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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.negative_binomial_dist"></a><a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
  28         Binomial 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">negative_binomial</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">negative_binomial_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">negative_binomial_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">negative_binomial</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">negative_binomial_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    <span class="comment">// Constructor from successes and success_fraction:</span>
  46    <span class="identifier">negative_binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
  47
  48    <span class="comment">// Parameter accessors:</span>
  49    <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  50    <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  51
  52    <span class="comment">// Bounds on success fraction:</span>
  53    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
  54       <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
  55       <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
  56       <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// alpha</span>
  57    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
  58       <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
  59       <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
  60       <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// alpha</span>
  61
  62    <span class="comment">// Estimate min/max number of trials:</span>
  63    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span>
  64       <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span>     <span class="comment">// Number of failures.</span>
  65       <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span>     <span class="comment">// Success fraction.</span>
  66       <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// Probability threshold alpha.</span>
  67    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span>
  68       <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span>     <span class="comment">// Number of failures.</span>
  69       <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span>     <span class="comment">// Success fraction.</span>
  70       <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// Probability threshold alpha.</span>
  71 <span class="special">};</span>
  72
  73 <span class="special">}}</span> <span class="comment">// namespaces</span>
  74 </pre>
  75 <p>
  76           The class type <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span></code>
  77           represents a <a href="http://en.wikipedia.org/wiki/Negative_binomial_distribution" target="_top">negative_binomial
  78           distribution</a>: it is used when there are exactly two mutually exclusive
  79           outcomes of a <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli
  80           trial</a>: these outcomes are labelled "success" and "failure".
  81         </p>
  82 <p>
  83           For k + r Bernoulli trials each with success fraction p, the negative_binomial
  84           distribution gives the probability of observing k failures and r successes
  85           with success on the last trial. The negative_binomial distribution assumes
  86           that success_fraction p is fixed for all (k + r) trials.
  87         </p>
  88 <div class="note"><table border="0" summary="Note">
  89 <tr>
  90 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
  91 <th align="left">Note</th>
  92 </tr>
  93 <tr><td align="left" valign="top"><p>
  94             The random variable for the negative binomial distribution is the number
  95             of trials, (the number of successes is a fixed property of the distribution)
  96             whereas for the binomial, the random variable is the number of successes,
  97             for a fixed number of trials.
  98           </p></td></tr>
  99 </table></div>
 100 <p>
 101           It has the PDF:
 102         </p>
 103 <p>
 104           <span class="inlinemediaobject"><img src="../../../../equations/neg_binomial_ref.svg"></span>
 105         </p>
 106 <p>
 107           The following graph illustrate how the PDF varies as the success fraction
 108           <span class="emphasis"><em>p</em></span> changes:
 109         </p>
 110 <p>
 111           <span class="inlinemediaobject"><img src="../../../../graphs/negative_binomial_pdf_1.svg" align="middle"></span>
 112         </p>
 113 <p>
 114           Alternatively, this graph shows how the shape of the PDF varies as the
 115           number of successes changes:
 116         </p>
 117 <p>
 118           <span class="inlinemediaobject"><img src="../../../../graphs/negative_binomial_pdf_2.svg" align="middle"></span>
 119         </p>
 120 <h5>
 121 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h0"></a>
 122           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.related_distributions"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.related_distributions">Related
 123           Distributions</a>
 124         </h5>
 125 <p>
 126           The name negative binomial distribution is reserved by some to the case
 127           where the successes parameter r is an integer. This integer version is
 128           also called the <a href="http://mathworld.wolfram.com/PascalDistribution.html" target="_top">Pascal
 129           distribution</a>.
 130         </p>
 131 <p>
 132           This implementation uses real numbers for the computation throughout (because
 133           it uses the <span class="bold"><strong>real-valued</strong></span> incomplete beta
 134           function family of functions). This real-valued version is also called
 135           the Polya Distribution.
 136         </p>
 137 <p>
 138           The Poisson distribution is a generalization of the Pascal distribution,
 139           where the success parameter r is an integer: to obtain the Pascal distribution
 140           you must ensure that an integer value is provided for r, and take integer
 141           values (floor or ceiling) from functions that return a number of successes.
 142         </p>
 143 <p>
 144           For large values of r (successes), the negative binomial distribution converges
 145           to the Poisson distribution.
 146         </p>
 147 <p>
 148           The geometric distribution is a special case where the successes parameter
 149           r = 1, so only a first and only success is required. geometric(p) = negative_binomial(1,
 150           p).
 151         </p>
 152 <p>
 153           The Poisson distribution is a special case for large successes
 154         </p>
 155 <p>
 156           poisson(&#955;) = lim <sub>r &#8594; &#8734;</sub>  &#160; negative_binomial(r, r / (&#955; + r)))
 157         </p>
 158 <div class="caution"><table border="0" summary="Caution">
 159 <tr>
 160 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
 161 <th align="left">Caution</th>
 162 </tr>
 163 <tr><td align="left" valign="top">
 164 <p>
 165             The Negative Binomial distribution is a discrete distribution: internally,
 166             functions like the <code class="computeroutput"><span class="identifier">cdf</span></code>
 167             and <code class="computeroutput"><span class="identifier">pdf</span></code> are treated "as
 168             if" they are continuous functions, but in reality the results returned
 169             from these functions only have meaning if an integer value is provided
 170             for the random variate argument.
 171           </p>
 172 <p>
 173             The quantile function will by default return an integer result that has
 174             been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles
 175             (where the probability is less than 0.5) are rounded downward, and upper
 176             quantiles (where the probability is greater than 0.5) are rounded upwards.
 177             This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at
 178             least</em></span> the requested coverage will be present in the central
 179             region, and <span class="emphasis"><em>no more than</em></span> the requested coverage
 180             will be present in the tails.
 181           </p>
 182 <p>
 183             This behaviour can be changed so that the quantile functions are rounded
 184             differently, or even return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>.
 185             It is strongly recommended that you read the tutorial <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
 186             Quantiles of Discrete Distributions</a> before using the quantile
 187             function on the Negative Binomial distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
 188             docs</a> describe how to change the rounding policy for these distributions.
 189           </p>
 190 </td></tr>
 191 </table></div>
 192 <h5>
 193 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h1"></a>
 194           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.member_functions"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.member_functions">Member
 195           Functions</a>
 196         </h5>
 197 <h6>
 198 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h2"></a>
 199           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.construct"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.construct">Construct</a>
 200         </h6>
 201 <pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
 202 </pre>
 203 <p>
 204           Constructor: <span class="emphasis"><em>r</em></span> is the total number of successes,
 205           <span class="emphasis"><em>p</em></span> is the probability of success of a single trial.
 206         </p>
 207 <p>
 208           Requires: <code class="computeroutput"><span class="identifier">r</span> <span class="special">&gt;</span>
 209           <span class="number">0</span></code> and <code class="computeroutput"><span class="number">0</span>
 210           <span class="special">&lt;=</span> <span class="identifier">p</span>
 211           <span class="special">&lt;=</span> <span class="number">1</span></code>.
 212         </p>
 213 <h6>
 214 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h3"></a>
 215           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.accessors"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.accessors">Accessors</a>
 216         </h6>
 217 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> <span class="comment">// successes / trials (0 &lt;= p &lt;= 1)</span>
 218 </pre>
 219 <p>
 220           Returns the parameter <span class="emphasis"><em>p</em></span> from which this distribution
 221           was constructed.
 222         </p>
 223 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> <span class="comment">// required successes (r &gt; 0)</span>
 224 </pre>
 225 <p>
 226           Returns the parameter <span class="emphasis"><em>r</em></span> from which this distribution
 227           was constructed.
 228         </p>
 229 <p>
 230           The best method of calculation for the following functions is disputed:
 231           see <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
 232           Distribution</a> for more discussion.
 233         </p>
 234 <h6>
 235 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h4"></a>
 236           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.lower_bound_on_parameter_p"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.lower_bound_on_parameter_p">Lower
 237           Bound on Parameter p</a>
 238         </h6>
 239 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
 240   <span class="identifier">RealType</span> <span class="identifier">failures</span><span class="special">,</span>
 241   <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
 242   <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">)</span> <span class="comment">// (0 &lt;= alpha &lt;= 1), 0.05 equivalent to 95% confidence.</span>
 243 </pre>
 244 <p>
 245           Returns a <span class="bold"><strong>lower bound</strong></span> on the success fraction:
 246         </p>
 247 <div class="variablelist">
 248 <p class="title"><b></b></p>
 249 <dl class="variablelist">
 250 <dt><span class="term">failures</span></dt>
 251 <dd><p>
 252                 The total number of failures before the <span class="emphasis"><em>r</em></span>th
 253                 success.
 254               </p></dd>
 255 <dt><span class="term">successes</span></dt>
 256 <dd><p>
 257                 The number of successes required.
 258               </p></dd>
 259 <dt><span class="term">alpha</span></dt>
 260 <dd><p>
 261                 The largest acceptable probability that the true value of the success
 262                 fraction is <span class="bold"><strong>less than</strong></span> the value
 263                 returned.
 264               </p></dd>
 265 </dl>
 266 </div>
 267 <p>
 268           For example, if you observe <span class="emphasis"><em>k</em></span> failures and <span class="emphasis"><em>r</em></span>
 269           successes from <span class="emphasis"><em>n</em></span> = k + r trials the best estimate
 270           for the success fraction is simply <span class="emphasis"><em>r/n</em></span>, but if you
 271           want to be 95% sure that the true value is <span class="bold"><strong>greater
 272           than</strong></span> some value, <span class="emphasis"><em>p<sub>min</sub></em></span>, then:
 273         </p>
 274 <pre class="programlisting"><span class="identifier">p</span><sub>min</sub> <span class="special">=</span> <span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
 275                     <span class="identifier">failures</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
 276 </pre>
 277 <p>
 278           <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See
 279           negative binomial confidence interval example.</a>
 280         </p>
 281 <p>
 282           This function uses the Clopper-Pearson method of computing the lower bound
 283           on the success fraction, whilst many texts refer to this method as giving
 284           an "exact" result in practice it produces an interval that guarantees
 285           <span class="emphasis"><em>at least</em></span> the coverage required, and may produce pessimistic
 286           estimates for some combinations of <span class="emphasis"><em>failures</em></span> and <span class="emphasis"><em>successes</em></span>.
 287           See:
 288         </p>
 289 <p>
 290           <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong
 291           Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some
 292           Discrete Distributions. Computational statistics and data analysis, 2005,
 293           vol. 48, no3, 605-621</a>.
 294         </p>
 295 <h6>
 296 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h5"></a>
 297           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.upper_bound_on_parameter_p"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.upper_bound_on_parameter_p">Upper
 298           Bound on Parameter p</a>
 299         </h6>
 300 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
 301    <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
 302    <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
 303    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// (0 &lt;= alpha &lt;= 1), 0.05 equivalent to 95% confidence.</span>
 304 </pre>
 305 <p>
 306           Returns an <span class="bold"><strong>upper bound</strong></span> on the success
 307           fraction:
 308         </p>
 309 <div class="variablelist">
 310 <p class="title"><b></b></p>
 311 <dl class="variablelist">
 312 <dt><span class="term">trials</span></dt>
 313 <dd><p>
 314                 The total number of trials conducted.
 315               </p></dd>
 316 <dt><span class="term">successes</span></dt>
 317 <dd><p>
 318                 The number of successes that occurred.
 319               </p></dd>
 320 <dt><span class="term">alpha</span></dt>
 321 <dd><p>
 322                 The largest acceptable probability that the true value of the success
 323                 fraction is <span class="bold"><strong>greater than</strong></span> the value
 324                 returned.
 325               </p></dd>
 326 </dl>
 327 </div>
 328 <p>
 329           For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span>
 330           trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>,
 331           but if you want to be 95% sure that the true value is <span class="bold"><strong>less
 332           than</strong></span> some value, <span class="emphasis"><em>p<sub>max</sub></em></span>, then:
 333         </p>
 334 <pre class="programlisting"><span class="identifier">p</span><sub>max</sub> <span class="special">=</span> <span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
 335                     <span class="identifier">r</span><span class="special">,</span> <span class="identifier">k</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
 336 </pre>
 337 <p>
 338           <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See
 339           negative binomial confidence interval example.</a>
 340         </p>
 341 <p>
 342           This function uses the Clopper-Pearson method of computing the lower bound
 343           on the success fraction, whilst many texts refer to this method as giving
 344           an "exact" result in practice it produces an interval that guarantees
 345           <span class="emphasis"><em>at least</em></span> the coverage required, and may produce pessimistic
 346           estimates for some combinations of <span class="emphasis"><em>failures</em></span> and <span class="emphasis"><em>successes</em></span>.
 347           See:
 348         </p>
 349 <p>
 350           <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong
 351           Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some
 352           Discrete Distributions. Computational statistics and data analysis, 2005,
 353           vol. 48, no3, 605-621</a>.
 354         </p>
 355 <h6>
 356 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h6"></a>
 357           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.estimating_number_of_trials_to_e"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.estimating_number_of_trials_to_e">Estimating
 358           Number of Trials to Ensure at Least a Certain Number of Failures</a>
 359         </h6>
 360 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span>
 361    <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span>     <span class="comment">// number of failures.</span>
 362    <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span>     <span class="comment">// success fraction.</span>
 363    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold (0.05 equivalent to 95%).</span>
 364 </pre>
 365 <p>
 366           This functions estimates the number of trials required to achieve a certain
 367           probability that <span class="bold"><strong>more than k failures will be observed</strong></span>.
 368         </p>
 369 <div class="variablelist">
 370 <p class="title"><b></b></p>
 371 <dl class="variablelist">
 372 <dt><span class="term">k</span></dt>
 373 <dd><p>
 374                 The target number of failures to be observed.
 375               </p></dd>
 376 <dt><span class="term">p</span></dt>
 377 <dd><p>
 378                 The probability of <span class="emphasis"><em>success</em></span> for each trial.
 379               </p></dd>
 380 <dt><span class="term">alpha</span></dt>
 381 <dd><p>
 382                 The maximum acceptable risk that only k failures or fewer will be
 383                 observed.
 384               </p></dd>
 385 </dl>
 386 </div>
 387 <p>
 388           For example:
 389         </p>
 390 <pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span><span class="number">10</span><span class="special">,</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
 391 </pre>
 392 <p>
 393           Returns the smallest number of trials we must conduct to be 95% sure of
 394           seeing 10 failures that occur with frequency one half.
 395         </p>
 396 <p>
 397           <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html" title="Estimating Sample Sizes for the Negative Binomial.">Worked
 398           Example.</a>
 399         </p>
 400 <p>
 401           This function uses numeric inversion of the negative binomial distribution
 402           to obtain the result: another interpretation of the result, is that it
 403           finds the number of trials (success+failures) that will lead to an <span class="emphasis"><em>alpha</em></span>
 404           probability of observing k failures or fewer.
 405         </p>
 406 <h6>
 407 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h7"></a>
 408           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.estimating_number_of_trials_to_0"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.estimating_number_of_trials_to_0">Estimating
 409           Number of Trials to Ensure a Maximum Number of Failures or Less</a>
 410         </h6>
 411 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span>
 412    <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span>     <span class="comment">// number of failures.</span>
 413    <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span>     <span class="comment">// success fraction.</span>
 414    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold (0.05 equivalent to 95%).</span>
 415 </pre>
 416 <p>
 417           This functions estimates the maximum number of trials we can conduct and
 418           achieve a certain probability that <span class="bold"><strong>k failures or
 419           fewer will be observed</strong></span>.
 420         </p>
 421 <div class="variablelist">
 422 <p class="title"><b></b></p>
 423 <dl class="variablelist">
 424 <dt><span class="term">k</span></dt>
 425 <dd><p>
 426                 The maximum number of failures to be observed.
 427               </p></dd>
 428 <dt><span class="term">p</span></dt>
 429 <dd><p>
 430                 The probability of <span class="emphasis"><em>success</em></span> for each trial.
 431               </p></dd>
 432 <dt><span class="term">alpha</span></dt>
 433 <dd><p>
 434                 The maximum acceptable <span class="emphasis"><em>risk</em></span> that more than k
 435                 failures will be observed.
 436               </p></dd>
 437 </dl>
 438 </div>
 439 <p>
 440           For example:
 441         </p>
 442 <pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1.0</span><span class="special">-</span><span class="number">1.0</span><span class="special">/</span><span class="number">1000000</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
 443 </pre>
 444 <p>
 445           Returns the largest number of trials we can conduct and still be 95% sure
 446           of seeing no failures that occur with frequency one in one million.
 447         </p>
 448 <p>
 449           This function uses numeric inversion of the negative binomial distribution
 450           to obtain the result: another interpretation of the result, is that it
 451           finds the number of trials (success+failures) that will lead to an <span class="emphasis"><em>alpha</em></span>
 452           probability of observing more than k failures.
 453         </p>
 454 <h5>
 455 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h8"></a>
 456           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.non_member_accessors"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.non_member_accessors">Non-member
 457           Accessors</a>
 458         </h5>
 459 <p>
 460           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 461           functions</a> that are generic to all distributions are supported:
 462           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 463           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 464           <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>,
 465           <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>,
 466           <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>,
 467           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 468           <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>,
 469           <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>.
 470         </p>
 471 <p>
 472           However it's worth taking a moment to define what these actually mean in
 473           the context of this distribution:
 474         </p>
 475 <div class="table">
 476 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table&#160;5.3.&#160;Meaning of the non-member accessors.</b></p>
 477 <div class="table-contents"><table class="table" summary="Meaning of the non-member accessors.">
 478 <colgroup>
 479 <col>
 480 <col>
 481 </colgroup>
 482 <thead><tr>
 483 <th>
 484                   <p>
 485                     Function
 486                   </p>
 487                 </th>
 488 <th>
 489                   <p>
 490                     Meaning
 491                   </p>
 492                 </th>
 493 </tr></thead>
 494 <tbody>
 495 <tr>
 496 <td>
 497                   <p>
 498                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
 499                     Function</a>
 500                   </p>
 501                 </td>
 502 <td>
 503                   <p>
 504                     The probability of obtaining <span class="bold"><strong>exactly k
 505                     failures</strong></span> from k+r trials with success fraction p.
 506                     For example:
 507                   </p>
 508 <pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">)</span></pre>
 509                 </td>
 510 </tr>
 511 <tr>
 512 <td>
 513                   <p>
 514                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
 515                     Function</a>
 516                   </p>
 517                 </td>
 518 <td>
 519                   <p>
 520                     The probability of obtaining <span class="bold"><strong>k failures
 521                     or fewer</strong></span> from k+r trials with success fraction p and
 522                     success on the last trial. For example:
 523                   </p>
 524 <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">)</span></pre>
 525                 </td>
 526 </tr>
 527 <tr>
 528 <td>
 529                   <p>
 530                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of
 531                     the Cumulative Distribution Function</a>
 532                   </p>
 533                 </td>
 534 <td>
 535                   <p>
 536                     The probability of obtaining <span class="bold"><strong>more than
 537                     k failures</strong></span> from k+r trials with success fraction p
 538                     and success on the last trial. For example:
 539                   </p>
 540 <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">))</span></pre>
 541                 </td>
 542 </tr>
 543 <tr>
 544 <td>
 545                   <p>
 546                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>
 547                   </p>
 548                 </td>
 549 <td>
 550                   <p>
 551                     The <span class="bold"><strong>greatest</strong></span> number of failures
 552                     k expected to be observed from k+r trials with success fraction
 553                     p, at probability P. Note that the value returned is a real-number,
 554                     and not an integer. Depending on the use case you may want to
 555                     take either the floor or ceiling of the real result. For example:
 556                   </p>
 557 <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">P</span><span class="special">)</span></pre>
 558                 </td>
 559 </tr>
 560 <tr>
 561 <td>
 562                   <p>
 563                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile
 564                     from the complement of the probability</a>
 565                   </p>
 566                 </td>
 567 <td>
 568                   <p>
 569                     The <span class="bold"><strong>smallest</strong></span> number of failures
 570                     k expected to be observed from k+r trials with success fraction
 571                     p, at probability P. Note that the value returned is a real-number,
 572                     and not an integer. Depending on the use case you may want to
 573                     take either the floor or ceiling of the real result. For example:
 574                   </p>
 575 <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="identifier">r</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">P</span><span class="special">))</span></pre>
 576                 </td>
 577 </tr>
 578 </tbody>
 579 </table></div>
 580 </div>
 581 <br class="table-break"><h5>
 582 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h9"></a>
 583           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.accuracy"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.accuracy">Accuracy</a>
 584         </h5>
 585 <p>
 586           This distribution is implemented using the incomplete beta functions <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>:
 587           please refer to these functions for information on accuracy.
 588         </p>
 589 <h5>
 590 <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h10"></a>
 591           <span class="phrase"><a name="math_toolkit.dist_ref.dists.negative_binomial_dist.implementation"></a></span><a class="link" href="negative_binomial_dist.html#math_toolkit.dist_ref.dists.negative_binomial_dist.implementation">Implementation</a>
 592         </h5>
 593 <p>
 594           In the following table, <span class="emphasis"><em>p</em></span> is the probability that
 595           any one trial will be successful (the success fraction), <span class="emphasis"><em>r</em></span>
 596           is the number of successes, <span class="emphasis"><em>k</em></span> is the number of failures,
 597           <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 598         </p>
 599 <div class="informaltable"><table class="table">
 600 <colgroup>
 601 <col>
 602 <col>
 603 </colgroup>
 604 <thead><tr>
 605 <th>
 606                   <p>
 607                     Function
 608                   </p>
 609                 </th>
 610 <th>
 611                   <p>
 612                     Implementation Notes
 613                   </p>
 614                 </th>
 615 </tr></thead>
 616 <tbody>
 617 <tr>
 618 <td>
 619                   <p>
 620                     pdf
 621                   </p>
 622                 </td>
 623 <td>
 624                   <p>
 625                     pdf = exp(lgamma(r + k) - lgamma(r) - lgamma(k+1)) * pow(p, r)
 626                     * pow((1-p), k)
 627                   </p>
 628                   <p>
 629                     Implementation is in terms of <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>:
 630                   </p>
 631                   <p>
 632                     (p/(r + k)) * ibeta_derivative(r, static_cast&lt;RealType&gt;(k+1),
 633                     p) The function <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>
 634                     is used here, since it has already been optimised for the lowest
 635                     possible error - indeed this is really just a thin wrapper around
 636                     part of the internals of the incomplete beta function.
 637                   </p>
 638                 </td>
 639 </tr>
 640 <tr>
 641 <td>
 642                   <p>
 643                     cdf
 644                   </p>
 645                 </td>
 646 <td>
 647                   <p>
 648                     Using the relation:
 649                   </p>
 650                   <p>
 651                     cdf = I<sub>p</sub>(r, k+1) = ibeta(r, k+1, p)
 652                   </p>
 653                   <p>
 654                     = ibeta(r, static_cast&lt;RealType&gt;(k+1), p)
 655                   </p>
 656                 </td>
 657 </tr>
 658 <tr>
 659 <td>
 660                   <p>
 661                     cdf complement
 662                   </p>
 663                 </td>
 664 <td>
 665                   <p>
 666                     Using the relation:
 667                   </p>
 668                   <p>
 669                     1 - cdf = I<sub>p</sub>(k+1, r)
 670                   </p>
 671                   <p>
 672                     = ibetac(r, static_cast&lt;RealType&gt;(k+1), p)
 673                   </p>
 674                 </td>
 675 </tr>
 676 <tr>
 677 <td>
 678                   <p>
 679                     quantile
 680                   </p>
 681                 </td>
 682 <td>
 683                   <p>
 684                     ibeta_invb(r, p, P) - 1
 685                   </p>
 686                 </td>
 687 </tr>
 688 <tr>
 689 <td>
 690                   <p>
 691                     quantile from the complement
 692                   </p>
 693                 </td>
 694 <td>
 695                   <p>
 696                     ibetac_invb(r, p, Q) -1)
 697                   </p>
 698                 </td>
 699 </tr>
 700 <tr>
 701 <td>
 702                   <p>
 703                     mean
 704                   </p>
 705                 </td>
 706 <td>
 707                   <p>
 708                     <code class="computeroutput"><span class="identifier">r</span><span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">)/</span><span class="identifier">p</span></code>
 709                   </p>
 710                 </td>
 711 </tr>
 712 <tr>
 713 <td>
 714                   <p>
 715                     variance
 716                   </p>
 717                 </td>
 718 <td>
 719                   <p>
 720                     <code class="computeroutput"><span class="identifier">r</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">)</span>
 721                     <span class="special">/</span> <span class="identifier">p</span>
 722                     <span class="special">*</span> <span class="identifier">p</span></code>
 723                   </p>
 724                 </td>
 725 </tr>
 726 <tr>
 727 <td>
 728                   <p>
 729                     mode
 730                   </p>
 731                 </td>
 732 <td>
 733                   <p>
 734                     <code class="computeroutput"><span class="identifier">floor</span><span class="special">((</span><span class="identifier">r</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span> <span class="identifier">p</span><span class="special">)/</span><span class="identifier">p</span><span class="special">)</span></code>
 735                   </p>
 736                 </td>
 737 </tr>
 738 <tr>
 739 <td>
 740                   <p>
 741                     skewness
 742                   </p>
 743                 </td>
 744 <td>
 745                   <p>
 746                     <code class="computeroutput"><span class="special">(</span><span class="number">2</span>
 747                     <span class="special">-</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span>
 748                     <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">r</span> <span class="special">*</span>
 749                     <span class="special">(</span><span class="number">1</span>
 750                     <span class="special">-</span> <span class="identifier">p</span><span class="special">))</span></code>
 751                   </p>
 752                 </td>
 753 </tr>
 754 <tr>
 755 <td>
 756                   <p>
 757                     kurtosis
 758                   </p>
 759                 </td>
 760 <td>
 761                   <p>
 762                     <code class="computeroutput"><span class="number">6</span> <span class="special">/</span>
 763                     <span class="identifier">r</span> <span class="special">+</span>
 764                     <span class="special">(</span><span class="identifier">p</span>
 765                     <span class="special">*</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span>
 766                     <span class="identifier">r</span> <span class="special">*</span>
 767                     <span class="special">(</span><span class="number">1</span>
 768                     <span class="special">-</span> <span class="identifier">p</span>
 769                     <span class="special">)</span></code>
 770                   </p>
 771                 </td>
 772 </tr>
 773 <tr>
 774 <td>
 775                   <p>
 776                     kurtosis excess
 777                   </p>
 778                 </td>
 779 <td>
 780                   <p>
 781                     <code class="computeroutput"><span class="number">6</span> <span class="special">/</span>
 782                     <span class="identifier">r</span> <span class="special">+</span>
 783                     <span class="special">(</span><span class="identifier">p</span>
 784                     <span class="special">*</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span>
 785                     <span class="identifier">r</span> <span class="special">*</span>
 786                     <span class="special">(</span><span class="number">1</span>
 787                     <span class="special">-</span> <span class="identifier">p</span>
 788                     <span class="special">)</span> <span class="special">-</span><span class="number">3</span></code>
 789                   </p>
 790                 </td>
 791 </tr>
 792 <tr>
 793 <td>
 794                   <p>
 795                     parameter estimation member functions
 796                   </p>
 797                 </td>
 798 <td>
 799                 </td>
 800 </tr>
 801 <tr>
 802 <td>
 803                   <p>
 804                     <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code>
 805                   </p>
 806                 </td>
 807 <td>
 808                   <p>
 809                     ibeta_inv(successes, failures + 1, alpha)
 810                   </p>
 811                 </td>
 812 </tr>
 813 <tr>
 814 <td>
 815                   <p>
 816                     <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code>
 817                   </p>
 818                 </td>
 819 <td>
 820                   <p>
 821                     ibetac_inv(successes, failures, alpha) plus see comments in code.
 822                   </p>
 823                 </td>
 824 </tr>
 825 <tr>
 826 <td>
 827                   <p>
 828                     <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code>
 829                   </p>
 830                 </td>
 831 <td>
 832                   <p>
 833                     ibeta_inva(k + 1, p, alpha)
 834                   </p>
 835                 </td>
 836 </tr>
 837 <tr>
 838 <td>
 839                   <p>
 840                     <code class="computeroutput"><span class="identifier">find_maximum_number_of_trials</span></code>
 841                   </p>
 842                 </td>
 843 <td>
 844                   <p>
 845                     ibetac_inva(k + 1, p, alpha)
 846                   </p>
 847                 </td>
 848 </tr>
 849 </tbody>
 850 </table></div>
 851 <p>
 852           Implementation notes:
 853         </p>
 854 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 855 <li class="listitem">
 856               The real concept type (that deliberately lacks the Lanczos approximation),
 857               was found to take several minutes to evaluate some extreme test values,
 858               so the test has been disabled for this type.
 859             </li>
 860 <li class="listitem">
 861               Much greater speed, and perhaps greater accuracy, might be achieved
 862               for extreme values by using a normal approximation. This is NOT been
 863               tested or implemented.
 864             </li>
 865 </ul></div>
 866 </div>
 867 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 868 <td align="left"></td>
 869 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 870       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 871       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 872       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 873         Distributed under the Boost Software License, Version 1.0. (See accompanying
 874         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>)
 875       </p>
 876 </div></td>
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