mean_if: Mean identification function

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

Mean identification function

Description

The function mean_if computes the mean identification function , when y materialises and x is the predictive mean.

The mean identification function is defined in Table 9 in Gneiting (2011).

Usage

mean_if(x, y)

Arguments

x	Predictive mean (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).

Details

The mean identification function is defined by:

V(x, y) := x - y

Domain of function:

x \in \mathbb{R}

y \in \mathbb{R}

Range of function:

V(x, y) \in \mathbb{R}

Value

Vector of values of the mean identification function.

Note

The mean functional is the mean \textnormal{E}_F[Y] of the probability distribution F of y (Gneiting 2011).

The mean identification function is a strict \mathbb{F}-identification function for the mean functional. (Gneiting 2011; Fissler and Ziegel 2016; Dimitriadis et al. 2024).

\mathbb{F} is the family of probability distributions F for which \textnormal{E}_F[Y] exists and is finite (Gneiting 2011; Fissler and Ziegel 2016; Dimitriadis et al. 2024).

References

Dimitriadis T, Fissler T, Ziegel JF (2024) Osband's principle for identification functions. Statistical Papers 65:1125–1132. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s00362-023-01428-x")}.

Fissler T, Ziegel JF (2016) Higher order elicitability and Osband's principle. The Annals of Statistics 44(4):1680–1707. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/16-AOS1439")}.

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")}.

Newey WK, Powell JL (1987) Asymmetric least squares estimation and testing. Econometrica 55(4):819–847. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/1911031")}.

Examples

# Compute the mean identification function.

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

df$mean_if <- mean_if(x = df$x, y = df$y)

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