1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Noncentral T 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="nc_f_dist.html" title="Noncentral F Distribution"> 10<link rel="next" href="normal_dist.html" title="Normal (Gaussian) 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="nc_f_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="normal_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_t_dist"></a><a class="link" href="nc_t_dist.html" title="Noncentral T Distribution">Noncentral T 28 Distribution</a> 29</h4></div></div></div> 30<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">non_central_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></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_t_distribution</span><span class="special">;</span> 36 37<span class="keyword">typedef</span> <span class="identifier">non_central_t_distribution</span><span class="special"><></span> <span class="identifier">non_central_t</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_t_distribution</span> 41<span class="special">{</span> 42<span class="keyword">public</span><span class="special">:</span> 43 <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> 44 <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> 45 46 <span class="comment">// Constructor:</span> 47 <span class="identifier">non_central_t_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">delta</span><span class="special">);</span> 48 49 <span class="comment">// Accessor to degrees_of_freedom parameter v:</span> 50 <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 51 52 <span class="comment">// Accessor to non-centrality parameter delta:</span> 53 <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 54<span class="special">};</span> 55 56<span class="special">}}</span> <span class="comment">// namespaces</span> 57</pre> 58<p> 59 The noncentral T distribution is a generalization of the <a class="link" href="students_t_dist.html" title="Students t Distribution">Students 60 t Distribution</a>. Let X have a normal distribution with mean δ and variance 61 1, and let <span class="emphasis"><em>ν S<sup>2</sup></em></span> have a chi-squared distribution with 62 degrees of freedom ν. Assume that X and S<sup>2</sup> are independent. The distribution 63 of <span class="serif_italic">t<sub>ν</sub>(δ)=X/S</span> is called a noncentral 64 t distribution with degrees of freedom ν and noncentrality parameter δ. 65 </p> 66<p> 67 This gives the following PDF: 68 </p> 69<div class="blockquote"><blockquote class="blockquote"><p> 70 <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref1.svg"></span> 71 72 </p></blockquote></div> 73<p> 74 where <span class="serif_italic"><sub>1</sub>F<sub>1</sub>(a;b;x)</span> is a confluent hypergeometric 75 function. 76 </p> 77<p> 78 The following graph illustrates how the distribution changes for different 79 values of ν and δ: 80 </p> 81<div class="blockquote"><blockquote class="blockquote"><p> 82 <span class="inlinemediaobject"><img src="../../../../graphs/nc_t_pdf.svg" align="middle"></span> 83 84 </p></blockquote></div> 85<div class="blockquote"><blockquote class="blockquote"><p> 86 <span class="inlinemediaobject"><img src="../../../../graphs/nc_t_cdf.svg" align="middle"></span> 87 88 </p></blockquote></div> 89<h5> 90<a name="math_toolkit.dist_ref.dists.nc_t_dist.h0"></a> 91 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.member_functions"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.member_functions">Member 92 Functions</a> 93 </h5> 94<pre class="programlisting"><span class="identifier">non_central_t_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">delta</span><span class="special">);</span> 95</pre> 96<p> 97 Constructs a non-central t distribution with degrees of freedom parameter 98 <span class="emphasis"><em>v</em></span> and non-centrality parameter <span class="emphasis"><em>delta</em></span>. 99 </p> 100<p> 101 Requires <span class="emphasis"><em>v</em></span> > 0 (including positive infinity) and 102 finite <span class="emphasis"><em>delta</em></span>, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 103 </p> 104<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 105</pre> 106<p> 107 Returns the parameter <span class="emphasis"><em>v</em></span> from which this object was 108 constructed. 109 </p> 110<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> 111</pre> 112<p> 113 Returns the non-centrality parameter <span class="emphasis"><em>delta</em></span> from which 114 this object was constructed. 115 </p> 116<h5> 117<a name="math_toolkit.dist_ref.dists.nc_t_dist.h1"></a> 118 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.non_member_accessors"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.non_member_accessors">Non-member 119 Accessors</a> 120 </h5> 121<p> 122 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 123 functions</a> that are generic to all distributions are supported: 124 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 125 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 126 <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>, 127 <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>, 128 <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>, 129 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 130 <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>, 131 <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>. 132 </p> 133<p> 134 The domain of the random variable is [-∞, +∞]. 135 </p> 136<h5> 137<a name="math_toolkit.dist_ref.dists.nc_t_dist.h2"></a> 138 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.accuracy"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.accuracy">Accuracy</a> 139 </h5> 140<p> 141 The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>) 142 found on various platforms with various floating-point types. Unless otherwise 143 specified, any floating-point type that is narrower than the one shown 144 will have <a class="link" href="../../relative_error.html#math_toolkit.relative_error.zero_error">effectively 145 zero error</a>. 146 </p> 147<div class="table"> 148<a name="math_toolkit.dist_ref.dists.nc_t_dist.table_non_central_t_CDF"></a><p class="title"><b>Table 5.8. Error rates for non central t CDF</b></p> 149<div class="table-contents"><table class="table" summary="Error rates for non central t CDF"> 150<colgroup> 151<col> 152<col> 153<col> 154<col> 155<col> 156</colgroup> 157<thead><tr> 158<th> 159 </th> 160<th> 161 <p> 162 GNU C++ version 7.1.0<br> linux<br> double 163 </p> 164 </th> 165<th> 166 <p> 167 GNU C++ version 7.1.0<br> linux<br> long double 168 </p> 169 </th> 170<th> 171 <p> 172 Sun compiler version 0x5150<br> Sun Solaris<br> long double 173 </p> 174 </th> 175<th> 176 <p> 177 Microsoft Visual C++ version 14.1<br> Win32<br> double 178 </p> 179 </th> 180</tr></thead> 181<tbody> 182<tr> 183<td> 184 <p> 185 Non Central T 186 </p> 187 </td> 188<td> 189 <p> 190 <span class="blue">Max = 0.796ε (Mean = 0.0691ε)</span><br> 191 <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max 192 = 5.28e+15ε (Mean = 8.49e+14ε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_t_CDF_Rmath_3_2_3_Non_Central_T">And 193 other failures.</a>)</span> 194 </p> 195 </td> 196<td> 197 <p> 198 <span class="blue">Max = 139ε (Mean = 31ε)</span> 199 </p> 200 </td> 201<td> 202 <p> 203 <span class="blue">Max = 145ε (Mean = 30.9ε)</span> 204 </p> 205 </td> 206<td> 207 <p> 208 <span class="blue">Max = 135ε (Mean = 32.1ε)</span> 209 </p> 210 </td> 211</tr> 212<tr> 213<td> 214 <p> 215 Non Central T (small non-centrality) 216 </p> 217 </td> 218<td> 219 <p> 220 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath 221 3.2.3:</em></span> Max = 2.09e+03ε (Mean = 244ε)) 222 </p> 223 </td> 224<td> 225 <p> 226 <span class="blue">Max = 3.86ε (Mean = 1.4ε)</span> 227 </p> 228 </td> 229<td> 230 <p> 231 <span class="blue">Max = 9.15ε (Mean = 2.13ε)</span> 232 </p> 233 </td> 234<td> 235 <p> 236 <span class="blue">Max = 6.17ε (Mean = 1.45ε)</span> 237 </p> 238 </td> 239</tr> 240<tr> 241<td> 242 <p> 243 Non Central T (large parameters) 244 </p> 245 </td> 246<td> 247 <p> 248 <span class="blue">Max = 257ε (Mean = 72.1ε)</span><br> <br> 249 (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 2.46ε (Mean = 0.657ε)) 250 </p> 251 </td> 252<td> 253 <p> 254 <span class="blue">Max = 5.26e+05ε (Mean = 1.48e+05ε)</span> 255 </p> 256 </td> 257<td> 258 <p> 259 <span class="blue">Max = 5.24e+05ε (Mean = 1.47e+05ε)</span> 260 </p> 261 </td> 262<td> 263 <p> 264 <span class="blue">Max = 286ε (Mean = 62.8ε)</span> 265 </p> 266 </td> 267</tr> 268</tbody> 269</table></div> 270</div> 271<br class="table-break"><div class="table"> 272<a name="math_toolkit.dist_ref.dists.nc_t_dist.table_non_central_t_CDF_complement"></a><p class="title"><b>Table 5.9. Error rates for non central t CDF complement</b></p> 273<div class="table-contents"><table class="table" summary="Error rates for non central t CDF complement"> 274<colgroup> 275<col> 276<col> 277<col> 278<col> 279<col> 280</colgroup> 281<thead><tr> 282<th> 283 </th> 284<th> 285 <p> 286 GNU C++ version 7.1.0<br> linux<br> double 287 </p> 288 </th> 289<th> 290 <p> 291 GNU C++ version 7.1.0<br> linux<br> long double 292 </p> 293 </th> 294<th> 295 <p> 296 Sun compiler version 0x5150<br> Sun Solaris<br> long double 297 </p> 298 </th> 299<th> 300 <p> 301 Microsoft Visual C++ version 14.1<br> Win32<br> double 302 </p> 303 </th> 304</tr></thead> 305<tbody> 306<tr> 307<td> 308 <p> 309 Non Central T 310 </p> 311 </td> 312<td> 313 <p> 314 <span class="blue">Max = 0.707ε (Mean = 0.0497ε)</span><br> 315 <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> <span class="red">Max 316 = 6.19e+15ε (Mean = 6.72e+14ε) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_non_central_t_CDF_complement_Rmath_3_2_3_Non_Central_T">And 317 other failures.</a>)</span> 318 </p> 319 </td> 320<td> 321 <p> 322 <span class="blue">Max = 201ε (Mean = 31.7ε)</span> 323 </p> 324 </td> 325<td> 326 <p> 327 <span class="blue">Max = 340ε (Mean = 43.2ε)</span> 328 </p> 329 </td> 330<td> 331 <p> 332 <span class="blue">Max = 154ε (Mean = 32.1ε)</span> 333 </p> 334 </td> 335</tr> 336<tr> 337<td> 338 <p> 339 Non Central T (small non-centrality) 340 </p> 341 </td> 342<td> 343 <p> 344 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath 345 3.2.3:</em></span> Max = 1.87e+03ε (Mean = 263ε)) 346 </p> 347 </td> 348<td> 349 <p> 350 <span class="blue">Max = 10.5ε (Mean = 2.13ε)</span> 351 </p> 352 </td> 353<td> 354 <p> 355 <span class="blue">Max = 10.5ε (Mean = 2.39ε)</span> 356 </p> 357 </td> 358<td> 359 <p> 360 <span class="blue">Max = 4.6ε (Mean = 1.63ε)</span> 361 </p> 362 </td> 363</tr> 364<tr> 365<td> 366 <p> 367 Non Central T (large parameters) 368 </p> 369 </td> 370<td> 371 <p> 372 <span class="blue">Max = 478ε (Mean = 96.3ε)</span><br> <br> 373 (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 2.24ε (Mean = 0.945ε)) 374 </p> 375 </td> 376<td> 377 <p> 378 <span class="blue">Max = 9.79e+05ε (Mean = 1.97e+05ε)</span> 379 </p> 380 </td> 381<td> 382 <p> 383 <span class="blue">Max = 9.79e+05ε (Mean = 1.97e+05ε)</span> 384 </p> 385 </td> 386<td> 387 <p> 388 <span class="blue">Max = 227ε (Mean = 50.4ε)</span> 389 </p> 390 </td> 391</tr> 392</tbody> 393</table></div> 394</div> 395<br class="table-break"><div class="caution"><table border="0" summary="Caution"> 396<tr> 397<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td> 398<th align="left">Caution</th> 399</tr> 400<tr><td align="left" valign="top"><p> 401 The complexity of the current algorithm is dependent upon δ<sup>2</sup>: consequently 402 the time taken to evaluate the CDF increases rapidly for δ > 500, likewise 403 the accuracy decreases rapidly for very large δ. 404 </p></td></tr> 405</table></div> 406<p> 407 Accuracy for the quantile and PDF functions should be broadly similar. 408 The <span class="emphasis"><em>mode</em></span> is determined numerically and cannot in principal 409 be more accurate than the square root of floating-point type FPT epsilon, 410 accessed using <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">epsilon</span><span class="special"><</span><span class="identifier">FPT</span><span class="special">>()</span></code>. 411 For 64-bit <code class="computeroutput"><span class="keyword">double</span></code>, epsilon 412 is about 1e-16, so the fractional accuracy is limited to 1e-8. 413 </p> 414<h5> 415<a name="math_toolkit.dist_ref.dists.nc_t_dist.h3"></a> 416 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.tests"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.tests">Tests</a> 417 </h5> 418<p> 419 There are two sets of tests of this distribution: 420 </p> 421<p> 422 Basic sanity checks compare this implementation to the test values given 423 in "Computing discrete mixtures of continuous distributions: noncentral 424 chisquare, noncentral t and the distribution of the square of the sample 425 multiple correlation coefficient." Denise Benton, K. Krishnamoorthy, 426 Computational Statistics & Data Analysis 43 (2003) 249-267. 427 </p> 428<p> 429 Accuracy checks use test data computed with this implementation and arbitrary 430 precision interval arithmetic: this test data is believed to be accurate 431 to at least 50 decimal places. 432 </p> 433<p> 434 The cases of large (or infinite) ν and/or large δ has received special treatment 435 to avoid catastrophic loss of accuracy. New tests have been added to confirm 436 the improvement achieved. 437 </p> 438<p> 439 From Boost 1.52, degrees of freedom ν can be +∞ 440when the normal distribution 441 located at δ 442(equivalent to the central Student's t distribution) is used 443 in place for accuracy and speed. 444 </p> 445<h5> 446<a name="math_toolkit.dist_ref.dists.nc_t_dist.h4"></a> 447 <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_t_dist.implementation"></a></span><a class="link" href="nc_t_dist.html#math_toolkit.dist_ref.dists.nc_t_dist.implementation">Implementation</a> 448 </h5> 449<p> 450 The CDF is computed using a modification of the method described in "Computing 451 discrete mixtures of continuous distributions: noncentral chisquare, noncentral 452 t and the distribution of the square of the sample multiple correlation 453 coefficient." Denise Benton, K. Krishnamoorthy, Computational Statistics 454 & Data Analysis 43 (2003) 249-267. 455 </p> 456<p> 457 This uses the following formula for the CDF: 458 </p> 459<div class="blockquote"><blockquote class="blockquote"><p> 460 <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref2.svg"></span> 461 462 </p></blockquote></div> 463<p> 464 Where I<sub>x</sub>(a,b) is the incomplete beta function, and Φ(x) is the normal CDF 465 at x. 466 </p> 467<p> 468 Iteration starts at the largest of the Poisson weighting terms (at i = 469 δ<sup>2</sup> / 2) and then proceeds in both directions as per Benton and Krishnamoorthy's 470 paper. 471 </p> 472<p> 473 Alternatively, by considering what happens when t = ∞, we have x = 1, and 474 therefore I<sub>x</sub>(a,b) = 1 and: 475 </p> 476<div class="blockquote"><blockquote class="blockquote"><p> 477 <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref3.svg"></span> 478 479 </p></blockquote></div> 480<p> 481 From this we can easily show that: 482 </p> 483<div class="blockquote"><blockquote class="blockquote"><p> 484 <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref4.svg"></span> 485 486 </p></blockquote></div> 487<p> 488 and therefore we have a means to compute either the probability or its 489 complement directly without the risk of cancellation error. The crossover 490 criterion for choosing whether to calculate the CDF or its complement is 491 the same as for the <a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral 492 Beta Distribution</a>. 493 </p> 494<p> 495 The PDF can be computed by a very similar method using: 496 </p> 497<div class="blockquote"><blockquote class="blockquote"><p> 498 <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref5.svg"></span> 499 500 </p></blockquote></div> 501<p> 502 Where I<sub>x</sub><sup>'</sup>(a,b) is the derivative of the incomplete beta function. 503 </p> 504<p> 505 For both the PDF and CDF we switch to approximating the distribution by 506 a Student's t distribution centred on δ when ν is very large. The crossover 507 location appears to be when δ/(4ν) < ε, this location was estimated by 508 inspection of equation 2.6 in "A Comparison of Approximations To Percentiles 509 of the Noncentral t-Distribution". H. Sahai and M. M. Ojeda, Revista 510 Investigacion Operacional Vol 21, No 2, 2000, page 123. 511 </p> 512<p> 513 Equation 2.6 is a Fisher-Cornish expansion by Eeden and Johnson. The second 514 term includes the ratio δ/(4ν), so when this term become negligible, this 515 and following terms can be ignored, leaving just Student's t distribution 516 centred on δ. 517 </p> 518<p> 519 This was also confirmed by experimental testing. 520 </p> 521<p> 522 See also 523 </p> 524<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 525<li class="listitem"> 526 "Some Approximations to the Percentage Points of the Noncentral 527 t-Distribution". C. van Eeden. International Statistical Review, 528 29, 4-31. 529 </li> 530<li class="listitem"> 531 "Continuous Univariate Distributions". N.L. Johnson, S. Kotz 532 and N. Balkrishnan. 1995. John Wiley and Sons New York. 533 </li> 534</ul></div> 535<p> 536 The quantile is calculated via the usual <a class="link" href="../../roots_noderiv.html" title="Root Finding Without Derivatives">root-finding 537 without derivatives</a> method with the initial guess taken as the quantile 538 of a normal approximation to the noncentral T. 539 </p> 540<p> 541 There is no closed form for the mode, so this is computed via functional 542 maximisation of the PDF. 543 </p> 544<p> 545 The remaining functions (mean, variance etc) are implemented using the 546 formulas given in Weisstein, Eric W. "Noncentral Student's t-Distribution." 547 From MathWorld--A Wolfram Web Resource. <a href="http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html" target="_top">http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html</a> 548 and in the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralStudentTDistribution.html" target="_top">Mathematica 549 documentation</a>. 550 </p> 551<p> 552 Some analytic properties of noncentral distributions (particularly unimodality, 553 and monotonicity of their modes) are surveyed and summarized by: 554 </p> 555<p> 556 Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, 557 141 (2003) 3-12. 558 </p> 559</div> 560<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 561<td align="left"></td> 562<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 563 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 564 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 565 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 566 Daryle Walker and Xiaogang Zhang<p> 567 Distributed under the Boost Software License, Version 1.0. (See accompanying 568 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>) 569 </p> 570</div></td> 571</tr></table> 572<hr> 573<div class="spirit-nav"> 574<a accesskey="p" href="nc_f_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="normal_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 575</div> 576</body> 577</html> 578
