linex_sf: LINEX scoring function

In scoringfunctions: Loss Functions for Assessing Point Forecasts

LINEX scoring function

Description

The function linex_sf computes the LINEX scoring function with parameter a when y materialises and x is the predictive -(1/a) \log{\textnormal{E}_F[\textnormal{e}^{-a Y}]} entropic risk measure (Gerber 1974).

The LINEX scoring function is defined by Varian (1975).

Usage

linex_sf(x, y, a)

Arguments

x	Predictive -(1/a) \log{\textnormal{E}_F[\textnormal{e}^{-a Y}]} entropic risk measure (prediction). 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).
a	It can be a vector of length n (must have the same length as y).

Details

The LINEX scoring function is defined by:

S(x, y, a) := \textnormal{e}^{a (x - y)} - a (x - y) - 1

Domain of function:

x \in \mathbb{R}

y \in \mathbb{R}

a \neq 0

Range of function:

S(x, y, a) \geq 0, \forall x, y \in \mathbb{R}, a \neq 0

Value

Vector of LINEX losses.

Note

For details on the LINEX scoring function, see Varian (1975) and Zellner (1986).

The LINEX scoring function is negatively oriented (i.e. the smaller, the better).

The LINEX scoring function is strictly \mathbb{F}-consistent for the -(1/a) \log{\textnormal{E}_F[\textnormal{e}^{-a Y}]} entropic risk measure. \mathbb{F} is the family of probability distributions F for which \textnormal{E}_F[\textnormal{e}^{-a Y}] and \textnormal{E}_F[Y] exist and are finite (Varian 1975; Zellner 1986; Gneiting 2011).

The functional elicited here is the entropic risk measure of serrexp_sf at the opposite sign of the parameter: serrexp_sf elicits (1/a) \log(\textnormal{E}_F[\exp(aY)]), whereas the LINEX scoring function with parameter a elicits -(1/a) \log(\textnormal{E}_F[\exp(-aY)]), which is the same one-parameter family evaluated at -a.

References

Gerber HU (1974) On additive premium calculation principles. ASTIN Bulletin: The Journal of the IAA 7(3):215–222. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1017/S0515036100006061")}.

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")}.

Varian HR (1975) A Bayesian approach to real estate assessment. In: Fienberg SE, Zellner A (eds) Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage. Amsterdam: North-Holland, pp 195–208.

Zellner A (1986) Bayesian estimation and prediction using asymmetric loss functions. Journal of the American Statistical Association 81(394):446–451. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.1986.10478289")}.

See Also

linex_rs

Examples

# Compute the LINEX scoring function.

df <- data.frame(
    y = rep(x = 2, times = 3),
    x = 1:3,
    a = c(-1, 1, 2)
)

df$linex_loss <- linex_sf(x = df$x, y = df$y, a = df$a)

print(df)

scoringfunctions documentation built on Aug. 30, 2026, 5:07 p.m.