Extreme Value Distribution Functions
Source:R/extreme_value_distribution.R
extreme_value_distribution.RdFunctions to compute the probability density function, cumulative distribution function, and quantile function for the Extreme Value (Gumbel) distribution.
With location \(a\) and scale \(b > 0\), the PDF and CDF are
$$f(x; a, b) = \frac{1}{b}\exp\left(\frac{a-x}{b}\right)\exp\left(-\exp\left(\frac{a-x}{b}\right)\right)$$ $$F(x; a, b) = \exp\left(-\exp\left(\frac{a-x}{b}\right)\right)$$
and the quantile is
$$F^{-1}(p; a, b) = a - b\log\left(-\log(p)\right)$$.
Usage
extreme_value_distribution(location = 0, scale = 1)
extreme_value_pdf(x, location = 0, scale = 1)
extreme_value_lpdf(x, location = 0, scale = 1)
extreme_value_cdf(x, location = 0, scale = 1)
extreme_value_lcdf(x, location = 0, scale = 1)
extreme_value_quantile(p, location = 0, scale = 1)Value
A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.
See also
Boost Documentation for more details on the mathematical background.
Examples
# Extreme Value distribution with location = 0, scale = 1
dist <- extreme_value_distribution(0, 1)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.5452392
logcdf(dist, 0.5)
#> [1] -0.6065307
pdf(dist, 0.5)
#> [1] 0.3307043
logpdf(dist, 0.5)
#> [1] -1.106531
hazard(dist, 0.5)
#> [1] 0.727205
chf(dist, 0.5)
#> [1] 0.7879837
mean(dist)
#> [1] 0.5772157
median(dist)
#> [1] 0.3665129
mode(dist)
#> [1] 0
range(dist)
#> [1] -Inf Inf
quantile(dist, 0.2)
#> [1] -0.475885
standard_deviation(dist)
#> [1] 1.28255
support(dist)
#> [1] -1.797693e+308 1.797693e+308
variance(dist)
#> [1] 1.644934
skewness(dist)
#> [1] 1.139547
kurtosis(dist)
#> [1] 5.4
kurtosis_excess(dist)
#> [1] 2.4
# Convenience functions
extreme_value_pdf(0)
#> [1] 0.3678794
extreme_value_lpdf(0)
#> [1] -1
extreme_value_cdf(0)
#> [1] 0.3678794
extreme_value_lcdf(0)
#> [1] -1
extreme_value_quantile(0.5)
#> [1] 0.3665129