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| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.students_t_dist"></a><a class="link" href="students_t_dist.html" title="Students t Distribution"> |
| Students t 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">students_t</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">students_t_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">students_t_distribution</span><span class="special"><></span> <span class="identifier">students_t</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">students_t_distribution</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">// Construct: |
| </span> <span class="identifier">students_t_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">v</span><span class="special">);</span> |
| |
| <span class="comment">// Accessor: |
| </span> <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| |
| <span class="comment">// degrees of freedom estimation: |
| </span> <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">difference_from_mean</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">sd</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span> |
| <span class="special">};</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| A statistical distribution published by William Gosset in 1908. His employer, |
| Guinness Breweries, required him to publish under a pseudonym, so he |
| chose "Student". Given N independent measurements, let |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/students_t_dist.png"></span> |
| </p> |
| <p> |
| where <span class="emphasis"><em>M</em></span> is the population mean,<span class="emphasis"><em>μ</em></span> |
| is the sample mean, and <span class="emphasis"><em>s</em></span> is the sample variance. |
| </p> |
| <p> |
| Student's t-distribution is defined as the distribution of the random |
| variable t which is - very loosely - the "best" that we can |
| do not knowing the true standard deviation of the sample. It has the |
| PDF: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/students_t_ref1.png"></span> |
| </p> |
| <p> |
| The Student's t-distribution takes a single parameter: the number of |
| degrees of freedom of the sample. When the degrees of freedom is <span class="emphasis"><em>one</em></span> |
| then this distribution is the same as the Cauchy-distribution. As the |
| number of degrees of freedom tends towards infinity, then this distribution |
| approaches the normal-distribution. The following graph illustrates how |
| the PDF varies with the degrees of freedom ν: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/students_t_pdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.students_t_dist.member_functions"></a><h5> |
| <a name="id1057694"></a> |
| <a class="link" href="students_t_dist.html#math_toolkit.dist.dist_ref.dists.students_t_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">students_t_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">v</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a Student's t-distribution with <span class="emphasis"><em>v</em></span> degrees |
| of freedom. |
| </p> |
| <p> |
| Requires v > 0, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| Note that non-integral degrees of freedom are supported, and meaningful |
| under certain circumstances. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the number of degrees of freedom of this distribution. |
| </p> |
| <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">difference_from_mean</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">sd</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the number of degrees of freedom required to observe a significant |
| result in the Student's t test when the mean differs from the "true" |
| mean by <span class="emphasis"><em>difference_from_mean</em></span>. |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl> |
| <dt><span class="term">difference_from_mean</span></dt> |
| <dd><p> |
| The difference between the true mean and the sample mean that we |
| wish to show is significant. |
| </p></dd> |
| <dt><span class="term">alpha</span></dt> |
| <dd><p> |
| The maximum acceptable probability of rejecting the null hypothesis |
| when it is in fact true. |
| </p></dd> |
| <dt><span class="term">beta</span></dt> |
| <dd><p> |
| The maximum acceptable probability of failing to reject the null |
| hypothesis when it is in fact false. |
| </p></dd> |
| <dt><span class="term">sd</span></dt> |
| <dd><p> |
| The sample standard deviation. |
| </p></dd> |
| <dt><span class="term">hint</span></dt> |
| <dd><p> |
| A hint for the location to start looking for the result, a good |
| choice for this would be the sample size of a previous borderline |
| Student's t test. |
| </p></dd> |
| </dl> |
| </div> |
| <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> |
| Remember that for a two-sided test, you must divide alpha by two before |
| calling this function. |
| </p></td></tr> |
| </table></div> |
| <p> |
| For more information on this function see the <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm" target="_top">NIST |
| Engineering Statistics Handbook</a>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.students_t_dist.non_member_accessors"></a><h5> |
| <a name="id1057985"></a> |
| <a class="link" href="students_t_dist.html#math_toolkit.dist.dist_ref.dists.students_t_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 [-∞, +∞]. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.students_t_dist.examples"></a><h5> |
| <a name="id1058084"></a> |
| <a class="link" href="students_t_dist.html#math_toolkit.dist.dist_ref.dists.students_t_dist.examples">Examples</a> |
| </h5> |
| <p> |
| Various <a class="link" href="../../stat_tut/weg/st_eg.html" title="Student's t Distribution Examples">worked examples</a> |
| are available illustrating the use of the Student's t distribution. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.students_t_dist.accuracy"></a><h5> |
| <a name="id1058111"></a> |
| <a class="link" href="students_t_dist.html#math_toolkit.dist.dist_ref.dists.students_t_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The normal distribution is implemented in terms of the <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete |
| beta function</a> and <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">it's |
| inverses</a>, refer to accuracy data on those functions for more information. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.students_t_dist.implementation"></a><h5> |
| <a name="id1058142"></a> |
| <a class="link" href="students_t_dist.html#math_toolkit.dist.dist_ref.dists.students_t_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| In the following table <span class="emphasis"><em>v</em></span> is the degrees of freedom |
| of the distribution, <span class="emphasis"><em>t</em></span> is the random variate, <span class="emphasis"><em>p</em></span> |
| is the probability and <span class="emphasis"><em>q = 1-p</em></span>. |
| </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 the relation: pdf = (v / (v + t<sup>2</sup>))<sup>(1+v)/2 </sup> / (sqrt(v) |
| * <a class="link" href="../../../special/sf_beta/beta_function.html" title="Beta">beta</a>(v/2, |
| 0.5)) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relations: |
| </p> |
| <p> |
| p = 1 - z <span class="emphasis"><em>iff t > 0</em></span> |
| </p> |
| <p> |
| p = z <span class="emphasis"><em>otherwise</em></span> |
| </p> |
| <p> |
| where z is given by: |
| </p> |
| <p> |
| <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v |
| / 2, 0.5, v / (v + t<sup>2</sup>)) / 2 <span class="emphasis"><em>iff v < 2t<sup>2</sup></em></span> |
| </p> |
| <p> |
| <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(0.5, |
| v / 2, t<sup>2 </sup> / (v + t<sup>2</sup>) / 2 <span class="emphasis"><em>otherwise</em></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: q = cdf(-t) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: t = sign(p - 0.5) * sqrt(v * y / x) |
| </p> |
| <p> |
| where: |
| </p> |
| <p> |
| x = <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v |
| / 2, 0.5, 2 * min(p, q)) |
| </p> |
| <p> |
| y = 1 - x |
| </p> |
| <p> |
| The quantities <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>y</em></span> |
| are both returned by <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a> |
| without the subtraction implied above. |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: t = -quantile(q) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| v / (v - 2) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| 3 * (v - 2) / (v - 4) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| 6 / (df - 4) |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<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> |
| </div></td> |
| </tr></table> |
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