serrexp_sf: Squared error exp scoring function

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Squared error exp scoring function

Description

The function serrexp_sf computes the squared error exp scoring function when y materialises and x is the (1/a) \log(\textnormal{E}_F[\exp(aY)]) predictive entropic risk measure (Gerber 1974).

The squared error exp scoring function is defined in Fissler and Pesenti (2023).

Usage

serrexp_sf(x, y, a)

Arguments

x	Predictive (1/a) \log(\textnormal{E}_F[\exp(aY)]) functional (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 squared error exp scoring function is defined by:

S(x, y, a) := (\exp(ax) - \exp(ay))^2

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 squared errors of exp-transformed variables.

Note

For details on the squared error exp scoring function, see Fissler and Pesenti (2023).

Appendix A1 in Fissler and Pesenti (2023) is given for a > 0. The domain a \neq 0 used here is the one given for g(t) = \exp(at) in Table 1 in Tyralis and Papacharalampous (2026), where the transformation is a bijection for every a \neq 0.

The squared error exp scoring function is negatively oriented (i.e. the smaller, the better).

The squared error exp scoring function is strictly \mathbb{F}-consistent for the (1/a) \log(\textnormal{E}_F[\exp(aY)]) entropic risk measure functional. \mathbb{F} is the family of probability distributions F for which \textnormal{E}_F[\exp(2 a Y)] exists and is finite (Fissler and Pesenti 2023; Tyralis and Papacharalampous 2026). The squared transform doubles the moment requirement: \textnormal{E}_F[\exp(aY)] < \infty alone makes the functional well defined, but the expected score is finite only under \textnormal{E}_F[\exp(2 a Y)] < \infty.

linex_sf parameterises the same one-parameter family of entropic risk measures at the opposite sign: it elicits its functional as -(1/a) \log(\textnormal{E}_F[\exp(-aY)]), which equals the functional documented here evaluated at -a.

References

Fissler T, Pesenti SM (2023) Sensitivity measures based on scoring functions. European Journal of Operational Research 307(3):1408–1423. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.ejor.2022.10.002")}.

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")}.

Tyralis H, Papacharalampous G (2026) Variable transformations in consistent loss functions. Knowledge-Based Systems 336:115202. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.knosys.2025.115202")}.

See Also

serrexp_rs, meanexp_if

Examples

# Compute the squared error exp scoring function.

df <- data.frame(
    y = rep(x = 0, times = 5),
    x = -2:2,
    a = c(-2, -1, 1, 2, 3)
)

df$squaredexp_error <- serrexp_sf(x = df$x, y = df$y, a = df$a)

print(df)

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