scoringfunctions (version 1.1)

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:

$$P(x, y) := (1/n) \sum_{i = 1}^{n} V(x_i, y_i)$$

where

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

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

and

$$V(x, y) := \textbf{1} \lbrace x \geq y \rbrace$$

Domain of function:

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

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

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))