...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#include <boost/math/distributions/skew_normal.hpp>
namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > class skew_normal_distribution; typedef skew_normal_distribution<> normal; template <class RealType, class Policy> class skew_normal_distribution { public: typedef RealType value_type; typedef Policy policy_type; // Constructor: skew_normal_distribution(RealType location = 0, RealType scale = 1, RealType shape = 0); // Accessors: RealType location()const; // mean if normal. RealType scale()const; // width, standard deviation if normal. RealType shape()const; // The distribution is right skewed if shape > 0 and is left skewed if shape < 0. // The distribution is normal if shape is zero. }; }} // namespaces
The skew normal distribution is a variant of the most well known Gaussian statistical distribution.
The skew normal distribution with shape zero resembles the Normal Distribution, hence the latter can be regarded as a special case of the more generic skew normal distribution.
If the standard (mean = 0, scale = 1) normal distribution probability density function is
and the cumulative distribution function
then the PDF of the skew normal distribution with shape parameter α, defined by O'Hagan and Leonhard (1976) is
Given location ξ, scale ω, and shape α, it can be transformed, to the form:
and CDF:
where T(h,a) is Owen's T function, and Φ(x) is the normal distribution.
The variation the PDF and CDF with its parameters is illustrated in the following graphs:
skew_normal_distribution(RealType location = 0, RealType scale = 1, RealType shape = 0);
Constructs a skew_normal distribution with location ξ, scale ω and shape α.
Requires scale > 0, otherwise domain_error is called.
RealType location()const;
returns the location ξ of this distribution,
RealType scale()const;
returns the scale ω of this distribution,
RealType shape()const;
returns the shape α of this distribution.
(Location and scale function match other similar distributions, allowing
the functions find_location
and find_scale
to be used
generically).
Note  

While the shape parameter may be chosen arbitrarily (finite), the resulting skewness of the distribution is in fact limited to about (1, 1); strictly, the interval is (0.9952717, 0.9952717). A parameter δ is related to the shape α by δ = α / (1 + α²), and used in the expression for skewness 
All the usual nonmember accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.
The domain of the random variable is [max_value], +[min_value]. Infinite values are not supported.
There are no closedform expression known for the mode and median, but these are computed for the
quantile(1/2)
.
The maximum of the PDF is sought through searching the root of f'(x)=0.
Both involve iterative methods that will have lower accuracy than other estimates.
The R Project for Statistical Computing using library(sn) described at SkewNormal Probability Distribution, and at R skewnormal(sn) package.
Package sn provides functions related to the skewnormal (SN) and the skewt (ST) probability distributions, both for the univariate and for the the multivariate case, including regression models.
Wolfram Mathematica was also used to generate some more accurate spot test data.
The skew_normal distribution with shape = zero is implemented as a special case, equivalent to the normal distribution in terms of the error function, and therefore should have excellent accuracy.
The PDF and mean, variance, skewness and kurtosis are also accurately evaluated using analytical expressions. The CDF requires Owen's T function that is evaluated using a Boost C++ Owens T implementation of the algorithms of M. Patefield and D. Tandy, Journal of Statistical Software, 5(5), 125 (2000); the complicated accuracy of this function is discussed in detail at Owens T.
The median and mode are calculated by iterative root finding, and both will be less accurate.
In the following table, ξ is the location of the distribution, and ω is its scale, and α is its shape.
Function 
Implementation Notes 


Using: 
cdf 
Using:

cdf complement 
Using: complement of normal distribution + 2 * Owens_t 
quantile 
Maximum of the pdf is sought through searching the root of f'(x)=0 
quantile from the complement 
quantile(SN(location ξ, scale ω, shapeα), p) 
location 
location ξ 
scale 
scale ω 
shape 
shape α 
median 
quantile(1/2) 
mean 

mode 
Maximum of the pdf is sought through searching the root of f'(x)=0 
variance 

skewness 

kurtosis 
kurtosis excess3 
kurtosis excess 
