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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.nc_f_dist"></a><a class="link" href="nc_f_dist.html" title="Noncentral F Distribution">Noncentral F
  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_f</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">non_central_f_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">non_central_f_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">non_central_f</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">non_central_f_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_f_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">v2</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
  48
  49    <span class="comment">// Accessor to degrees_of_freedom parameters v1 &amp; v2:</span>
  50    <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  51    <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  52
  53    <span class="comment">// Accessor to non-centrality parameter lambda:</span>
  54    <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  55 <span class="special">};</span>
  56
  57 <span class="special">}}</span> <span class="comment">// namespaces</span>
  58 </pre>
  59 <p>
  60           The noncentral F distribution is a generalization of the <a class="link" href="f_dist.html" title="F Distribution">Fisher
  61           F Distribution</a>. It is defined as the ratio
  62         </p>
  63 <div class="blockquote"><blockquote class="blockquote"><p>
  64             <span class="serif_italic">F = (X/v1) / (Y/v2)</span>
  65           </p></blockquote></div>
  66 <p>
  67           where X is a noncentral &#967;<sup>2</sup>
  68 random variable with <span class="emphasis"><em>v1</em></span> degrees
  69           of freedom and non-centrality parameter &#955;, and Y is a central &#967;<sup>2</sup> random variable
  70           with <span class="emphasis"><em>v2</em></span> degrees of freedom.
  71         </p>
  72 <p>
  73           This gives the following PDF:
  74         </p>
  75 <div class="blockquote"><blockquote class="blockquote"><p>
  76             <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref1.svg"></span>
  77
  78           </p></blockquote></div>
  79 <p>
  80           where <span class="emphasis"><em>L<sub>a</sub><sup>b</sup>(c)</em></span> is a generalised Laguerre polynomial
  81           and <span class="emphasis"><em>B(a,b)</em></span> is the <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>
  82           function, or
  83         </p>
  84 <div class="blockquote"><blockquote class="blockquote"><p>
  85             <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref2.svg"></span>
  86
  87           </p></blockquote></div>
  88 <p>
  89           The following graph illustrates how the distribution changes for different
  90           values of &#955;:
  91         </p>
  92 <div class="blockquote"><blockquote class="blockquote"><p>
  93             <span class="inlinemediaobject"><img src="../../../../graphs/nc_f_pdf.svg" align="middle"></span>
  94
  95           </p></blockquote></div>
  96 <h5>
  97 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h0"></a>
  98           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.member_functions"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.member_functions">Member
  99           Functions</a>
 100         </h5>
 101 <pre class="programlisting"><span class="identifier">non_central_f_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">v2</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
 102 </pre>
 103 <p>
 104           Constructs a non-central beta distribution with parameters <span class="emphasis"><em>v1</em></span>
 105           and <span class="emphasis"><em>v2</em></span> and non-centrality parameter <span class="emphasis"><em>lambda</em></span>.
 106         </p>
 107 <p>
 108           Requires <span class="emphasis"><em>v1</em></span> &gt; 0, <span class="emphasis"><em>v2</em></span> &gt; 0
 109           and lambda &gt;= 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 110         </p>
 111 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 112 </pre>
 113 <p>
 114           Returns the parameter <span class="emphasis"><em>v1</em></span> from which this object was
 115           constructed.
 116         </p>
 117 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 118 </pre>
 119 <p>
 120           Returns the parameter <span class="emphasis"><em>v2</em></span> from which this object was
 121           constructed.
 122         </p>
 123 <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>
 124 </pre>
 125 <p>
 126           Returns the non-centrality parameter <span class="emphasis"><em>lambda</em></span> from which
 127           this object was constructed.
 128         </p>
 129 <h5>
 130 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h1"></a>
 131           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.non_member_accessors"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.non_member_accessors">Non-member
 132           Accessors</a>
 133         </h5>
 134 <p>
 135           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 136           functions</a> that are generic to all distributions are supported:
 137           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 138           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 139           <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>,
 140           <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>,
 141           <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>,
 142           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 143           <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>,
 144           <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>.
 145         </p>
 146 <p>
 147           The domain of the random variable is [0, +&#8734;].
 148         </p>
 149 <h5>
 150 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h2"></a>
 151           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.accuracy"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.accuracy">Accuracy</a>
 152         </h5>
 153 <p>
 154           This distribution is implemented in terms of the <a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral
 155           Beta Distribution</a>: refer to that distribution for accuracy data.
 156         </p>
 157 <h5>
 158 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h3"></a>
 159           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.tests"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.tests">Tests</a>
 160         </h5>
 161 <p>
 162           Since this distribution is implemented by adapting another distribution,
 163           the tests consist of basic sanity checks computed by the <a href="http://www.r-project.org/" target="_top">R-2.5.1
 164           Math library statistical package</a> and its pbeta and dbeta functions.
 165         </p>
 166 <h5>
 167 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h4"></a>
 168           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.implementation"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.implementation">Implementation</a>
 169         </h5>
 170 <p>
 171           In the following table <span class="emphasis"><em>v1</em></span> and <span class="emphasis"><em>v2</em></span>
 172           are the first and second degrees of freedom parameters of the distribution,
 173           &#955;
 174 is the non-centrality parameter, <span class="emphasis"><em>x</em></span> is the random variate,
 175           <span class="emphasis"><em>p</em></span> is the probability, and <span class="emphasis"><em>q = 1-p</em></span>.
 176         </p>
 177 <div class="informaltable"><table class="table">
 178 <colgroup>
 179 <col>
 180 <col>
 181 </colgroup>
 182 <thead><tr>
 183 <th>
 184                   <p>
 185                     Function
 186                   </p>
 187                 </th>
 188 <th>
 189                   <p>
 190                     Implementation Notes
 191                   </p>
 192                 </th>
 193 </tr></thead>
 194 <tbody>
 195 <tr>
 196 <td>
 197                   <p>
 198                     pdf
 199                   </p>
 200                 </td>
 201 <td>
 202                   <p>
 203                     Implemented in terms of the non-central beta PDF using the relation:
 204                   </p>
 205                   <p>
 206                     <span class="serif_italic">f(x;v1,v2;&#955;) = (v1/v2) / ((1+y)*(1+y))
 207                     * g(y/(1+y);v1/2,v2/2;&#955;)</span>
 208                   </p>
 209                   <p>
 210                     where <span class="serif_italic">g(x; a, b; &#955;)</span> is the
 211                     non central beta PDF, and:
 212                   </p>
 213                   <p>
 214                     <span class="serif_italic">y = x * v1 / v2</span>
 215                   </p>
 216                 </td>
 217 </tr>
 218 <tr>
 219 <td>
 220                   <p>
 221                     cdf
 222                   </p>
 223                 </td>
 224 <td>
 225                   <p>
 226                     Using the relation:
 227                   </p>
 228                   <p>
 229                     <span class="serif_italic">p = B<sub>y</sub>(v1/2, v2/2; &#955;)</span>
 230                   </p>
 231                   <p>
 232                     where <span class="serif_italic">B<sub>x</sub>(a, b; &#955;)</span> is the
 233                     noncentral beta distribution CDF and
 234                   </p>
 235                   <p>
 236                     <span class="serif_italic">y = x * v1 / v2</span>
 237                   </p>
 238                 </td>
 239 </tr>
 240 <tr>
 241 <td>
 242                   <p>
 243                     cdf complement
 244                   </p>
 245                 </td>
 246 <td>
 247                   <p>
 248                     Using the relation:
 249                   </p>
 250                   <p>
 251                     <span class="serif_italic">q = 1 - B<sub>y</sub>(v1/2, v2/2; &#955;)</span>
 252                   </p>
 253                   <p>
 254                     where <span class="serif_italic">1 - B<sub>x</sub>(a, b; &#955;)</span> is
 255                     the complement of the noncentral beta distribution CDF and
 256                   </p>
 257                   <p>
 258                     <span class="serif_italic">y = x * v1 / v2</span>
 259                   </p>
 260                 </td>
 261 </tr>
 262 <tr>
 263 <td>
 264                   <p>
 265                     quantile
 266                   </p>
 267                 </td>
 268 <td>
 269                   <p>
 270                     Using the relation:
 271                   </p>
 272                   <p>
 273                     <span class="serif_italic">x = (bx / (1-bx)) * (v1 / v2)</span>
 274                   </p>
 275                   <p>
 276                     where
 277                   </p>
 278                   <p>
 279                     <span class="serif_italic">bx = Q<sub>p</sub><sup>-1</sup>(v1/2, v2/2; &#955;)</span>
 280                   </p>
 281                   <p>
 282                     and
 283                   </p>
 284                   <p>
 285                     <span class="serif_italic">Q<sub>p</sub><sup>-1</sup>(v1/2, v2/2; &#955;)</span>
 286                   </p>
 287                   <p>
 288                     is the noncentral beta quantile.
 289                   </p>
 290                 </td>
 291 </tr>
 292 <tr>
 293 <td>
 294                   <p>
 295                     quantile
 296                   </p>
 297                   <p>
 298                     from the complement
 299                   </p>
 300                 </td>
 301 <td>
 302                   <p>
 303                     Using the relation:
 304                   </p>
 305                   <p>
 306                     <span class="serif_italic">x = (bx / (1-bx)) * (v1 / v2)</span>
 307                   </p>
 308                   <p>
 309                     where
 310                   </p>
 311                   <p>
 312                     <span class="serif_italic">bx = QC<sub>q</sub><sup>-1</sup>(v1/2, v2/2; &#955;)</span>
 313                   </p>
 314                   <p>
 315                     and
 316                   </p>
 317                   <p>
 318                     <span class="serif_italic">QC<sub>q</sub><sup>-1</sup>(v1/2, v2/2; &#955;)</span>
 319                   </p>
 320                   <p>
 321                     is the noncentral beta quantile from the complement.
 322                   </p>
 323                 </td>
 324 </tr>
 325 <tr>
 326 <td>
 327                   <p>
 328                     mean
 329                   </p>
 330                 </td>
 331 <td>
 332                   <p>
 333                     <span class="serif_italic">v2 * (v1 + l) / (v1 * (v2 - 2))</span>
 334                   </p>
 335                 </td>
 336 </tr>
 337 <tr>
 338 <td>
 339                   <p>
 340                     mode
 341                   </p>
 342                 </td>
 343 <td>
 344                   <p>
 345                     By numeric maximalisation of the PDF.
 346                   </p>
 347                 </td>
 348 </tr>
 349 <tr>
 350 <td>
 351                   <p>
 352                     variance
 353                   </p>
 354                 </td>
 355 <td>
 356                   <p>
 357                     Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein,
 358                     Eric W. "Noncentral F-Distribution." From MathWorld--A
 359                     Wolfram Web Resource.</a>
 360                   </p>
 361                 </td>
 362 </tr>
 363 <tr>
 364 <td>
 365                   <p>
 366                     skewness
 367                   </p>
 368                 </td>
 369 <td>
 370                   <p>
 371                     Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein,
 372                     Eric W. "Noncentral F-Distribution." From MathWorld--A
 373                     Wolfram Web Resource.</a>, and to the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html" target="_top">Mathematica
 374                     documentation</a>
 375                   </p>
 376                 </td>
 377 </tr>
 378 <tr>
 379 <td>
 380                   <p>
 381                     kurtosis and kurtosis excess
 382                   </p>
 383                 </td>
 384 <td>
 385                   <p>
 386                     Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein,
 387                     Eric W. "Noncentral F-Distribution." From MathWorld--A
 388                     Wolfram Web Resource.</a>, and to the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html" target="_top">Mathematica
 389                     documentation</a>
 390                   </p>
 391                 </td>
 392 </tr>
 393 </tbody>
 394 </table></div>
 395 <p>
 396           Some analytic properties of noncentral distributions (particularly unimodality,
 397           and monotonicity of their modes) are surveyed and summarized by:
 398         </p>
 399 <p>
 400           Andrea van Aubel &amp; Wolfgang Gawronski, Applied Mathematics and Computation,
 401           141 (2003) 3-12.
 402         </p>
 403 </div>
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