scoringfunctions (version 1.1)

nmoment_if: \(n\)-th moment identification function

Description

The function nmoment_if computes the \(n\)-th moment identification function, when \(y\) materialises and \(x\) is the predictive \(n\)-th moment.

The expectile identification function is defined in Table 9 in Gneiting (2011) by setting \(r(t) = t^n\) and \(s(t) = 1\).

Usage

nmoment_if(x, y, n)

Value

Vector of values of the \(n\)-th moment identification function.

Arguments

x

Predictive \(n\)-th moment. It can be a vector of length \(m\) (must have the same length as \(y\)).

y

Realisation (true value) of process. It can be a vector of length \(m\) (must have the same length as \(x\)).

n

\(n\)) is the moment order. It can be a vector of length \(m\) (must have the same length as \(x\)).

Details

The \(n\)-th moment identification function is defined by:

$$V(x, y, n) := x - y^n$$

Domain of function:

$$x \in \mathbb{R}$$

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

$$n \in \mathbb{N}$$

References

Fissler T, Ziegel JF (2016) Higher order elicitability and Osband's principle. The Annals of Statistics 44(4):1680--1707. 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. tools:::Rd_expr_doi("10.1198/jasa.2011.r10138").

Examples

# Compute the n-th moment scoring function.

df <- data.frame(
    y = rep(x = 2, times = 6),
    x = c(1, 2, 3, 1, 2, 3),
    n = c(2, 2, 2, 3, 3, 3)
)

df$nmoment_if <- nmoment_if(x = df$x, y = df$y, n = df$n)

print(df)