quantile_level: Sample quantile level function

In scoringfunctions: A Collection of Loss Functions for Assessing Point Forecasts

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)

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

Value

Value of the sample quantile level.

Note

The sample quantile level is directly related to the quantile identification function quantile_if.

If \textbf{\textit{y}} materialises and \textbf{\textit{x}} is the predictive quantile at level p, then ideally, the sample quantile level should be equal to the nominal quantile level p.

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))

scoringfunctions documentation built on April 4, 2025, 12:28 a.m.