1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Noncentral Beta Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="negative_binomial_dist.html" title="Negative Binomial Distribution"> 10<link rel="next" href="nc_chi_squared_dist.html" title="Noncentral Chi-Squared Distribution"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="negative_binomial_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nc_chi_squared_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h4 class="title"> 27<a name="math_toolkit.dist_ref.dists.nc_beta_dist"></a><a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral 28 Beta Distribution</a> 29</h4></div></div></div> 30<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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_beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></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"><</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 21. 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<></a> <span class="special">></span> 35<span class="keyword">class</span> <span class="identifier">non_central_beta_distribution</span><span class="special">;</span> 36 37<span class="keyword">typedef</span> <span class="identifier">non_central_beta_distribution</span><span class="special"><></span> <span class="identifier">non_central_beta</span><span class="special">;</span> 38 39<span class="keyword">template</span> <span class="special"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 40<span class="keyword">class</span> <span class="identifier">non_central_beta_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_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">beta</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 shape parameters:</span> 50 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 51 <span class="identifier">RealType</span> <span class="identifier">beta</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 beta distribution is a generalization of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta 61 Distribution</a>. 62 </p> 63<p> 64 It is defined as the ratio 65 </p> 66<div class="blockquote"><blockquote class="blockquote"><p> 67 <span class="serif_italic">X = χ<sub>m</sub><sup>2</sup>(λ) / (χ<sub>m</sub><sup>2</sup>(λ) + χ<sub>n</sub><sup>2</sup>)</span> 68 </p></blockquote></div> 69<p> 70 where <span class="serif_italic">χ<sub>m</sub><sup>2</sup>(λ)</span> is a noncentral <span class="serif_italic">χ<sup>2</sup></span> random variable with <span class="emphasis"><em>m</em></span> 71 degrees of freedom, and χ<sub>n</sub><sup>2</sup> 72is a central <span class="serif_italic">χ<sup>2</sup> </span> 73 random variable with <span class="emphasis"><em>n</em></span> degrees of freedom. 74 </p> 75<p> 76 This gives a PDF that can be expressed as a Poisson mixture of beta distribution 77 PDFs: 78 </p> 79<div class="blockquote"><blockquote class="blockquote"><p> 80 <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span> 81 82 </p></blockquote></div> 83<p> 84 where P(i;λ/2) is the discrete Poisson probability at <span class="emphasis"><em>i</em></span>, 85 with mean λ/2, and I<sub>x</sub><sup>'</sup>(α, β) is the derivative of the incomplete beta function. 86 This leads to the usual form of the CDF as: 87 </p> 88<div class="blockquote"><blockquote class="blockquote"><p> 89 <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref2.svg"></span> 90 91 </p></blockquote></div> 92<p> 93 The following graph illustrates how the distribution changes for different 94 values of λ: 95 </p> 96<div class="blockquote"><blockquote class="blockquote"><p> 97 <span class="inlinemediaobject"><img src="../../../../graphs/nc_beta_pdf.svg" align="middle"></span> 98 99 </p></blockquote></div> 100<h5> 101<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h0"></a> 102 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.member_functions"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.member_functions">Member 103 Functions</a> 104 </h5> 105<pre class="programlisting"><span class="identifier">non_central_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span> 106</pre> 107<p> 108 Constructs a noncentral beta distribution with shape parameters <span class="emphasis"><em>a</em></span> 109 and <span class="emphasis"><em>b</em></span> and non-centrality parameter <span class="emphasis"><em>lambda</em></span>. 110 </p> 111<p> 112 Requires a > 0, b > 0 and lambda >= 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 113 </p> 114<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 115</pre> 116<p> 117 Returns the parameter <span class="emphasis"><em>a</em></span> from which this object was 118 constructed. 119 </p> 120<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 121</pre> 122<p> 123 Returns the parameter <span class="emphasis"><em>b</em></span> from which this object was 124 constructed. 125 </p> 126<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> 127</pre> 128<p> 129 Returns the parameter <span class="emphasis"><em>lambda</em></span> from which this object 130 was constructed. 131 </p> 132<h5> 133<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h1"></a> 134 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.non_member_accessors"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.non_member_accessors">Non-member 135 Accessors</a> 136 </h5> 137<p> 138 Most of the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member 139 accessor functions</a> are supported: <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative 140 Distribution Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability 141 Density Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, 142 <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.variance">variance</a>, 143 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 144 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, 145 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, 146 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>, 147 <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>. 148 </p> 149<p> 150 Mean and variance are implemented using hypergeometric pfq functions and 151 relations given in <a href="http://reference.wolfram.com/mathematica/ref/NoncentralBetaDistribution.html" target="_top">Wolfram 152 Noncentral Beta Distribution</a>. 153 </p> 154<p> 155 However, the following are not currently implemented: <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, 156 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a> and 157 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>. 158 </p> 159<p> 160 The domain of the random variable is [0, 1]. 161 </p> 162<h5> 163<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h2"></a> 164 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.accuracy"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.accuracy">Accuracy</a> 165 </h5> 166<p> 167 The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>) 168 found on various platforms with various floating point types. The failures 169 in the comparison to the <a href="http://www.r-project.org/" target="_top">R Math 170 library</a>, seem to be mostly in the corner cases when the probability 171 would be very small. Unless otherwise specified any floating-point type 172 that is narrower than the one shown will have <a class="link" href="../../relative_error.html#math_toolkit.relative_error.zero_error">effectively 173 zero error</a>. 174 </p> 175<div class="table"> 176<a name="math_toolkit.dist_ref.dists.nc_beta_dist.table_non_central_beta_CDF"></a><p class="title"><b>Table 5.4. Error rates for non central beta CDF</b></p> 177<div class="table-contents"><table class="table" summary="Error rates for non central beta CDF"> 178<colgroup> 179<col> 180<col> 181<col> 182<col> 183<col> 184</colgroup> 185<thead><tr> 186<th> 187 </th> 188<th> 189 <p> 190 GNU C++ version 7.1.0<br> linux<br> double 191 </p> 192 </th> 193<th> 194 <p> 195 GNU C++ version 7.1.0<br> linux<br> long double 196 </p> 197 </th> 198<th> 199 <p> 200 Sun compiler version 0x5150<br> Sun Solaris<br> long double 201 </p> 202 </th> 203<th> 204 <p> 205 Microsoft Visual C++ version 14.1<br> Win32<br> double 206 </p> 207 </th> 208</tr></thead> 209<tbody> 210<tr> 211<td> 212 <p> 213 Non Central Beta, medium parameters 214 </p> 215 </td> 216<td> 217 <p> 218 <span class="blue">Max = 0.998ε (Mean = 0.0649ε)</span><br> 219 <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max 220 = 1.46e+26ε (Mean = 3.5e+24ε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_Rmath_3_2_3_Non_Central_Beta_medium_parameters">And 221 other failures.</a>)</span> 222 </p> 223 </td> 224<td> 225 <p> 226 <span class="blue">Max = 824ε (Mean = 27.4ε)</span> 227 </p> 228 </td> 229<td> 230 <p> 231 <span class="blue">Max = 832ε (Mean = 38.1ε)</span> 232 </p> 233 </td> 234<td> 235 <p> 236 <span class="blue">Max = 242ε (Mean = 31ε)</span> 237 </p> 238 </td> 239</tr> 240<tr> 241<td> 242 <p> 243 Non Central Beta, large parameters 244 </p> 245 </td> 246<td> 247 <p> 248 <span class="blue">Max = 1.18ε (Mean = 0.175ε)</span><br> 249 <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max 250 = 1.01e+36ε (Mean = 1.19e+35ε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_Rmath_3_2_3_Non_Central_Beta_large_parameters">And 251 other failures.</a>)</span> 252 </p> 253 </td> 254<td> 255 <p> 256 <span class="blue">Max = 2.5e+04ε (Mean = 3.78e+03ε)</span> 257 </p> 258 </td> 259<td> 260 <p> 261 <span class="blue">Max = 2.57e+04ε (Mean = 4.45e+03ε)</span> 262 </p> 263 </td> 264<td> 265 <p> 266 <span class="blue">Max = 3.66e+03ε (Mean = 500ε)</span> 267 </p> 268 </td> 269</tr> 270</tbody> 271</table></div> 272</div> 273<br class="table-break"><div class="table"> 274<a name="math_toolkit.dist_ref.dists.nc_beta_dist.table_non_central_beta_CDF_complement"></a><p class="title"><b>Table 5.5. Error rates for non central beta CDF complement</b></p> 275<div class="table-contents"><table class="table" summary="Error rates for non central beta CDF complement"> 276<colgroup> 277<col> 278<col> 279<col> 280<col> 281<col> 282</colgroup> 283<thead><tr> 284<th> 285 </th> 286<th> 287 <p> 288 GNU C++ version 7.1.0<br> linux<br> double 289 </p> 290 </th> 291<th> 292 <p> 293 GNU C++ version 7.1.0<br> linux<br> long double 294 </p> 295 </th> 296<th> 297 <p> 298 Sun compiler version 0x5150<br> Sun Solaris<br> long double 299 </p> 300 </th> 301<th> 302 <p> 303 Microsoft Visual C++ version 14.1<br> Win32<br> double 304 </p> 305 </th> 306</tr></thead> 307<tbody> 308<tr> 309<td> 310 <p> 311 Non Central Beta, medium parameters 312 </p> 313 </td> 314<td> 315 <p> 316 <span class="blue">Max = 0.998ε (Mean = 0.0936ε)</span><br> 317 <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max 318 = 7.5e+97ε (Mean = 1.37e+96ε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_complement_Rmath_3_2_3_Non_Central_Beta_medium_parameters">And 319 other failures.</a>)</span> 320 </p> 321 </td> 322<td> 323 <p> 324 <span class="blue">Max = 396ε (Mean = 50.7ε)</span> 325 </p> 326 </td> 327<td> 328 <p> 329 <span class="blue">Max = 554ε (Mean = 57.2ε)</span> 330 </p> 331 </td> 332<td> 333 <p> 334 <span class="blue">Max = 624ε (Mean = 62.7ε)</span> 335 </p> 336 </td> 337</tr> 338<tr> 339<td> 340 <p> 341 Non Central Beta, large parameters 342 </p> 343 </td> 344<td> 345 <p> 346 <span class="blue">Max = 0.986ε (Mean = 0.188ε)</span><br> 347 <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max 348 = +INFε (Mean = +INFε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_beta_CDF_complement_Rmath_3_2_3_Non_Central_Beta_large_parameters">And 349 other failures.</a>)</span> 350 </p> 351 </td> 352<td> 353 <p> 354 <span class="blue">Max = 6.83e+03ε (Mean = 993ε)</span> 355 </p> 356 </td> 357<td> 358 <p> 359 <span class="blue">Max = 3.56e+03ε (Mean = 707ε)</span> 360 </p> 361 </td> 362<td> 363 <p> 364 <span class="blue">Max = 1.25e+04ε (Mean = 1.49e+03ε)</span> 365 </p> 366 </td> 367</tr> 368</tbody> 369</table></div> 370</div> 371<br class="table-break"><p> 372 Error rates for the PDF, the complement of the CDF and for the quantile 373 functions are broadly similar. 374 </p> 375<h5> 376<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h3"></a> 377 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.tests"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.tests">Tests</a> 378 </h5> 379<p> 380 There are two sets of test data used to verify this implementation: firstly 381 we can compare with a few sample values generated by the <a href="http://www.r-project.org/" target="_top">R 382 library</a>. Secondly, we have tables of test data, computed with this 383 implementation and using interval arithmetic - this data should be accurate 384 to at least 50 decimal digits - and is the used for our accuracy tests. 385 </p> 386<h5> 387<a name="math_toolkit.dist_ref.dists.nc_beta_dist.h4"></a> 388 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.implementation"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.implementation">Implementation</a> 389 </h5> 390<p> 391 The CDF and its complement are evaluated as follows: 392 </p> 393<p> 394 First we determine which of the two values (the CDF or its complement) 395 is likely to be the smaller, the crossover point is taken to be the mean 396 of the distribution: for this we use the approximation due to: R. Chattamvelli 397 and R. Shanmugam, "Algorithm AS 310: Computing the Non-Central Beta 398 Distribution Function", Applied Statistics, Vol. 46, No. 1. (1997), 399 pp. 146-156. 400 </p> 401<div class="blockquote"><blockquote class="blockquote"><p> 402 <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref3.svg"></span> 403 404 </p></blockquote></div> 405<p> 406 Then either the CDF or its complement is computed using the relations: 407 </p> 408<div class="blockquote"><blockquote class="blockquote"><p> 409 <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref4.svg"></span> 410 411 </p></blockquote></div> 412<p> 413 The summation is performed by starting at i = λ/2, and then recursing in 414 both directions, using the usual recurrence relations for the Poisson PDF 415 and incomplete beta functions. This is the "Method 2" described 416 by: 417 </p> 418<p> 419 Denise Benton and K. Krishnamoorthy, "Computing discrete mixtures 420 of continuous distributions: noncentral chisquare, noncentral t and the 421 distribution of the square of the sample multiple correlation coefficient", 422 Computational Statistics & Data Analysis 43 (2003) 249-267. 423 </p> 424<p> 425 Specific applications of the above formulae to the noncentral beta distribution 426 can be found in: 427 </p> 428<p> 429 Russell V. Lenth, "Algorithm AS 226: Computing Noncentral Beta Probabilities", 430 Applied Statistics, Vol. 36, No. 2. (1987), pp. 241-244. 431 </p> 432<p> 433 H. Frick, "Algorithm AS R84: A Remark on Algorithm AS 226: Computing 434 Non-Central Beta Probabilities", Applied Statistics, Vol. 39, No. 435 2. (1990), pp. 311-312. 436 </p> 437<p> 438 Ming Long Lam, "Remark AS R95: A Remark on Algorithm AS 226: Computing 439 Non-Central Beta Probabilities", Applied Statistics, Vol. 44, No. 440 4. (1995), pp. 551-552. 441 </p> 442<p> 443 Harry O. Posten, "An Effective Algorithm for the Noncentral Beta Distribution 444 Function", The American Statistician, Vol. 47, No. 2. (May, 1993), 445 pp. 129-131. 446 </p> 447<p> 448 R. Chattamvelli, "A Note on the Noncentral Beta Distribution Function", 449 The American Statistician, Vol. 49, No. 2. (May, 1995), pp. 231-234. 450 </p> 451<p> 452 Of these, the Posten reference provides the most complete overview, and 453 includes the modification starting iteration at λ/2. 454 </p> 455<p> 456 The main difference between this implementation and the above references 457 is the direct computation of the complement when most efficient to do so, 458 and the accumulation of the sum to -1 rather than subtracting the result 459 from 1 at the end: this can substantially reduce the number of iterations 460 required when the result is near 1. 461 </p> 462<p> 463 The PDF is computed using the methodology of Benton and Krishnamoorthy 464 and the relation: 465 </p> 466<div class="blockquote"><blockquote class="blockquote"><p> 467 <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span> 468 469 </p></blockquote></div> 470<p> 471 Quantiles are computed using a specially modified version of <a class="link" href="../../roots_noderiv/bracket_solve.html" title="Bracket and Solve Root">bracket 472 and solve</a>, starting the search for the root at the mean of the distribution. 473 (A Cornish-Fisher type expansion was also tried, but while this gets quite 474 close to the root in many cases, when it is wrong it tends to introduce 475 quite pathological behaviour: more investigation in this area is probably 476 warranted). 477 </p> 478</div> 479<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 480<td align="left"></td> 481<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 482 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 483 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 484 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 485 Daryle Walker and Xiaogang Zhang<p> 486 Distributed under the Boost Software License, Version 1.0. (See accompanying 487 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>) 488 </p> 489</div></td> 490</tr></table> 491<hr> 492<div class="spirit-nav"> 493<a accesskey="p" href="negative_binomial_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nc_chi_squared_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 494</div> 495</body> 496</html> 497
