scoringfunctions (version 1.2)

quantile_level: Sample quantile level function

Description

The function quantile_level computes the sample quantile level, when \(\textbf{\textit{y}}\) materialises and \(\textbf{\textit{x}}\) is the predictive quantile at level \(p\).

Usage

quantile_level(x, y)

Value

Value of the sample quantile level.

Arguments

x

Predictive quantile (prediction) at level \(p\). 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 \(\textbf{\textit{x}}\)).

Details

The sample quantile level function is defined by:

$$P(\textbf{\textit{x}}, \textbf{\textit{y}}) := (1/n) \sum_{i = 1}^{n} \textbf{1} \lbrace x_i \geq y_i \rbrace$$

where

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

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

Domain of function:

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

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

Range of function:

$$0 \leq P(\textbf{\textit{x}}, \textbf{\textit{y}}) \leq 1, \forall \textbf{\textit{x}}, \textbf{\textit{y}} \in \mathbb{R}^n$$

References

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").

Koenker R, Bassett Jr G (1978) Regression quantiles. Econometrica 46(1):33--50. tools:::Rd_expr_doi("10.2307/1913643").

See Also

quantile_sf, quantile_rs, quantile_if

Examples

# Compute the sample quantile level.

set.seed(12345)

x <- qnorm(p = 0.75, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

y <- rnorm(n = 1000, mean = 0, sd = 1)

print(quantile_level(x = x, y = y))