)

expectile_rs: 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)

Value

Value of the realised expectile score.

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:

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

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166--1211. 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. 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))