nmoment_rs: Realised n-th moment score

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Realised n-th moment score

Description

The function nmoment_rs computes the realised n-th moment score, when \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the prediction.

Realised n-th moment score is a realised score corresponding to the n-th moment scoring function nmoment_sf.

Usage

nmoment_rs(x, y, n)

Arguments

x	Prediction. It can be a vector of length m (must have the same length as \textbf{\textit{y}}).
y	Realisation (true value) of process. It can be a vector of length m (must have the same length as \textbf{\textit{x}}).
n	Moment order. It can be a scalar.

Details

The realised n-th moment score is defined by:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, n) := (1/m) \sum_{i = 1}^{m} L(x_i, y_i, n)

where

\textbf{\textit{x}} = (x_1, ..., x_m)^\mathsf{T}

\textbf{\textit{y}} = (y_1, ..., y_m)^\mathsf{T}

and

L(x, y, n) := -x^2 - 2 x (y^n - x)

Domain of function:

\textbf{\textit{x}} \in \mathbb{R}^m

\textbf{\textit{y}} \in \mathbb{R}^m

n \in \mathbb{N}

Range of function:

S(\textbf{\textit{x}}, \textbf{\textit{y}}, n) \geq -(1/m) \sum_{i = 1}^{m} y_i^{2 n}, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^m, n \in \mathbb{N}

Value

Value of the realised n-th moment score.

Note

For details on the n-th moment scoring function, see nmoment_sf.

The concept of realised (average) scores is defined by Gneiting (2011) and Fissler and Ziegel (2019).

The realised n-th moment score is the realised (average) score corresponding to the n-th moment scoring function.

References

Fissler T, Ziegel JF (2019) Order-sensitivity and equivariance of scoring functions. Electronic Journal of Statistics 13(1):1166–1211. \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/jasa.2011.r10138")}.

See Also

nmoment_sf, nmoment_if

Examples

# Compute the realised n-th moment score.
# x = 1 is the predictive 2nd moment E[Y^2] of a standard normal distribution.

set.seed(12345)

n <- 2

x <- 1

y <- rnorm(n = 100, mean = 0, sd = 1)

print(nmoment_rs(x = x, y = y, n = n))

print(nmoment_rs(x = rep(x = x, times = 100), y = y, n = n))

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