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  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.nc_t_dist"></a><a class="link" href="nc_t_dist.html" title="Noncentral T Distribution">Noncentral T
  28         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">non_central_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;15.&#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">non_central_t_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">non_central_t_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">non_central_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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  40 <span class="keyword">class</span> <span class="identifier">non_central_t_distribution</span>
  41 <span class="special">{</span>
  42 <span class="keyword">public</span><span class="special">:</span>
  43    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  44    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  45
  46    <span class="comment">// Constructor:</span>
  47    <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>
  48
  49    <span class="comment">// Accessor to degrees_of_freedom parameter v:</span>
  50    <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>
  51
  52    <span class="comment">// Accessor to non-centrality parameter delta:</span>
  53    <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  54 <span class="special">};</span>
  55
  56 <span class="special">}}</span> <span class="comment">// namespaces</span>
  57 </pre>
  58 <p>
  59           The noncentral T distribution is a generalization of the <a class="link" href="students_t_dist.html" title="Students t Distribution">Students
  60           t Distribution</a>. Let X have a normal distribution with mean &#948; and variance
  61           1, and let &#957; S<sup>2</sup> have a chi-squared distribution with degrees of freedom &#957;.
  62           Assume that X and S<sup>2</sup> are independent. The distribution of t<sub>&#957;</sub>(&#948;)=X/S is called
  63           a noncentral t distribution with degrees of freedom &#957; and noncentrality parameter
  64           &#948;.
  65         </p>
  66 <p>
  67           This gives the following PDF:
  68         </p>
  69 <p>
  70           <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref1.svg"></span>
  71         </p>
  72 <p>
  73           where <sub>1</sub>F<sub>1</sub>(a;b;x) is a confluent hypergeometric function.
  74         </p>
  75 <p>
  76           The following graph illustrates how the distribution changes for different
  77           values of &#957; and &#948;:
  78         </p>
  79 <p>
  80           <span class="inlinemediaobject"><img src="../../../../graphs/nc_t_pdf.svg" align="middle"></span>
  81   <span class="inlinemediaobject"><img src="../../../../graphs/nc_t_cdf.svg" align="middle"></span>
  82         </p>
  83 <h5>
  84 <a name="math_toolkit.dist_ref.dists.nc_t_dist.h0"></a>
  85           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.member_functions"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.member_functions">Member
  86           Functions</a>
  87         </h5>
  88 <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">delta</span><span class="special">);</span>
  89 </pre>
  90 <p>
  91           Constructs a non-central t distribution with degrees of freedom parameter
  92           <span class="emphasis"><em>v</em></span> and non-centrality parameter <span class="emphasis"><em>delta</em></span>.
  93         </p>
  94 <p>
  95           Requires <span class="emphasis"><em>v</em></span> &gt; 0 (including positive infinity) and
  96           finite <span class="emphasis"><em>delta</em></span>, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
  97         </p>
  98 <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>
  99 </pre>
 100 <p>
 101           Returns the parameter <span class="emphasis"><em>v</em></span> from which this object was
 102           constructed.
 103         </p>
 104 <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>
 105 </pre>
 106 <p>
 107           Returns the non-centrality parameter <span class="emphasis"><em>delta</em></span> from which
 108           this object was constructed.
 109         </p>
 110 <h5>
 111 <a name="math_toolkit.dist_ref.dists.nc_t_dist.h1"></a>
 112           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.non_member_accessors"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.non_member_accessors">Non-member
 113           Accessors</a>
 114         </h5>
 115 <p>
 116           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 117           functions</a> that are generic to all distributions are supported:
 118           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 119           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 120           <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>,
 121           <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>,
 122           <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>,
 123           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 124           <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>,
 125           <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>.
 126         </p>
 127 <p>
 128           The domain of the random variable is [-&#8734;, +&#8734;].
 129         </p>
 130 <h5>
 131 <a name="math_toolkit.dist_ref.dists.nc_t_dist.h2"></a>
 132           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.accuracy"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.accuracy">Accuracy</a>
 133         </h5>
 134 <p>
 135           The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>)
 136           found on various platforms with various floating-point types. Unless otherwise
 137           specified, any floating-point type that is narrower than the one shown
 138           will have <a class="link" href="../../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 139           zero error</a>.
 140         </p>
 141 <div class="table">
 142 <a name="math_toolkit.dist_ref.dists.nc_t_dist.table_non_central_t_CDF"></a><p class="title"><b>Table&#160;5.8.&#160;Error rates for non central t CDF</b></p>
 143 <div class="table-contents"><table class="table" summary="Error rates for non central t CDF">
 144 <colgroup>
 145 <col>
 146 <col>
 147 <col>
 148 <col>
 149 <col>
 150 </colgroup>
 151 <thead><tr>
 152 <th>
 153                 </th>
 154 <th>
 155                   <p>
 156                     Microsoft Visual C++ version 12.0<br> Win32<br> double
 157                   </p>
 158                 </th>
 159 <th>
 160                   <p>
 161                     GNU C++ version 5.1.0<br> linux<br> double
 162                   </p>
 163                 </th>
 164 <th>
 165                   <p>
 166                     GNU C++ version 5.1.0<br> linux<br> long double
 167                   </p>
 168                 </th>
 169 <th>
 170                   <p>
 171                     Sun compiler version 0x5130<br> Sun Solaris<br> long double
 172                   </p>
 173                 </th>
 174 </tr></thead>
 175 <tbody>
 176 <tr>
 177 <td>
 178                   <p>
 179                     Non Central T
 180                   </p>
 181                 </td>
 182 <td>
 183                   <p>
 184                     <span class="blue">Max = 138&#949; (Mean = 31.5&#949;)</span>
 185                   </p>
 186                 </td>
 187 <td>
 188                   <p>
 189                     <span class="blue">Max = 0.796&#949; (Mean = 0.0691&#949;)</span><br>
 190                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
 191                     = 5.28e+15&#949; (Mean = 8.49e+14&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_non_central_t_CDF_Rmath_3_0_2_Non_Central_T">And
 192                     other failures.</a>)</span>
 193                   </p>
 194                 </td>
 195 <td>
 196                   <p>
 197                     <span class="blue">Max = 141&#949; (Mean = 31.1&#949;)</span>
 198                   </p>
 199                 </td>
 200 <td>
 201                   <p>
 202                     <span class="blue">Max = 145&#949; (Mean = 30.2&#949;)</span>
 203                   </p>
 204                 </td>
 205 </tr>
 206 <tr>
 207 <td>
 208                   <p>
 209                     Non Central T (small non-centrality)
 210                   </p>
 211                 </td>
 212 <td>
 213                   <p>
 214                     <span class="blue">Max = 3.61&#949; (Mean = 1.03&#949;)</span>
 215                   </p>
 216                 </td>
 217 <td>
 218                   <p>
 219                     <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
 220                     3.0.2:</em></span> Max = 2.09e+03&#949; (Mean = 244&#949;))
 221                   </p>
 222                 </td>
 223 <td>
 224                   <p>
 225                     <span class="blue">Max = 7.86&#949; (Mean = 1.69&#949;)</span>
 226                   </p>
 227                 </td>
 228 <td>
 229                   <p>
 230                     <span class="blue">Max = 9.15&#949; (Mean = 2.25&#949;)</span>
 231                   </p>
 232                 </td>
 233 </tr>
 234 <tr>
 235 <td>
 236                   <p>
 237                     Non Central T (large parameters)
 238                   </p>
 239                 </td>
 240 <td>
 241                   <p>
 242                     <span class="blue">Max = 286&#949; (Mean = 62.8&#949;)</span>
 243                   </p>
 244                 </td>
 245 <td>
 246                   <p>
 247                     <span class="blue">Max = 257&#949; (Mean = 72.1&#949;)</span><br> <br>
 248                     (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 2.46&#949; (Mean = 0.657&#949;))
 249                   </p>
 250                 </td>
 251 <td>
 252                   <p>
 253                     <span class="blue">Max = 5.26e+05&#949; (Mean = 1.48e+05&#949;)</span>
 254                   </p>
 255                 </td>
 256 <td>
 257                   <p>
 258                     <span class="blue">Max = 5.24e+05&#949; (Mean = 1.47e+05&#949;)</span>
 259                   </p>
 260                 </td>
 261 </tr>
 262 </tbody>
 263 </table></div>
 264 </div>
 265 <br class="table-break"><div class="table">
 266 <a name="math_toolkit.dist_ref.dists.nc_t_dist.table_non_central_t_CDF_complement"></a><p class="title"><b>Table&#160;5.9.&#160;Error rates for non central t CDF complement</b></p>
 267 <div class="table-contents"><table class="table" summary="Error rates for non central t CDF complement">
 268 <colgroup>
 269 <col>
 270 <col>
 271 <col>
 272 <col>
 273 <col>
 274 </colgroup>
 275 <thead><tr>
 276 <th>
 277                 </th>
 278 <th>
 279                   <p>
 280                     Microsoft Visual C++ version 12.0<br> Win32<br> double
 281                   </p>
 282                 </th>
 283 <th>
 284                   <p>
 285                     GNU C++ version 5.1.0<br> linux<br> double
 286                   </p>
 287                 </th>
 288 <th>
 289                   <p>
 290                     GNU C++ version 5.1.0<br> linux<br> long double
 291                   </p>
 292                 </th>
 293 <th>
 294                   <p>
 295                     Sun compiler version 0x5130<br> Sun Solaris<br> long double
 296                   </p>
 297                 </th>
 298 </tr></thead>
 299 <tbody>
 300 <tr>
 301 <td>
 302                   <p>
 303                     Non Central T
 304                   </p>
 305                 </td>
 306 <td>
 307                   <p>
 308                     <span class="blue">Max = 150&#949; (Mean = 32.3&#949;)</span>
 309                   </p>
 310                 </td>
 311 <td>
 312                   <p>
 313                     <span class="blue">Max = 0.707&#949; (Mean = 0.0497&#949;)</span><br>
 314                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
 315                     = 6.19e+15&#949; (Mean = 6.72e+14&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_non_central_t_CDF_complement_Rmath_3_0_2_Non_Central_T">And
 316                     other failures.</a>)</span>
 317                   </p>
 318                 </td>
 319 <td>
 320                   <p>
 321                     <span class="blue">Max = 203&#949; (Mean = 31.8&#949;)</span>
 322                   </p>
 323                 </td>
 324 <td>
 325                   <p>
 326                     <span class="blue">Max = 340&#949; (Mean = 43.6&#949;)</span>
 327                   </p>
 328                 </td>
 329 </tr>
 330 <tr>
 331 <td>
 332                   <p>
 333                     Non Central T (small non-centrality)
 334                   </p>
 335                 </td>
 336 <td>
 337                   <p>
 338                     <span class="blue">Max = 5.21&#949; (Mean = 1.43&#949;)</span>
 339                   </p>
 340                 </td>
 341 <td>
 342                   <p>
 343                     <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
 344                     3.0.2:</em></span> Max = 1.87e+03&#949; (Mean = 263&#949;))
 345                   </p>
 346                 </td>
 347 <td>
 348                   <p>
 349                     <span class="blue">Max = 7.48&#949; (Mean = 1.86&#949;)</span>
 350                   </p>
 351                 </td>
 352 <td>
 353                   <p>
 354                     <span class="blue">Max = 10.9&#949; (Mean = 2.43&#949;)</span>
 355                   </p>
 356                 </td>
 357 </tr>
 358 <tr>
 359 <td>
 360                   <p>
 361                     Non Central T (large parameters)
 362                   </p>
 363                 </td>
 364 <td>
 365                   <p>
 366                     <span class="blue">Max = 227&#949; (Mean = 50.4&#949;)</span>
 367                   </p>
 368                 </td>
 369 <td>
 370                   <p>
 371                     <span class="blue">Max = 478&#949; (Mean = 96.3&#949;)</span><br> <br>
 372                     (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 2.24&#949; (Mean = 0.945&#949;))
 373                   </p>
 374                 </td>
 375 <td>
 376                   <p>
 377                     <span class="blue">Max = 9.79e+05&#949; (Mean = 1.97e+05&#949;)</span>
 378                   </p>
 379                 </td>
 380 <td>
 381                   <p>
 382                     <span class="blue">Max = 9.79e+05&#949; (Mean = 1.97e+05&#949;)</span>
 383                   </p>
 384                 </td>
 385 </tr>
 386 </tbody>
 387 </table></div>
 388 </div>
 389 <br class="table-break"><div class="caution"><table border="0" summary="Caution">
 390 <tr>
 391 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
 392 <th align="left">Caution</th>
 393 </tr>
 394 <tr><td align="left" valign="top"><p>
 395             The complexity of the current algorithm is dependent upon &#948;<sup>2</sup>: consequently
 396             the time taken to evaluate the CDF increases rapidly for &#948; &gt; 500, likewise
 397             the accuracy decreases rapidly for very large &#948;.
 398           </p></td></tr>
 399 </table></div>
 400 <p>
 401           Accuracy for the quantile and PDF functions should be broadly similar.
 402           The <span class="emphasis"><em>mode</em></span> is determined numerically and cannot in principal
 403           be more accurate than the square root of floating-point type FPT epsilon,
 404           accessed using <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">epsilon</span><span class="special">&lt;</span><span class="identifier">FPT</span><span class="special">&gt;()</span></code>.
 405           For 64-bit <code class="computeroutput"><span class="keyword">double</span></code>, epsilon
 406           is about 1e-16, so the fractional accuracy is limited to 1e-8.
 407         </p>
 408 <h5>
 409 <a name="math_toolkit.dist_ref.dists.nc_t_dist.h3"></a>
 410           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.tests"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.tests">Tests</a>
 411         </h5>
 412 <p>
 413           There are two sets of tests of this distribution:
 414         </p>
 415 <p>
 416           Basic sanity checks compare this implementation to the test values given
 417           in "Computing discrete mixtures of continuous distributions: noncentral
 418           chisquare, noncentral t and the distribution of the square of the sample
 419           multiple correlation coefficient." Denise Benton, K. Krishnamoorthy,
 420           Computational Statistics &amp; Data Analysis 43 (2003) 249-267.
 421         </p>
 422 <p>
 423           Accuracy checks use test data computed with this implementation and arbitary
 424           precision interval arithmetic: this test data is believed to be accurate
 425           to at least 50 decimal places.
 426         </p>
 427 <p>
 428           The cases of large (or infinite) &#957; and/or large &#948; has received special treatment
 429           to avoid catastrophic loss of accuracy. New tests have been added to confirm
 430           the improvement achieved.
 431         </p>
 432 <p>
 433           From Boost 1.52, degrees of freedom &#957; can be +&#8734;
 434 when the normal distribution
 435           located at &#948;
 436 (equivalent to the central Student's t distribution) is used
 437           in place for accuracy and speed.
 438         </p>
 439 <h5>
 440 <a name="math_toolkit.dist_ref.dists.nc_t_dist.h4"></a>
 441           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.implementation"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.implementation">Implementation</a>
 442         </h5>
 443 <p>
 444           The CDF is computed using a modification of the method described in "Computing
 445           discrete mixtures of continuous distributions: noncentral chisquare, noncentral
 446           t and the distribution of the square of the sample multiple correlation
 447           coefficient." Denise Benton, K. Krishnamoorthy, Computational Statistics
 448           &amp; Data Analysis 43 (2003) 249-267.
 449         </p>
 450 <p>
 451           This uses the following formula for the CDF:
 452         </p>
 453 <p>
 454           <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref2.svg"></span>
 455         </p>
 456 <p>
 457           Where I<sub>x</sub>(a,b) is the incomplete beta function, and &#934;(x) is the normal CDF
 458           at x.
 459         </p>
 460 <p>
 461           Iteration starts at the largest of the Poisson weighting terms (at i =
 462           &#948;<sup>2</sup> / 2) and then proceeds in both directions as per Benton and Krishnamoorthy's
 463           paper.
 464         </p>
 465 <p>
 466           Alternatively, by considering what happens when t = &#8734;, we have x = 1, and
 467           therefore I<sub>x</sub>(a,b) = 1 and:
 468         </p>
 469 <p>
 470           <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref3.svg"></span>
 471         </p>
 472 <p>
 473           From this we can easily show that:
 474         </p>
 475 <p>
 476           <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref4.svg"></span>
 477         </p>
 478 <p>
 479           and therefore we have a means to compute either the probability or its
 480           complement directly without the risk of cancellation error. The crossover
 481           criterion for choosing whether to calculate the CDF or its complement is
 482           the same as for the <a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral
 483           Beta Distribution</a>.
 484         </p>
 485 <p>
 486           The PDF can be computed by a very similar method using:
 487         </p>
 488 <p>
 489           <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref5.svg"></span>
 490         </p>
 491 <p>
 492           Where I<sub>x</sub><sup>'</sup>(a,b) is the derivative of the incomplete beta function.
 493         </p>
 494 <p>
 495           For both the PDF and CDF we switch to approximating the distribution by
 496           a Student's t distribution centred on &#948; when &#957; is very large. The crossover
 497           location appears to be when &#948;/(4&#957;) &lt; &#949;, this location was estimated by
 498           inspection of equation 2.6 in "A Comparison of Approximations To Percentiles
 499           of the Noncentral t-Distribution". H. Sahai and M. M. Ojeda, Revista
 500           Investigacion Operacional Vol 21, No 2, 2000, page 123.
 501         </p>
 502 <p>
 503           Equation 2.6 is a Fisher-Cornish expansion by Eeden and Johnson. The second
 504           term includes the ratio &#948;/(4&#957;), so when this term become negligible, this
 505           and following terms can be ignored, leaving just Student's t distribution
 506           centred on &#948;.
 507         </p>
 508 <p>
 509           This was also confirmed by experimental testing.
 510         </p>
 511 <p>
 512           See also
 513         </p>
 514 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 515 <li class="listitem">
 516               "Some Approximations to the Percentage Points of the Noncentral
 517               t-Distribution". C. van Eeden. International Statistical Review,
 518               29, 4-31.
 519             </li>
 520 <li class="listitem">
 521               "Continuous Univariate Distributions". N.L. Johnson, S. Kotz
 522               and N. Balkrishnan. 1995. John Wiley and Sons New York.
 523             </li>
 524 </ul></div>
 525 <p>
 526           The quantile is calculated via the usual <a class="link" href="../../roots/roots_noderiv.html" title="Root Finding Without Derivatives">root-finding
 527           without derivatives</a> method with the initial guess taken as the quantile
 528           of a normal approximation to the noncentral T.
 529         </p>
 530 <p>
 531           There is no closed form for the mode, so this is computed via functional
 532           maximisation of the PDF.
 533         </p>
 534 <p>
 535           The remaining functions (mean, variance etc) are implemented using the
 536           formulas given in Weisstein, Eric W. "Noncentral Student's t-Distribution."
 537           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>
 538           and in the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralStudentTDistribution.html" target="_top">Mathematica
 539           documentation</a>.
 540         </p>
 541 <p>
 542           Some analytic properties of noncentral distributions (particularly unimodality,
 543           and monotonicity of their modes) are surveyed and summarized by:
 544         </p>
 545 <p>
 546           Andrea van Aubel &amp; Wolfgang Gawronski, Applied Mathematics and Computation,
 547           141 (2003) 3-12.
 548         </p>
 549 </div>
 550 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 551 <td align="left"></td>
 552 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 553       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 554       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 555       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 556         Distributed under the Boost Software License, Version 1.0. (See accompanying
 557         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>)
 558       </p>
 559 </div></td>
 560 </tr></table>
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