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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.students_t_dist"></a><a class="link" href="students_t_dist.html" title="Students t Distribution">Students
  28         t Distribution</a>
  29 </h4></div></div></div>
  30 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span></pre>
  31 <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>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  34           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">students_t_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">students_t_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">students_t</span><span class="special">;</span>
  38
  39 <span class="keyword">template</span> <span class="special">&lt;</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="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  40 <span class="keyword">class</span> <span class="identifier">students_t_distribution</span>
  41 <span class="special">{</span>
  42    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  43    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  44
  45    <span class="comment">// Constructor:</span>
  46    <span class="identifier">students_t_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
  47
  48    <span class="comment">// Accessor:</span>
  49    <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>
  50
  51    <span class="comment">// degrees of freedom estimation:</span>
  52    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
  53       <span class="identifier">RealType</span> <span class="identifier">difference_from_mean</span><span class="special">,</span>
  54       <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
  55       <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span>
  56       <span class="identifier">RealType</span> <span class="identifier">sd</span><span class="special">,</span>
  57       <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span>
  58 <span class="special">};</span>
  59
  60 <span class="special">}}</span> <span class="comment">// namespaces</span>
  61 </pre>
  62 <p>
  63           Student's t-distribution is a statistical distribution published by William
  64           Gosset in 1908. His employer, Guinness Breweries, required him to publish
  65           under a pseudonym (possibly to hide that they were using statistics to
  66           improve beer quality), so he chose "Student".
  67         </p>
  68 <p>
  69           Given N independent measurements, let
  70         </p>
  71 <div class="blockquote"><blockquote class="blockquote"><p>
  72             <span class="inlinemediaobject"><img src="../../../../equations/students_t_dist.svg"></span>
  73
  74           </p></blockquote></div>
  75 <p>
  76           where <span class="emphasis"><em>M</em></span> is the population mean, &#956; is the sample mean,
  77           and <span class="emphasis"><em>s</em></span> is the sample variance.
  78         </p>
  79 <p>
  80           <a href="https://en.wikipedia.org/wiki/Student%27s_t-distribution" target="_top">Student's
  81           t-distribution</a> is defined as the distribution of the random variable
  82           t which is - very loosely - the "best" that we can do while not
  83           knowing the true standard deviation of the sample. It has the PDF:
  84         </p>
  85 <div class="blockquote"><blockquote class="blockquote"><p>
  86             <span class="inlinemediaobject"><img src="../../../../equations/students_t_ref1.svg"></span>
  87
  88           </p></blockquote></div>
  89 <p>
  90           The Student's t-distribution takes a single parameter: the number of degrees
  91           of freedom of the sample. When the degrees of freedom is <span class="emphasis"><em>one</em></span>
  92           then this distribution is the same as the Cauchy-distribution. As the number
  93           of degrees of freedom tends towards infinity, then this distribution approaches
  94           the normal-distribution. The following graph illustrates how the PDF varies
  95           with the degrees of freedom &#957;:
  96         </p>
  97 <div class="blockquote"><blockquote class="blockquote"><p>
  98             <span class="inlinemediaobject"><img src="../../../../graphs/students_t_pdf.svg" align="middle"></span>
  99
 100           </p></blockquote></div>
 101 <h5>
 102 <a name="math_toolkit.dist_ref.dists.students_t_dist.h0"></a>
 103           <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.member_functions"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.member_functions">Member
 104           Functions</a>
 105         </h5>
 106 <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">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
 107 </pre>
 108 <p>
 109           Constructs a Student's t-distribution with <span class="emphasis"><em>v</em></span> degrees
 110           of freedom.
 111         </p>
 112 <p>
 113           Requires <span class="emphasis"><em>v</em></span> &gt; 0, including infinity (if RealType
 114           permits), otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 115           Note that non-integral degrees of freedom are supported, and are meaningful
 116           under certain circumstances.
 117         </p>
 118 <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>
 119 </pre>
 120 <p>
 121           returns the number of degrees of freedom of this distribution.
 122         </p>
 123 <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>
 124    <span class="identifier">RealType</span> <span class="identifier">difference_from_mean</span><span class="special">,</span>
 125    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
 126    <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span>
 127    <span class="identifier">RealType</span> <span class="identifier">sd</span><span class="special">,</span>
 128    <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span>
 129 </pre>
 130 <p>
 131           returns the number of degrees of freedom required to observe a significant
 132           result in the Student's t test when the mean differs from the "true"
 133           mean by <span class="emphasis"><em>difference_from_mean</em></span>.
 134         </p>
 135 <div class="variablelist">
 136 <p class="title"><b></b></p>
 137 <dl class="variablelist">
 138 <dt><span class="term">difference_from_mean</span></dt>
 139 <dd><p>
 140                 The difference between the true mean and the sample mean that we
 141                 wish to show is significant.
 142               </p></dd>
 143 <dt><span class="term">alpha</span></dt>
 144 <dd><p>
 145                 The maximum acceptable probability of rejecting the null hypothesis
 146                 when it is in fact true.
 147               </p></dd>
 148 <dt><span class="term">beta</span></dt>
 149 <dd><p>
 150                 The maximum acceptable probability of failing to reject the null
 151                 hypothesis when it is in fact false.
 152               </p></dd>
 153 <dt><span class="term">sd</span></dt>
 154 <dd><p>
 155                 The sample standard deviation.
 156               </p></dd>
 157 <dt><span class="term">hint</span></dt>
 158 <dd><p>
 159                 A hint for the location to start looking for the result, a good choice
 160                 for this would be the sample size of a previous borderline Student's
 161                 t test.
 162               </p></dd>
 163 </dl>
 164 </div>
 165 <div class="note"><table border="0" summary="Note">
 166 <tr>
 167 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 168 <th align="left">Note</th>
 169 </tr>
 170 <tr><td align="left" valign="top"><p>
 171             Remember that for a two-sided test, you must divide alpha by two before
 172             calling this function.
 173           </p></td></tr>
 174 </table></div>
 175 <p>
 176           For more information on this function see the <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm" target="_top">NIST
 177           Engineering Statistics Handbook</a>.
 178         </p>
 179 <h5>
 180 <a name="math_toolkit.dist_ref.dists.students_t_dist.h1"></a>
 181           <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.non_member_accessors"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.non_member_accessors">Non-member
 182           Accessors</a>
 183         </h5>
 184 <p>
 185           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 186           functions</a> that are generic to all distributions are supported:
 187           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 188           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 189           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 190           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
 191           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 192           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 193           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
 194           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 195         </p>
 196 <p>
 197           The domain of the random variable is [-&#8734;, +&#8734;].
 198         </p>
 199 <h5>
 200 <a name="math_toolkit.dist_ref.dists.students_t_dist.h2"></a>
 201           <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.examples"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.examples">Examples</a>
 202         </h5>
 203 <p>
 204           Various <a class="link" href="../../stat_tut/weg/st_eg.html" title="Student's t Distribution Examples">worked examples</a>
 205           are available illustrating the use of the Student's t distribution.
 206         </p>
 207 <h5>
 208 <a name="math_toolkit.dist_ref.dists.students_t_dist.h3"></a>
 209           <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.accuracy"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.accuracy">Accuracy</a>
 210         </h5>
 211 <p>
 212           The normal distribution is implemented in terms of the <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete
 213           beta function</a> and <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">its
 214           inverses</a>, refer to accuracy data on those functions for more information.
 215         </p>
 216 <h5>
 217 <a name="math_toolkit.dist_ref.dists.students_t_dist.h4"></a>
 218           <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.implementation0"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.implementation0">Implementation</a>
 219         </h5>
 220 <p>
 221           In the following table <span class="emphasis"><em>v</em></span> is the degrees of freedom
 222           of the distribution, <span class="emphasis"><em>t</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
 223           is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 224         </p>
 225 <div class="informaltable"><table class="table">
 226 <colgroup>
 227 <col>
 228 <col>
 229 </colgroup>
 230 <thead><tr>
 231 <th>
 232                   <p>
 233                     Function
 234                   </p>
 235                 </th>
 236 <th>
 237                   <p>
 238                     Implementation Notes
 239                   </p>
 240                 </th>
 241 </tr></thead>
 242 <tbody>
 243 <tr>
 244 <td>
 245                   <p>
 246                     pdf
 247                   </p>
 248                 </td>
 249 <td>
 250                   <p>
 251                     Using the relation: <span class="serif_italic">pdf = (v / (v
 252                     + t<sup>2</sup>))<sup>(1+v)/2 </sup> / (sqrt(v) * <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>(v/2,
 253                     0.5))</span>
 254                   </p>
 255                 </td>
 256 </tr>
 257 <tr>
 258 <td>
 259                   <p>
 260                     cdf
 261                   </p>
 262                 </td>
 263 <td>
 264                   <p>
 265                     Using the relations:
 266                   </p>
 267                   <p>
 268                     <span class="serif_italic">p = 1 - z <span class="emphasis"><em>iff t &gt; 0</em></span></span>
 269                   </p>
 270                   <p>
 271                     <span class="serif_italic">p = z <span class="emphasis"><em>otherwise</em></span></span>
 272                   </p>
 273                   <p>
 274                     where z is given by:
 275                   </p>
 276                   <p>
 277                     <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v
 278                     / 2, 0.5, v / (v + t<sup>2</sup>)) / 2 <span class="emphasis"><em>iff v &lt; 2t<sup>2</sup></em></span>
 279                   </p>
 280                   <p>
 281                     <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(0.5,
 282                     v / 2, t<sup>2 </sup> / (v + t<sup>2</sup>) / 2 <span class="emphasis"><em>otherwise</em></span>
 283                   </p>
 284                 </td>
 285 </tr>
 286 <tr>
 287 <td>
 288                   <p>
 289                     cdf complement
 290                   </p>
 291                 </td>
 292 <td>
 293                   <p>
 294                     Using the relation: q = cdf(-t)
 295                   </p>
 296                 </td>
 297 </tr>
 298 <tr>
 299 <td>
 300                   <p>
 301                     quantile
 302                   </p>
 303                 </td>
 304 <td>
 305                   <p>
 306                     Using the relation: <span class="serif_italic">t = sign(p -
 307                     0.5) * sqrt(v * y / x)</span>
 308                   </p>
 309                   <p>
 310                     where:
 311                   </p>
 312                   <p>
 313                     <span class="serif_italic">x = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v
 314                     / 2, 0.5, 2 * min(p, q)) </span>
 315                   </p>
 316                   <p>
 317                     <span class="serif_italic">y = 1 - x</span>
 318                   </p>
 319                   <p>
 320                     The quantities <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>y</em></span>
 321                     are both returned by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>
 322                     without the subtraction implied above.
 323                   </p>
 324                 </td>
 325 </tr>
 326 <tr>
 327 <td>
 328                   <p>
 329                     quantile from the complement
 330                   </p>
 331                 </td>
 332 <td>
 333                   <p>
 334                     Using the relation: t = -quantile(q)
 335                   </p>
 336                 </td>
 337 </tr>
 338 <tr>
 339 <td>
 340                   <p>
 341                     mode
 342                   </p>
 343                 </td>
 344 <td>
 345                   <p>
 346                     0
 347                   </p>
 348                 </td>
 349 </tr>
 350 <tr>
 351 <td>
 352                   <p>
 353                     mean
 354                   </p>
 355                 </td>
 356 <td>
 357                   <p>
 358                     0
 359                   </p>
 360                 </td>
 361 </tr>
 362 <tr>
 363 <td>
 364                   <p>
 365                     variance
 366                   </p>
 367                 </td>
 368 <td>
 369                   <p>
 370                     if (v &gt; 2) v / (v - 2) else NaN
 371                   </p>
 372                 </td>
 373 </tr>
 374 <tr>
 375 <td>
 376                   <p>
 377                     skewness
 378                   </p>
 379                 </td>
 380 <td>
 381                   <p>
 382                     if (v &gt; 3) 0 else NaN
 383                   </p>
 384                 </td>
 385 </tr>
 386 <tr>
 387 <td>
 388                   <p>
 389                     kurtosis
 390                   </p>
 391                 </td>
 392 <td>
 393                   <p>
 394                     if (v &gt; 4) 3 * (v - 2) / (v - 4) else NaN
 395                   </p>
 396                 </td>
 397 </tr>
 398 <tr>
 399 <td>
 400                   <p>
 401                     kurtosis excess
 402                   </p>
 403                 </td>
 404 <td>
 405                   <p>
 406                     if (v &gt; 4) 6 / (df - 4) else NaN
 407                   </p>
 408                 </td>
 409 </tr>
 410 </tbody>
 411 </table></div>
 412 <p>
 413           If the moment index <span class="emphasis"><em>k</em></span> is less than <span class="emphasis"><em>v</em></span>,
 414           then the moment is undefined. Evaluating the moment will throw a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 415           unless ignored by a policy, when it will return <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">();</span></code>
 416         </p>
 417 <h6>
 418 <a name="math_toolkit.dist_ref.dists.students_t_dist.h5"></a>
 419           <span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.implementation"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.implementation">Implementation</a>
 420         </h6>
 421 <p>
 422           (By popular demand, we now support infinite argument and random deviate.
 423           But we have not implemented the return of infinity as suggested by <a href="http://en.wikipedia.org/wiki/Student%27s_t-distribution" target="_top">Wikipedia
 424           Student's t</a>, instead throwing a domain error or return NaN. See
 425           also <a href="https://svn.boost.org/trac/boost/ticket/7177" target="_top">https://svn.boost.org/trac/boost/ticket/7177</a>.)
 426         </p>
 427 </div>
 428 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 429 <td align="left"></td>
 430 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 431       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 432       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 433       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 434       Daryle Walker and Xiaogang Zhang<p>
 435         Distributed under the Boost Software License, Version 1.0. (See accompanying
 436         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>)
 437       </p>
 438 </div></td>
 439 </tr></table>
 440 <hr>
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