Functions to compute the probability density function, cumulative distribution function, and quantile function for the Skew Normal distribution.
The skew normal distribution is a variant of the most well known Gaussian statistical distribution. If the standard (mean = 0, scale = 1) normal distribution probability density function is \(\phi(x)\) and the cumulative distribution function is \(\Phi(x)\), then the PDF of the skew normal distribution with shape parameter \(\alpha\) is: $$f(x;\alpha) = 2\phi(x)\Phi(\alpha x)$$
Given location \(\xi\), scale \(\omega\), and shape \(\alpha\), it can be transformed to: $$f(x) = \frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\Phi\left(\alpha\frac{x-\xi}{\omega}\right)$$
Accuracy and Implementation Notes: The skew_normal distribution with shape = zero is equivalent to the normal distribution and uses the error function for excellent accuracy. The CDF requires Owen's T function, which is evaluated using algoritms by Patefield and Tandy. The median and mode are calculated by iterative root finding and may be less accurate than other estimates.
Usage
skew_normal_distribution(location = 0, scale = 1, shape = 0)
skew_normal_pdf(x, location = 0, scale = 1, shape = 0)
skew_normal_lpdf(x, location = 0, scale = 1, shape = 0)
skew_normal_cdf(x, location = 0, scale = 1, shape = 0)
skew_normal_lcdf(x, location = 0, scale = 1, shape = 0)
skew_normal_quantile(p, location = 0, scale = 1, shape = 0)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
# Skew Normal distribution with location = 0, scale = 1, shape = 0
dist <- skew_normal_distribution(0, 1, 0)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.6914625
logcdf(dist, 0.5)
#> [1] -0.3689464
pdf(dist, 0.5)
#> [1] 0.3520653
logpdf(dist, 0.5)
#> [1] -1.043939
hazard(dist, 0.5)
#> [1] 1.141078
chf(dist, 0.5)
#> [1] 1.175912
mean(dist)
#> [1] 0
median(dist)
#> [1] 0
mode(dist)
#> [1] 0
range(dist)
#> [1] -Inf Inf
quantile(dist, 0.2)
#> [1] -0.8416212
standard_deviation(dist)
#> [1] 1
support(dist)
#> [1] -1.797693e+308 1.797693e+308
variance(dist)
#> [1] 1
skewness(dist)
#> [1] 0
kurtosis(dist)
#> [1] 3
kurtosis_excess(dist)
#> [1] 0
# Convenience functions
skew_normal_pdf(0)
#> [1] 0.3989423
skew_normal_lpdf(0)
#> [1] -0.9189385
skew_normal_cdf(0)
#> [1] 0.5
skew_normal_lcdf(0)
#> [1] -0.6931472
skew_normal_quantile(0.5)
#> [1] 0