lqmean_rs: Realised $L_q$-mean score

Description

The function lqmean_rs computes the realised $L_q$-mean score with parameter $q$, when $\textbf{\textit{y}}$ materialises and $\textbf{\textit{x}}$ is the prediction.

Realised $L_q$-mean score is a realised score corresponding to the $L_q$-mean scoring function lqmean_sf.

Usage

lqmean_rs(x, y, q)

Value

Value of the realised $L_q$-mean score.

Arguments

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

Details

The realised $L_q$-mean score is defined by:

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

where

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

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

and

$$L(x, y, q) := |x - y|^q$$

Domain of function:

$$\textbf{\textit{x}} \in \mathbb{R}^n$$

$$\textbf{\textit{y}} \in \mathbb{R}^n$$

$$q > 1$$

Range of function:

$$S(\textbf{\textit{x}}, \textbf{\textit{y}}, q) \geq 0, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n, q > 1$$

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").

Examples

Run this code

# Compute the realised Lq-mean score.

set.seed(12345)

q <- 2

x <- 0

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

print(lqmean_rs(x = x, y = y, q = q))

print(lqmean_rs(x = rep(x = x, times = 100), y = y, q = q))

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lqmean_rs: Realised \(L_q\)-mean score