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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_t_dist"></a><a class="link" href="nc_t_dist.html" title="Noncentral T Distribution"> Noncentral |
| 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">non_central_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">non_central_t_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">non_central_t_distribution</span><span class="special"><></span> <span class="identifier">non_central_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">non_central_t_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">non_central_t_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">delta</span><span class="special">);</span> |
| |
| <span class="comment">// Accessor to degrees_of_freedom parameter v: |
| </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">// Accessor to non-centrality parameter lambda: |
| </span> <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| The noncentral T distribution is a generalization of the <a class="link" href="students_t_dist.html" title="Students t Distribution">Students |
| t Distribution</a>. Let X have a normal distribution with mean δ and |
| variance 1, and let ν S<sup>2</sup> have a chi-squared distribution with degrees of |
| freedom ν. Assume that X and S<sup>2</sup> are independent. The distribution of t<sub>ν</sub>(δ)=X/S |
| is called a noncentral t distribution with degrees of freedom ν and noncentrality |
| parameter δ. |
| </p> |
| <p> |
| This gives the following PDF: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_t_ref1.png"></span> |
| </p> |
| <p> |
| where <sub>1</sub>F<sub>1</sub>(a;b;x) is a confluent hypergeometric function. |
| </p> |
| <p> |
| The following graph illustrates how the distribution changes for different |
| values of δ: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/nc_t_pdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_t_dist.member_functions"></a><h5> |
| <a name="id1050010"></a> |
| <a class="link" href="nc_t_dist.html#math_toolkit.dist.dist_ref.dists.nc_t_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">non_central_t_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a non-central t distribution with degrees of freedom parameter |
| <span class="emphasis"><em>v</em></span> and non-centrality parameter <span class="emphasis"><em>delta</em></span>. |
| </p> |
| <p> |
| Requires v > 0 and finite delta, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </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 parameter <span class="emphasis"><em>v</em></span> from which this object was |
| constructed. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the non-centrality parameter <span class="emphasis"><em>delta</em></span> from |
| which this object was constructed. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_t_dist.non_member_accessors"></a><h5> |
| <a name="id1050149"></a> |
| <a class="link" href="nc_t_dist.html#math_toolkit.dist.dist_ref.dists.nc_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.nc_t_dist.accuracy"></a><h5> |
| <a name="id1050249"></a> |
| <a class="link" href="nc_t_dist.html#math_toolkit.dist.dist_ref.dists.nc_t_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>) |
| found on various platforms with various floating-point types. Unless |
| otherwise specified, any floating-point type that is narrower than the |
| one shown will have <a class="link" href="../../../backgrounders/relative_error.html#zero_error">effectively zero error</a>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_t_dist.errors_in_cdf_of_the_noncentral_t_distribution"></a><p class="title"><b>Table 15. Errors In CDF of the Noncentral T Distribution</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In CDF of the Noncentral T Distribution"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| ν,δ < 600 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=120 Mean=26 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA32, gcc-4.1.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=121 Mean=26 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=122 Mean=25 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=115 Mean=24 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="caution"><table border="0" summary="Caution"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../../doc/src/images/caution.png"></td> |
| <th align="left">Caution</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| The complexity of the current algorithm is dependent upon δ<sup>2</sup>: consequently |
| the time taken to evaluate the CDF increases rapidly for δ > 500, |
| likewise the accuracy decreases rapidly for very large δ. |
| </p></td></tr> |
| </table></div> |
| <p> |
| Accuracy for the quantile and PDF functions should be broadly similar, |
| note however that the <span class="emphasis"><em>mode</em></span> is determined numerically |
| and can not in principal be more accurate than the square root of machine |
| epsilon. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_t_dist.tests"></a><h5> |
| <a name="id1050449"></a> |
| <a class="link" href="nc_t_dist.html#math_toolkit.dist.dist_ref.dists.nc_t_dist.tests">Tests</a> |
| </h5> |
| <p> |
| There are two sets of tests of this distribution: basic sanity checks |
| compare this implementation to the test values given in "Computing |
| discrete mixtures of continuous distributions: noncentral chisquare, |
| noncentral t and the distribution of the square of the sample multiple |
| correlation coefficient." Denise Benton, K. Krishnamoorthy, Computational |
| Statistics & Data Analysis 43 (2003) 249-267. While accuracy checks |
| use test data computed with this implementation and arbitary precision |
| interval arithmetic: this test data is believed to be accurate to at |
| least 50 decimal places. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_t_dist.implementation"></a><h5> |
| <a name="id1050468"></a> |
| <a class="link" href="nc_t_dist.html#math_toolkit.dist.dist_ref.dists.nc_t_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| The CDF is computed using a modification of the method described in "Computing |
| discrete mixtures of continuous distributions: noncentral chisquare, |
| noncentral t and the distribution of the square of the sample multiple |
| correlation coefficient." Denise Benton, K. Krishnamoorthy, Computational |
| Statistics & Data Analysis 43 (2003) 249-267. |
| </p> |
| <p> |
| This uses the following formula for the CDF: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_t_ref2.png"></span> |
| </p> |
| <p> |
| Where I<sub>x</sub>(a,b) is the incomplete beta function, and Φ(x) is the normal |
| CDF at x. |
| </p> |
| <p> |
| Iteration starts at the largest of the Poisson weighting terms (at i |
| = δ<sup>2</sup> / 2) and then proceeds in both directions as per Benton and Krishnamoorthy's |
| paper. |
| </p> |
| <p> |
| Alternatively, by considering what happens when t = ∞, we have x = 1, |
| and therefore I<sub>x</sub>(a,b) = 1 and: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_t_ref3.png"></span> |
| </p> |
| <p> |
| From this we can easily show that: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_t_ref4.png"></span> |
| </p> |
| <p> |
| and therefore we have a means to compute either the probability or its |
| complement directly without the risk of cancellation error. The crossover |
| criterion for choosing whether to calculate the CDF or it's complement |
| is the same as for the <a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral |
| Beta Distribution</a>. |
| </p> |
| <p> |
| The PDF can be computed by a very similar method using: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_t_ref5.png"></span> |
| </p> |
| <p> |
| Where I<sub>x</sub><sup>'</sup>(a,b) is the derivative of the incomplete beta function. |
| </p> |
| <p> |
| The quantile is calculated via the usual <a class="link" href="../../../toolkit/internals1/roots2.html" title="Root Finding Without Derivatives">derivative-free |
| root-finding techniques</a>, with the initial guess taken as the quantile |
| of a normal approximation to the noncentral T. |
| </p> |
| <p> |
| There is no closed form for the mode, so this is computed via functional |
| maximisation of the PDF. |
| </p> |
| <p> |
| The remaining functions (mean, variance etc) are implemented using the |
| formulas given in Weisstein, Eric W. "Noncentral Student's t-Distribution." |
| From MathWorld--A Wolfram Web Resource. <a href="http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html" target="_top">http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html</a> |
| and in the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralStudentTDistribution.html" target="_top">Mathematica |
| documentation</a>. |
| </p> |
| <p> |
| Some analytic properties of noncentral distributions (particularly unimodality, |
| and monotonicity of their modes) are surveyed and summarized by: |
| </p> |
| <p> |
| Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, |
| 141 (2003) 3-12. |
| </p> |
| </div> |
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| <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>) |
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