serrexp_sf: Squared error exp scoring function

In scoringfunctions: A Collection of 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) := (\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) \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).

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(aY)] exists and is finite (Fissler and Pesenti 2023; Tyralis and Papacharalampous 2025).

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 (2025) Transformations of predictions and realizations in consistent scoring functions. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.48550/arXiv.2502.16542")}.

Examples

# Compute the squarer 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 April 4, 2025, 12:28 a.m.