scoringfunctions (version 1.1)

mv_sf: Mean - variance scoring function

Description

The function mv_sf computes the mean - variance scoring function, when \(y\) materialises, \(x_1\) is the predictive mean and \(x_2\) is the predictive variance.

The mean - variance scoring function is defined by eq. (3.11) in Fissler and Ziegel (2019).

Usage

mv_sf(x1, x2, y)

Value

Vector of mean - variance losses.

Arguments

x1

Predictive mean (prediction). It can be a vector of length \(n\) (must have the same length as \(y\)).

x2

Predictive variance (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 \(x_1\)).

Details

The mean - variance scoring function is defined by:

$$ S(x_1, x_2, y) := x_2^{-2} (x_1^2 - 2 x_2 - 2 x_1 y + y^2) $$

Domain of function:

$$x_1 \in \mathbb{R}$$

$$x_2 > 0$$

$$y \in \mathbb{R}$$

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").

Osband KH (1985) Providing Incentives for Better Cost Forecasting. PhD thesis, University of California, Berkeley. tools:::Rd_expr_doi("10.5281/zenodo.4355667").

Examples

# Compute the mean - variance scoring function.

df <- data.frame(
    y = rep(x = 0, times = 6),
    x1 = c(2, 2, -2, -2, 0, 0),
    x2 = c(1, 2, 1, 2, 1, 2)
)

df$mv_penalty <- mv_sf(x1 = df$x1, x2 = df$x2, y = df$y)

print(df)