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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;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_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;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_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<pre class="programlisting"><span class="identifier">F</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">X</span><span class="special">/</span><span class="identifier">v1</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">Y</span><span class="special">/</span><span class="identifier">v2</span><span class="special">)</span>
64</pre>
65<p>
66 where X is a noncentral &#967;<sup>2</sup>
67random variable with <span class="emphasis"><em>v1</em></span> degrees
68 of freedom and non-centrality parameter &#955;, and Y is a central &#967;<sup>2</sup> random variable
69 with <span class="emphasis"><em>v2</em></span> degrees of freedom.
70 </p>
71<p>
72 This gives the following PDF:
73 </p>
74<p>
75 <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref1.svg"></span>
76 </p>
77<p>
78 where L<sub>a</sub><sup>b</sup>(c) is a generalised Laguerre polynomial and B(a,b) is the <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a> function, or
79 </p>
80<p>
81 <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref2.svg"></span>
82 </p>
83<p>
84 The following graph illustrates how the distribution changes for different
85 values of &#955;:
86 </p>
87<p>
88 <span class="inlinemediaobject"><img src="../../../../graphs/nc_f_pdf.svg" align="middle"></span>
89 </p>
90<h5>
91<a name="math_toolkit.dist_ref.dists.nc_f_dist.h0"></a>
92 <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
93 Functions</a>
94 </h5>
95<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>
96</pre>
97<p>
98 Constructs a non-central beta distribution with parameters <span class="emphasis"><em>v1</em></span>
99 and <span class="emphasis"><em>v2</em></span> and non-centrality parameter <span class="emphasis"><em>lambda</em></span>.
100 </p>
101<p>
102 Requires v1 &gt; 0, v2 &gt; 0 and lambda &gt;= 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
103 </p>
104<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>
105</pre>
106<p>
107 Returns the parameter <span class="emphasis"><em>v1</em></span> from which this object was
108 constructed.
109 </p>
110<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>
111</pre>
112<p>
113 Returns the parameter <span class="emphasis"><em>v2</em></span> from which this object was
114 constructed.
115 </p>
116<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>
117</pre>
118<p>
119 Returns the non-centrality parameter <span class="emphasis"><em>lambda</em></span> from which
120 this object was constructed.
121 </p>
122<h5>
123<a name="math_toolkit.dist_ref.dists.nc_f_dist.h1"></a>
124 <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
125 Accessors</a>
126 </h5>
127<p>
128 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
129 functions</a> that are generic to all distributions are supported:
130 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
131 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
132 <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>,
133 <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>,
134 <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>,
135 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
136 <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>,
137 <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>.
138 </p>
139<p>
140 The domain of the random variable is [0, +&#8734;].
141 </p>
142<h5>
143<a name="math_toolkit.dist_ref.dists.nc_f_dist.h2"></a>
144 <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>
145 </h5>
146<p>
147 This distribution is implemented in terms of the <a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral
148 Beta Distribution</a>: refer to that distribution for accuracy data.
149 </p>
150<h5>
151<a name="math_toolkit.dist_ref.dists.nc_f_dist.h3"></a>
152 <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>
153 </h5>
154<p>
155 Since this distribution is implemented by adapting another distribution,
156 the tests consist of basic sanity checks computed by the <a href="http://www.r-project.org/" target="_top">R-2.5.1
157 Math library statistical package</a> and its pbeta and dbeta functions.
158 </p>
159<h5>
160<a name="math_toolkit.dist_ref.dists.nc_f_dist.h4"></a>
161 <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>
162 </h5>
163<p>
164 In the following table <span class="emphasis"><em>v1</em></span> and <span class="emphasis"><em>v2</em></span>
165 are the first and second degrees of freedom parameters of the distribution,
166 &#955;
167is the non-centrality parameter, <span class="emphasis"><em>x</em></span> is the random variate,
168 <span class="emphasis"><em>p</em></span> is the probability, and <span class="emphasis"><em>q = 1-p</em></span>.
169 </p>
170<div class="informaltable"><table class="table">
171<colgroup>
172<col>
173<col>
174</colgroup>
175<thead><tr>
176<th>
177 <p>
178 Function
179 </p>
180 </th>
181<th>
182 <p>
183 Implementation Notes
184 </p>
185 </th>
186</tr></thead>
187<tbody>
188<tr>
189<td>
190 <p>
191 pdf
192 </p>
193 </td>
194<td>
195 <p>
196 Implemented in terms of the non-central beta PDF using the relation:
197 </p>
198 <p>
199 f(x;v1,v2;&#955;) = (v1/v2) / ((1+y)*(1+y)) * g(y/(1+y);v1/2,v2/2;&#955;)
200 </p>
201 <p>
202 where g(x; a, b; &#955;) is the non central beta PDF, and:
203 </p>
204 <p>
205 y = x * v1 / v2
206 </p>
207 </td>
208</tr>
209<tr>
210<td>
211 <p>
212 cdf
213 </p>
214 </td>
215<td>
216 <p>
217 Using the relation:
218 </p>
219 <p>
220 p = B<sub>y</sub>(v1/2, v2/2; &#955;)
221 </p>
222 <p>
223 where B<sub>x</sub>(a, b; &#955;) is the noncentral beta distribution CDF and
224 </p>
225 <p>
226 y = x * v1 / v2
227 </p>
228 </td>
229</tr>
230<tr>
231<td>
232 <p>
233 cdf complement
234 </p>
235 </td>
236<td>
237 <p>
238 Using the relation:
239 </p>
240 <p>
241 q = 1 - B<sub>y</sub>(v1/2, v2/2; &#955;)
242 </p>
243 <p>
244 where 1 - B<sub>x</sub>(a, b; &#955;) is the complement of the noncentral beta
245 distribution CDF and
246 </p>
247 <p>
248 y = x * v1 / v2
249 </p>
250 </td>
251</tr>
252<tr>
253<td>
254 <p>
255 quantile
256 </p>
257 </td>
258<td>
259 <p>
260 Using the relation:
261 </p>
262 <p>
263 x = (bx / (1-bx)) * (v1 / v2)
264 </p>
265 <p>
266 where
267 </p>
268 <p>
269 bx = Q<sub>p</sub><sup>-1</sup>(v1/2, v2/2; &#955;)
270 </p>
271 <p>
272 and
273 </p>
274 <p>
275 Q<sub>p</sub><sup>-1</sup>(v1/2, v2/2; &#955;)
276 </p>
277 <p>
278 is the noncentral beta quantile.
279 </p>
280 </td>
281</tr>
282<tr>
283<td>
284 <p>
285 quantile
286 </p>
287 <p>
288 from the complement
289 </p>
290 </td>
291<td>
292 <p>
293 Using the relation:
294 </p>
295 <p>
296 x = (bx / (1-bx)) * (v1 / v2)
297 </p>
298 <p>
299 where
300 </p>
301 <p>
302 bx = QC<sub>q</sub><sup>-1</sup>(v1/2, v2/2; &#955;)
303 </p>
304 <p>
305 and
306 </p>
307 <p>
308 QC<sub>q</sub><sup>-1</sup>(v1/2, v2/2; &#955;)
309 </p>
310 <p>
311 is the noncentral beta quantile from the complement.
312 </p>
313 </td>
314</tr>
315<tr>
316<td>
317 <p>
318 mean
319 </p>
320 </td>
321<td>
322 <p>
323 v2 * (v1 + l) / (v1 * (v2 - 2))
324 </p>
325 </td>
326</tr>
327<tr>
328<td>
329 <p>
330 mode
331 </p>
332 </td>
333<td>
334 <p>
335 By numeric maximalisation of the PDF.
336 </p>
337 </td>
338</tr>
339<tr>
340<td>
341 <p>
342 variance
343 </p>
344 </td>
345<td>
346 <p>
347 Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein,
348 Eric W. "Noncentral F-Distribution." From MathWorld--A
349 Wolfram Web Resource.</a>
350 </p>
351 </td>
352</tr>
353<tr>
354<td>
355 <p>
356 skewness
357 </p>
358 </td>
359<td>
360 <p>
361 Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein,
362 Eric W. "Noncentral F-Distribution." From MathWorld--A
363 Wolfram Web Resource.</a>, and to the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html" target="_top">Mathematica
364 documentation</a>
365 </p>
366 </td>
367</tr>
368<tr>
369<td>
370 <p>
371 kurtosis and kurtosis excess
372 </p>
373 </td>
374<td>
375 <p>
376 Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein,
377 Eric W. "Noncentral F-Distribution." From MathWorld--A
378 Wolfram Web Resource.</a>, and to the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html" target="_top">Mathematica
379 documentation</a>
380 </p>
381 </td>
382</tr>
383</tbody>
384</table></div>
385<p>
386 Some analytic properties of noncentral distributions (particularly unimodality,
387 and monotonicity of their modes) are surveyed and summarized by:
388 </p>
389<p>
390 Andrea van Aubel &amp; Wolfgang Gawronski, Applied Mathematics and Computation,
391 141 (2003) 3-12.
392 </p>
393</div>
394<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
395<td align="left"></td>
396<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
397 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
398 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
399 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
400 Distributed under the Boost Software License, Version 1.0. (See accompanying
401 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>)
402 </p>
403</div></td>
404</tr></table>
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