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- [section:skew_normal_dist Skew Normal Distribution]
- ``#include <boost/math/distributions/skew_normal.hpp>``
- namespace boost{ namespace math{
- template <class RealType = double,
- class ``__Policy`` = ``__policy_class`` >
- 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
- [@http://en.wikipedia.org/wiki/Normal_distribution 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
- [equation normal01_pdf]
- and the cumulative distribution function
- [equation normal01_cdf]
- then the [@http://en.wikipedia.org/wiki/Probability_density_function PDF]
- of the [@http://en.wikipedia.org/wiki/Skew_normal_distribution skew normal distribution]
- with shape parameter [alpha], defined by O'Hagan and Leonhard (1976) is
- [equation skew_normal_pdf0]
- Given [@http://en.wikipedia.org/wiki/Location_parameter location] [xi],
- [@http://en.wikipedia.org/wiki/Scale_parameter scale] [omega],
- and [@http://en.wikipedia.org/wiki/Shape_parameter shape] [alpha],
- it can be
- [@http://en.wikipedia.org/wiki/Skew_normal_distribution transformed],
- to the form:
- [equation skew_normal_pdf]
- and [@http://en.wikipedia.org/wiki/Cumulative_distribution_function CDF]:
- [equation skew_normal_cdf]
- where ['T(h,a)] is Owen's T function, and ['[Phi](x)] is the normal distribution.
- The variation the PDF and CDF with its parameters is illustrated
- in the following graphs:
- [graph skew_normal_pdf]
- [graph skew_normal_cdf]
- [h4 Member Functions]
- skew_normal_distribution(RealType location = 0, RealType scale = 1, RealType shape = 0);
- Constructs a skew_normal distribution with location [xi],
- scale [omega] and shape [alpha].
- Requires scale > 0, otherwise __domain_error is called.
- RealType location()const;
- returns the location [xi] of this distribution,
- RealType scale()const;
- returns the scale [omega] of this distribution,
- RealType shape()const;
- returns the shape [alpha] 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 [delta] is related to the shape [alpha] by
- [delta] = [alpha] / (1 + [alpha][pow2]),
- and used in the expression for skewness
- [equation skew_normal_skewness]
- ] [/note]
- [h4 References]
- * [@http://azzalini.stat.unipd.it/SN/ Skew-Normal Probability Distribution] for many links and bibliography.
- * [@http://azzalini.stat.unipd.it/SN/Intro/intro.html A very brief introduction to the skew-normal distribution]
- by Adelchi Azzalini (2005-11-2).
- * See a [@http://www.tri.org.au/azzalini.html skew-normal function animation].
- [h4 Non-member Accessors]
- All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
- that are generic to all distributions are supported: __usual_accessors.
- The domain of the random variable is ['-[max_value], +[min_value]].
- Infinite values are not supported.
- There are no [@http://en.wikipedia.org/wiki/Closed-form_expression closed-form expression]
- known for the mode and median, but these are computed for the
- * mode - by finding the maximum of the PDF.
- * median - by computing `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.
- [h4 Testing]
- __R using library(sn) described at
- [@http://azzalini.stat.unipd.it/SN/ Skew-Normal Probability Distribution],
- and at [@http://cran.r-project.org/web/packages/sn/sn.pd R skew-normal(sn) package].
- Package sn provides functions related to the skew-normal (SN)
- and the skew-t (ST) probability distributions,
- both for the univariate and for the the multivariate case,
- including regression models.
- __Mathematica was also used to generate some more accurate spot test data.
- [h4 Accuracy]
- The skew_normal distribution with shape = zero is implemented as a special case,
- equivalent to the normal distribution in terms of the
- [link math_toolkit.sf_erf.error_function error function],
- and therefore should have excellent accuracy.
- The PDF and mean, variance, skewness and kurtosis are also accurately evaluated using
- [@http://en.wikipedia.org/wiki/Analytical_expression analytical expressions].
- The CDF requires [@http://en.wikipedia.org/wiki/Owen%27s_T_function 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), 1-25 (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.
- [h4 Implementation]
- In the following table, [xi] is the location of the distribution,
- and [omega] is its scale, and [alpha] is its shape.
- [table
- [[Function][Implementation Notes]]
- [[pdf][Using:[equation skew_normal_pdf] ]]
- [[cdf][Using: [equation skew_normal_cdf][br]
- where ['T(h,a)] is Owen's T function, and ['[Phi](x)] is the normal distribution. ]]
- [[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 [xi], scale [omega], -shape[alpha]), p)]]
- [[location][location [xi]]]
- [[scale][scale [omega]]]
- [[shape][shape [alpha]]]
- [[median][quantile(1/2)]]
- [[mean][[equation skew_normal_mean]]]
- [[mode][Maximum of the pdf is sought through searching the root of f'(x)=0]]
- [[variance][[equation skew_normal_variance] ]]
- [[skewness][[equation skew_normal_skewness] ]]
- [[kurtosis][kurtosis excess-3]]
- [[kurtosis excess] [ [equation skew_normal_kurt_ex] ]]
- ] [/table]
- [endsect] [/section:skew_normal_dist skew_Normal]
- [/ skew_normal.qbk
- Copyright 2012 Bejamin Sobotta, John Maddock and Paul A. Bristow.
- Distributed under the Boost Software License, Version 1.0.
- (See accompanying file LICENSE_1_0.txt or copy at
- http://www.boost.org/LICENSE_1_0.txt).
- ]
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