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<html>
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<title>Skew Normal Distribution</title>
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<div class="section math_toolkit_dist_dist_ref_dists_skew_normal_dist">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist"></a><a class="link" href="skew_normal_dist.html" title="Skew Normal Distribution">Skew
Normal Distribution</a>
</h5></div></div></div>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">skew_normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
<p>
</p>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span>
<span class="keyword">class</span> <span class="identifier">skew_normal_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">skew_normal_distribution</span><span class="special"><></span> <span class="identifier">normal</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span>
<span class="keyword">class</span> <span class="identifier">skew_normal_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
<span class="comment">// Constructor:</span>
<span class="identifier">skew_normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
<span class="comment">// Accessors:</span>
<span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// mean if normal.</span>
<span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// width, standard deviation if normal.</span>
<span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// The distribution is right skewed if shape > 0 and is left skewed if shape < 0.</span>
<span class="comment">// The distribution is normal if shape is zero.</span>
<span class="special">};</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
The skew normal distribution is a variant of the most well known Gaussian
statistical distribution.
</p>
<p>
The skew normal distribution with shape zero resembles the <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">Normal
Distribution</a>, hence the latter can be regarded as a special case
of the more generic skew normal distribution.
</p>
<p>
If the standard (mean = 0, scale = 1) normal distribution probability
density function is
</p>
<p>
    <span class="inlinemediaobject"><img src="../../../../../equations/normal01_pdf.png"></span>
</p>
<p>
and the cumulative distribution function
</p>
<p>
    <span class="inlinemediaobject"><img src="../../../../../equations/normal01_cdf.png"></span>
</p>
<p>
then the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">PDF</a>
of the <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">skew
normal distribution</a> with shape parameter α, defined by O'Hagan
and Leonhard (1976) is
</p>
<p>
    <span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_pdf0.png"></span>
</p>
<p>
Given <a href="http://en.wikipedia.org/wiki/Location_parameter" target="_top">location</a>
ξ, <a href="http://en.wikipedia.org/wiki/Scale_parameter" target="_top">scale</a>
ω, and <a href="http://en.wikipedia.org/wiki/Shape_parameter" target="_top">shape</a>
α, it can be <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">transformed</a>,
to the form:
</p>
<p>
    <span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_pdf.png"></span>
</p>
<p>
and <a href="http://en.wikipedia.org/wiki/Cumulative_distribution_function" target="_top">CDF</a>:
</p>
<p>
    <span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_cdf.png"></span>
</p>
<p>
where <span class="emphasis"><em>T(h,a)</em></span> is Owen's T function, and <span class="emphasis"><em>Φ(x)</em></span>
is the normal distribution.
</p>
<p>
The variation the PDF and CDF with its parameters is illustrated in the
following graphs:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../graphs/skew_normal_pdf.png" align="middle"></span>
<span class="inlinemediaobject"><img src="../../../../../graphs/skew_normal_cdf.png" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.h0"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.member_functions"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist.dist_ref.dists.skew_normal_dist.member_functions">Member
Functions</a>
</h5>
<pre class="programlisting"><span class="identifier">skew_normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
</pre>
<p>
Constructs a skew_normal distribution with location ξ, scale ω and shape
α.
</p>
<p>
Requires scale > 0, otherwise <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
is called.
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
returns the location ξ of this distribution,
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
returns the scale ω of this distribution,
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
returns the shape α of this distribution.
</p>
<p>
(Location and scale function match other similar distributions, allowing
the functions <code class="computeroutput"><span class="identifier">find_location</span></code>
and <code class="computeroutput"><span class="identifier">find_scale</span></code> to be
used generically).
</p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
While the shape parameter may be chosen arbitrarily (finite), the resulting
<span class="bold"><strong>skewness</strong></span> of the distribution is in
fact limited to about (-1, 1); strictly, the interval is (-0.9952717,
0.9952717).
</p>
<p>
A parameter δ is related to the shape α by δ = α / (1 + α²), and used in the
expression for skewness <span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_skewness.png"></span>
</p>
</td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.h1"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.references"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist.dist_ref.dists.skew_normal_dist.references">References</a>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
<li class="listitem">
<a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal Probability
Distribution</a> for many links and bibliography.
</li>
<li class="listitem">
<a href="http://azzalini.stat.unipd.it/SN/Intro/intro.html" target="_top">A
very brief introduction to the skew-normal distribution</a> by
Adelchi Azzalini (2005-11-2).
</li>
<li class="listitem">
See a <a href="http://www.tri.org.au/azzalini.html" target="_top">skew-normal
function animation</a>.
</li>
</ul></div>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.h2"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.non_member_accessors"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist.dist_ref.dists.skew_normal_dist.non_member_accessors">Non-member
Accessors</a>
</h5>
<p>
All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member
accessor functions</a> that are generic to all distributions are supported:
<a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard
Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>,
<a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>,
<a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>,
<a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>,
<a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>,
<a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>.
</p>
<p>
The domain of the random variable is <span class="emphasis"><em>-[max_value</em></span>,
+[min_value]]. Infinite values are not supported.
</p>
<p>
There are no <a href="http://en.wikipedia.org/wiki/Closed-form_expression" target="_top">closed-form
expression</a> known for the mode and median, but these are computed
for the
</p>
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
<li class="listitem">
mode - by finding the maximum of the PDF.
</li>
<li class="listitem">
median - by computing <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="number">1</span><span class="special">/</span><span class="number">2</span><span class="special">)</span></code>.
</li>
</ul></div>
<p>
The maximum of the PDF is sought through searching the root of f'(x)=0.
</p>
<p>
Both involve iterative methods that will have lower accuracy than other
estimates.
</p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.h3"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.testing"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist.dist_ref.dists.skew_normal_dist.testing">Testing</a>
</h5>
<p>
<a href="http://www.r-project.org/" target="_top">The R Project for Statistical
Computing</a> using library(sn) described at <a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal
Probability Distribution</a>, and at <a href="http://cran.r-project.org/web/packages/sn/sn.pd" target="_top">R
skew-normal(sn) package</a>.
</p>
<p>
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.
</p>
<p>
<a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram
Mathematica</a> was also used to generate some more accurate spot
test data.
</p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.h4"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.accuracy"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist.dist_ref.dists.skew_normal_dist.accuracy">Accuracy</a>
</h5>
<p>
The skew_normal distribution with shape = zero is implemented as a special
case, equivalent to the normal distribution in terms of the <a class="link" href="../../../special/sf_erf/error_function.html" title="Error Functions">error
function</a>, and therefore should have excellent accuracy.
</p>
<p>
The PDF and mean, variance, skewness and kurtosis are also accurately
evaluated using <a href="http://en.wikipedia.org/wiki/Analytical_expression" target="_top">analytical
expressions</a>. The CDF requires <a href="http://en.wikipedia.org/wiki/Owen%27s_T_function" target="_top">Owen's
T function</a> that is evaluated using a Boost C++ <a class="link" href="../../../special/owens_t.html" title="Owen's T function">Owens
T</a> 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 <a class="link" href="../../../special/owens_t.html" title="Owen's T function">Owens
T</a>.
</p>
<p>
The median and mode are calculated by iterative root finding, and both
will be less accurate.
</p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.h5"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.skew_normal_dist.implementation"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist.dist_ref.dists.skew_normal_dist.implementation">Implementation</a>
</h5>
<p>
In the following table, ξ is the location of the distribution, and ω is its
scale, and α is its shape.
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Function
</p>
</th>
<th>
<p>
Implementation Notes
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
pdf
</p>
</td>
<td>
<p>
Using: <span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_pdf.png"></span>
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf
</p>
</td>
<td>
<p>
Using: <span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_cdf.png"></span><br> where <span class="emphasis"><em>T(h,a)</em></span>
is Owen's T function, and <span class="emphasis"><em>Φ(x)</em></span> is the normal
distribution.
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf complement
</p>
</td>
<td>
<p>
Using: complement of normal distribution + 2 * Owens_t
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile
</p>
</td>
<td>
<p>
Maximum of the pdf is sought through searching the root of
f'(x)=0
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile from the complement
</p>
</td>
<td>
<p>
-quantile(SN(-location ξ, scale ω, -shapeα), p)
</p>
</td>
</tr>
<tr>
<td>
<p>
location
</p>
</td>
<td>
<p>
location ξ
</p>
</td>
</tr>
<tr>
<td>
<p>
scale
</p>
</td>
<td>
<p>
scale ω
</p>
</td>
</tr>
<tr>
<td>
<p>
shape
</p>
</td>
<td>
<p>
shape α
</p>
</td>
</tr>
<tr>
<td>
<p>
median
</p>
</td>
<td>
<p>
quantile(1/2)
</p>
</td>
</tr>
<tr>
<td>
<p>
mean
</p>
</td>
<td>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_mean.png"></span>
</p>
</td>
</tr>
<tr>
<td>
<p>
mode
</p>
</td>
<td>
<p>
Maximum of the pdf is sought through searching the root of
f'(x)=0
</p>
</td>
</tr>
<tr>
<td>
<p>
variance
</p>
</td>
<td>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_variance.png"></span>
</p>
</td>
</tr>
<tr>
<td>
<p>
skewness
</p>
</td>
<td>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_skewness.png"></span>
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis
</p>
</td>
<td>
<p>
kurtosis excess-3
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis excess
</p>
</td>
<td>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/skew_normal_kurt_ex.png"></span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
Lalande, Johan Råde, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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