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1 | // Copyright John Maddock 2010. |
2 | // Copyright Paul A. Bristow 2010. | |
3 | ||
4 | // Use, modification and distribution are subject to the | |
5 | // Boost Software License, Version 1.0. (See accompanying file | |
6 | // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) | |
7 | ||
8 | #ifndef BOOST_STATS_INVERSE_GAUSSIAN_HPP | |
9 | #define BOOST_STATS_INVERSE_GAUSSIAN_HPP | |
10 | ||
11 | #ifdef _MSC_VER | |
12 | #pragma warning(disable: 4512) // assignment operator could not be generated | |
13 | #endif | |
14 | ||
15 | // http://en.wikipedia.org/wiki/Normal-inverse_Gaussian_distribution | |
16 | // http://mathworld.wolfram.com/InverseGaussianDistribution.html | |
17 | ||
18 | // The normal-inverse Gaussian distribution | |
19 | // also called the Wald distribution (some sources limit this to when mean = 1). | |
20 | ||
21 | // It is the continuous probability distribution | |
22 | // that is defined as the normal variance-mean mixture where the mixing density is the | |
23 | // inverse Gaussian distribution. The tails of the distribution decrease more slowly | |
24 | // than the normal distribution. It is therefore suitable to model phenomena | |
25 | // where numerically large values are more probable than is the case for the normal distribution. | |
26 | ||
27 | // The Inverse Gaussian distribution was first studied in relationship to Brownian motion. | |
28 | // In 1956 M.C.K. Tweedie used the name 'Inverse Gaussian' because there is an inverse | |
29 | // relationship between the time to cover a unit distance and distance covered in unit time. | |
30 | ||
31 | // Examples are returns from financial assets and turbulent wind speeds. | |
32 | // The normal-inverse Gaussian distributions form | |
33 | // a subclass of the generalised hyperbolic distributions. | |
34 | ||
35 | // See also | |
36 | ||
37 | // http://en.wikipedia.org/wiki/Normal_distribution | |
38 | // http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm | |
39 | // Also: | |
40 | // Weisstein, Eric W. "Normal Distribution." | |
41 | // From MathWorld--A Wolfram Web Resource. | |
42 | // http://mathworld.wolfram.com/NormalDistribution.html | |
43 | ||
44 | // http://www.jstatsoft.org/v26/i04/paper General class of inverse Gaussian distributions. | |
45 | // ig package - withdrawn but at http://cran.r-project.org/src/contrib/Archive/ig/ | |
46 | ||
47 | // http://www.stat.ucl.ac.be/ISdidactique/Rhelp/library/SuppDists/html/inverse_gaussian.html | |
48 | // R package for dinverse_gaussian, ... | |
49 | ||
50 | // http://www.statsci.org/s/inverse_gaussian.s and http://www.statsci.org/s/inverse_gaussian.html | |
51 | ||
52 | //#include <boost/math/distributions/fwd.hpp> | |
53 | #include <boost/math/special_functions/erf.hpp> // for erf/erfc. | |
54 | #include <boost/math/distributions/complement.hpp> | |
55 | #include <boost/math/distributions/detail/common_error_handling.hpp> | |
56 | #include <boost/math/distributions/normal.hpp> | |
57 | #include <boost/math/distributions/gamma.hpp> // for gamma function | |
58 | // using boost::math::gamma_p; | |
59 | ||
60 | #include <boost/math/tools/tuple.hpp> | |
61 | //using std::tr1::tuple; | |
62 | //using std::tr1::make_tuple; | |
63 | #include <boost/math/tools/roots.hpp> | |
64 | //using boost::math::tools::newton_raphson_iterate; | |
65 | ||
66 | #include <utility> | |
67 | ||
68 | namespace boost{ namespace math{ | |
69 | ||
70 | template <class RealType = double, class Policy = policies::policy<> > | |
71 | class inverse_gaussian_distribution | |
72 | { | |
73 | public: | |
74 | typedef RealType value_type; | |
75 | typedef Policy policy_type; | |
76 | ||
77 | inverse_gaussian_distribution(RealType l_mean = 1, RealType l_scale = 1) | |
78 | : m_mean(l_mean), m_scale(l_scale) | |
79 | { // Default is a 1,1 inverse_gaussian distribution. | |
80 | static const char* function = "boost::math::inverse_gaussian_distribution<%1%>::inverse_gaussian_distribution"; | |
81 | ||
82 | RealType result; | |
83 | detail::check_scale(function, l_scale, &result, Policy()); | |
84 | detail::check_location(function, l_mean, &result, Policy()); | |
85 | detail::check_x_gt0(function, l_mean, &result, Policy()); | |
86 | } | |
87 | ||
88 | RealType mean()const | |
89 | { // alias for location. | |
90 | return m_mean; // aka mu | |
91 | } | |
92 | ||
93 | // Synonyms, provided to allow generic use of find_location and find_scale. | |
94 | RealType location()const | |
95 | { // location, aka mu. | |
96 | return m_mean; | |
97 | } | |
98 | RealType scale()const | |
99 | { // scale, aka lambda. | |
100 | return m_scale; | |
101 | } | |
102 | ||
103 | RealType shape()const | |
104 | { // shape, aka phi = lambda/mu. | |
105 | return m_scale / m_mean; | |
106 | } | |
107 | ||
108 | private: | |
109 | // | |
110 | // Data members: | |
111 | // | |
112 | RealType m_mean; // distribution mean or location, aka mu. | |
113 | RealType m_scale; // distribution standard deviation or scale, aka lambda. | |
114 | }; // class normal_distribution | |
115 | ||
116 | typedef inverse_gaussian_distribution<double> inverse_gaussian; | |
117 | ||
118 | template <class RealType, class Policy> | |
119 | inline const std::pair<RealType, RealType> range(const inverse_gaussian_distribution<RealType, Policy>& /*dist*/) | |
120 | { // Range of permissible values for random variable x, zero to max. | |
121 | using boost::math::tools::max_value; | |
122 | return std::pair<RealType, RealType>(static_cast<RealType>(0.), max_value<RealType>()); // - to + max value. | |
123 | } | |
124 | ||
125 | template <class RealType, class Policy> | |
126 | inline const std::pair<RealType, RealType> support(const inverse_gaussian_distribution<RealType, Policy>& /*dist*/) | |
127 | { // Range of supported values for random variable x, zero to max. | |
128 | // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero. | |
129 | using boost::math::tools::max_value; | |
130 | return std::pair<RealType, RealType>(static_cast<RealType>(0.), max_value<RealType>()); // - to + max value. | |
131 | } | |
132 | ||
133 | template <class RealType, class Policy> | |
134 | inline RealType pdf(const inverse_gaussian_distribution<RealType, Policy>& dist, const RealType& x) | |
135 | { // Probability Density Function | |
136 | BOOST_MATH_STD_USING // for ADL of std functions | |
137 | ||
138 | RealType scale = dist.scale(); | |
139 | RealType mean = dist.mean(); | |
140 | RealType result = 0; | |
141 | static const char* function = "boost::math::pdf(const inverse_gaussian_distribution<%1%>&, %1%)"; | |
142 | if(false == detail::check_scale(function, scale, &result, Policy())) | |
143 | { | |
144 | return result; | |
145 | } | |
146 | if(false == detail::check_location(function, mean, &result, Policy())) | |
147 | { | |
148 | return result; | |
149 | } | |
150 | if(false == detail::check_x_gt0(function, mean, &result, Policy())) | |
151 | { | |
152 | return result; | |
153 | } | |
154 | if(false == detail::check_positive_x(function, x, &result, Policy())) | |
155 | { | |
156 | return result; | |
157 | } | |
158 | ||
159 | if (x == 0) | |
160 | { | |
161 | return 0; // Convenient, even if not defined mathematically. | |
162 | } | |
163 | ||
164 | result = | |
165 | sqrt(scale / (constants::two_pi<RealType>() * x * x * x)) | |
166 | * exp(-scale * (x - mean) * (x - mean) / (2 * x * mean * mean)); | |
167 | return result; | |
168 | ||
169 | ||
170 | template <class RealType, class Policy> | |
171 | inline RealType cdf(const inverse_gaussian_distribution<RealType, Policy>& dist, const RealType& x) | |
172 | { // Cumulative Density Function. | |
173 | BOOST_MATH_STD_USING // for ADL of std functions. | |
174 | ||
175 | RealType scale = dist.scale(); | |
176 | RealType mean = dist.mean(); | |
177 | static const char* function = "boost::math::cdf(const inverse_gaussian_distribution<%1%>&, %1%)"; | |
178 | RealType result = 0; | |
179 | if(false == detail::check_scale(function, scale, &result, Policy())) | |
180 | { | |
181 | return result; | |
182 | } | |
183 | if(false == detail::check_location(function, mean, &result, Policy())) | |
184 | { | |
185 | return result; | |
186 | } | |
187 | if (false == detail::check_x_gt0(function, mean, &result, Policy())) | |
188 | { | |
189 | return result; | |
190 | } | |
191 | if(false == detail::check_positive_x(function, x, &result, Policy())) | |
192 | { | |
193 | return result; | |
194 | } | |
195 | if (x == 0) | |
196 | { | |
197 | return 0; // Convenient, even if not defined mathematically. | |
198 | } | |
199 | // Problem with this formula for large scale > 1000 or small x, | |
200 | //result = 0.5 * (erf(sqrt(scale / x) * ((x / mean) - 1) / constants::root_two<RealType>(), Policy()) + 1) | |
201 | // + exp(2 * scale / mean) / 2 | |
202 | // * (1 - erf(sqrt(scale / x) * (x / mean + 1) / constants::root_two<RealType>(), Policy())); | |
203 | // so use normal distribution version: | |
204 | // Wikipedia CDF equation http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution. | |
205 | ||
206 | normal_distribution<RealType> n01; | |
207 | ||
208 | RealType n0 = sqrt(scale / x); | |
209 | n0 *= ((x / mean) -1); | |
210 | RealType n1 = cdf(n01, n0); | |
211 | RealType expfactor = exp(2 * scale / mean); | |
212 | RealType n3 = - sqrt(scale / x); | |
213 | n3 *= (x / mean) + 1; | |
214 | RealType n4 = cdf(n01, n3); | |
215 | result = n1 + expfactor * n4; | |
216 | return result; | |
217 | } // cdf | |
218 | ||
219 | template <class RealType, class Policy> | |
220 | struct inverse_gaussian_quantile_functor | |
221 | { | |
222 | ||
223 | inverse_gaussian_quantile_functor(const boost::math::inverse_gaussian_distribution<RealType, Policy> dist, RealType const& p) | |
224 | : distribution(dist), prob(p) | |
225 | { | |
226 | } | |
227 | boost::math::tuple<RealType, RealType> operator()(RealType const& x) | |
228 | { | |
229 | RealType c = cdf(distribution, x); | |
230 | RealType fx = c - prob; // Difference cdf - value - to minimize. | |
231 | RealType dx = pdf(distribution, x); // pdf is 1st derivative. | |
232 | // return both function evaluation difference f(x) and 1st derivative f'(x). | |
233 | return boost::math::make_tuple(fx, dx); | |
234 | } | |
235 | private: | |
236 | const boost::math::inverse_gaussian_distribution<RealType, Policy> distribution; | |
237 | RealType prob; | |
238 | }; | |
239 | ||
240 | template <class RealType, class Policy> | |
241 | struct inverse_gaussian_quantile_complement_functor | |
242 | { | |
243 | inverse_gaussian_quantile_complement_functor(const boost::math::inverse_gaussian_distribution<RealType, Policy> dist, RealType const& p) | |
244 | : distribution(dist), prob(p) | |
245 | { | |
246 | } | |
247 | boost::math::tuple<RealType, RealType> operator()(RealType const& x) | |
248 | { | |
249 | RealType c = cdf(complement(distribution, x)); | |
250 | RealType fx = c - prob; // Difference cdf - value - to minimize. | |
251 | RealType dx = -pdf(distribution, x); // pdf is 1st derivative. | |
252 | // return both function evaluation difference f(x) and 1st derivative f'(x). | |
253 | //return std::tr1::make_tuple(fx, dx); if available. | |
254 | return boost::math::make_tuple(fx, dx); | |
255 | } | |
256 | private: | |
257 | const boost::math::inverse_gaussian_distribution<RealType, Policy> distribution; | |
258 | RealType prob; | |
259 | }; | |
260 | ||
261 | namespace detail | |
262 | { | |
263 | template <class RealType> | |
264 | inline RealType guess_ig(RealType p, RealType mu = 1, RealType lambda = 1) | |
265 | { // guess at random variate value x for inverse gaussian quantile. | |
266 | BOOST_MATH_STD_USING | |
267 | using boost::math::policies::policy; | |
268 | // Error type. | |
269 | using boost::math::policies::overflow_error; | |
270 | // Action. | |
271 | using boost::math::policies::ignore_error; | |
272 | ||
273 | typedef policy< | |
274 | overflow_error<ignore_error> // Ignore overflow (return infinity) | |
275 | > no_overthrow_policy; | |
276 | ||
277 | RealType x; // result is guess at random variate value x. | |
278 | RealType phi = lambda / mu; | |
279 | if (phi > 2.) | |
280 | { // Big phi, so starting to look like normal Gaussian distribution. | |
281 | // x=(qnorm(p,0,1,true,false) - 0.5 * sqrt(mu/lambda)) / sqrt(lambda/mu); | |
282 | // Whitmore, G.A. and Yalovsky, M. | |
283 | // A normalising logarithmic transformation for inverse Gaussian random variables, | |
284 | // Technometrics 20-2, 207-208 (1978), but using expression from | |
285 | // V Seshadri, Inverse Gaussian distribution (1998) ISBN 0387 98618 9, page 6. | |
286 | ||
287 | normal_distribution<RealType, no_overthrow_policy> n01; | |
288 | x = mu * exp(quantile(n01, p) / sqrt(phi) - 1/(2 * phi)); | |
289 | } | |
290 | else | |
291 | { // phi < 2 so much less symmetrical with long tail, | |
292 | // so use gamma distribution as an approximation. | |
293 | using boost::math::gamma_distribution; | |
294 | ||
295 | // Define the distribution, using gamma_nooverflow: | |
296 | typedef gamma_distribution<RealType, no_overthrow_policy> gamma_nooverflow; | |
297 | ||
298 | gamma_nooverflow g(static_cast<RealType>(0.5), static_cast<RealType>(1.)); | |
299 | ||
300 | // gamma_nooverflow g(static_cast<RealType>(0.5), static_cast<RealType>(1.)); | |
301 | // R qgamma(0.2, 0.5, 1) 0.0320923 | |
302 | RealType qg = quantile(complement(g, p)); | |
303 | //RealType qg1 = qgamma(1.- p, 0.5, 1.0, true, false); | |
304 | x = lambda / (qg * 2); | |
305 | // | |
306 | if (x > mu/2) // x > mu /2? | |
307 | { // x too large for the gamma approximation to work well. | |
308 | //x = qgamma(p, 0.5, 1.0); // qgamma(0.270614, 0.5, 1) = 0.05983807 | |
309 | RealType q = quantile(g, p); | |
310 | // x = mu * exp(q * static_cast<RealType>(0.1)); // Said to improve at high p | |
311 | // x = mu * x; // Improves at high p? | |
312 | x = mu * exp(q / sqrt(phi) - 1/(2 * phi)); | |
313 | } | |
314 | } | |
315 | return x; | |
316 | } // guess_ig | |
317 | } // namespace detail | |
318 | ||
319 | template <class RealType, class Policy> | |
320 | inline RealType quantile(const inverse_gaussian_distribution<RealType, Policy>& dist, const RealType& p) | |
321 | { | |
322 | BOOST_MATH_STD_USING // for ADL of std functions. | |
323 | // No closed form exists so guess and use Newton Raphson iteration. | |
324 | ||
325 | RealType mean = dist.mean(); | |
326 | RealType scale = dist.scale(); | |
327 | static const char* function = "boost::math::quantile(const inverse_gaussian_distribution<%1%>&, %1%)"; | |
328 | ||
329 | RealType result = 0; | |
330 | if(false == detail::check_scale(function, scale, &result, Policy())) | |
331 | return result; | |
332 | if(false == detail::check_location(function, mean, &result, Policy())) | |
333 | return result; | |
334 | if (false == detail::check_x_gt0(function, mean, &result, Policy())) | |
335 | return result; | |
336 | if(false == detail::check_probability(function, p, &result, Policy())) | |
337 | return result; | |
338 | if (p == 0) | |
339 | { | |
340 | return 0; // Convenient, even if not defined mathematically? | |
341 | } | |
342 | if (p == 1) | |
343 | { // overflow | |
344 | result = policies::raise_overflow_error<RealType>(function, | |
345 | "probability parameter is 1, but must be < 1!", Policy()); | |
346 | return result; // std::numeric_limits<RealType>::infinity(); | |
347 | } | |
348 | ||
349 | RealType guess = detail::guess_ig(p, dist.mean(), dist.scale()); | |
350 | using boost::math::tools::max_value; | |
351 | ||
352 | RealType min = 0.; // Minimum possible value is bottom of range of distribution. | |
353 | RealType max = max_value<RealType>();// Maximum possible value is top of range. | |
354 | // int digits = std::numeric_limits<RealType>::digits; // Maximum possible binary digits accuracy for type T. | |
355 | // digits used to control how accurate to try to make the result. | |
356 | // To allow user to control accuracy versus speed, | |
357 | int get_digits = policies::digits<RealType, Policy>();// get digits from policy, | |
358 | boost::uintmax_t m = policies::get_max_root_iterations<Policy>(); // and max iterations. | |
359 | using boost::math::tools::newton_raphson_iterate; | |
360 | result = | |
361 | newton_raphson_iterate(inverse_gaussian_quantile_functor<RealType, Policy>(dist, p), guess, min, max, get_digits, m); | |
362 | return result; | |
363 | } // quantile | |
364 | ||
365 | template <class RealType, class Policy> | |
366 | inline RealType cdf(const complemented2_type<inverse_gaussian_distribution<RealType, Policy>, RealType>& c) | |
367 | { | |
368 | BOOST_MATH_STD_USING // for ADL of std functions. | |
369 | ||
370 | RealType scale = c.dist.scale(); | |
371 | RealType mean = c.dist.mean(); | |
372 | RealType x = c.param; | |
373 | static const char* function = "boost::math::cdf(const complement(inverse_gaussian_distribution<%1%>&), %1%)"; | |
374 | // infinite arguments not supported. | |
375 | //if((boost::math::isinf)(x)) | |
376 | //{ | |
377 | // if(x < 0) return 1; // cdf complement -infinity is unity. | |
378 | // return 0; // cdf complement +infinity is zero | |
379 | //} | |
380 | // These produce MSVC 4127 warnings, so the above used instead. | |
381 | //if(std::numeric_limits<RealType>::has_infinity && x == std::numeric_limits<RealType>::infinity()) | |
382 | //{ // cdf complement +infinity is zero. | |
383 | // return 0; | |
384 | //} | |
385 | //if(std::numeric_limits<RealType>::has_infinity && x == -std::numeric_limits<RealType>::infinity()) | |
386 | //{ // cdf complement -infinity is unity. | |
387 | // return 1; | |
388 | //} | |
389 | RealType result = 0; | |
390 | if(false == detail::check_scale(function, scale, &result, Policy())) | |
391 | return result; | |
392 | if(false == detail::check_location(function, mean, &result, Policy())) | |
393 | return result; | |
394 | if (false == detail::check_x_gt0(function, mean, &result, Policy())) | |
395 | return result; | |
396 | if(false == detail::check_positive_x(function, x, &result, Policy())) | |
397 | return result; | |
398 | ||
399 | normal_distribution<RealType> n01; | |
400 | RealType n0 = sqrt(scale / x); | |
401 | n0 *= ((x / mean) -1); | |
402 | RealType cdf_1 = cdf(complement(n01, n0)); | |
403 | ||
404 | RealType expfactor = exp(2 * scale / mean); | |
405 | RealType n3 = - sqrt(scale / x); | |
406 | n3 *= (x / mean) + 1; | |
407 | ||
408 | //RealType n5 = +sqrt(scale/x) * ((x /mean) + 1); // note now positive sign. | |
409 | RealType n6 = cdf(complement(n01, +sqrt(scale/x) * ((x /mean) + 1))); | |
410 | // RealType n4 = cdf(n01, n3); // = | |
411 | result = cdf_1 - expfactor * n6; | |
412 | return result; | |
413 | } // cdf complement | |
414 | ||
415 | template <class RealType, class Policy> | |
416 | inline RealType quantile(const complemented2_type<inverse_gaussian_distribution<RealType, Policy>, RealType>& c) | |
417 | { | |
418 | BOOST_MATH_STD_USING // for ADL of std functions | |
419 | ||
420 | RealType scale = c.dist.scale(); | |
421 | RealType mean = c.dist.mean(); | |
422 | static const char* function = "boost::math::quantile(const complement(inverse_gaussian_distribution<%1%>&), %1%)"; | |
423 | RealType result = 0; | |
424 | if(false == detail::check_scale(function, scale, &result, Policy())) | |
425 | return result; | |
426 | if(false == detail::check_location(function, mean, &result, Policy())) | |
427 | return result; | |
428 | if (false == detail::check_x_gt0(function, mean, &result, Policy())) | |
429 | return result; | |
430 | RealType q = c.param; | |
431 | if(false == detail::check_probability(function, q, &result, Policy())) | |
432 | return result; | |
433 | ||
434 | RealType guess = detail::guess_ig(q, mean, scale); | |
435 | // Complement. | |
436 | using boost::math::tools::max_value; | |
437 | ||
438 | RealType min = 0.; // Minimum possible value is bottom of range of distribution. | |
439 | RealType max = max_value<RealType>();// Maximum possible value is top of range. | |
440 | // int digits = std::numeric_limits<RealType>::digits; // Maximum possible binary digits accuracy for type T. | |
441 | // digits used to control how accurate to try to make the result. | |
442 | int get_digits = policies::digits<RealType, Policy>(); | |
443 | boost::uintmax_t m = policies::get_max_root_iterations<Policy>(); | |
444 | using boost::math::tools::newton_raphson_iterate; | |
445 | result = | |
446 | newton_raphson_iterate(inverse_gaussian_quantile_complement_functor<RealType, Policy>(c.dist, q), guess, min, max, get_digits, m); | |
447 | return result; | |
448 | } // quantile | |
449 | ||
450 | template <class RealType, class Policy> | |
451 | inline RealType mean(const inverse_gaussian_distribution<RealType, Policy>& dist) | |
452 | { // aka mu | |
453 | return dist.mean(); | |
454 | } | |
455 | ||
456 | template <class RealType, class Policy> | |
457 | inline RealType scale(const inverse_gaussian_distribution<RealType, Policy>& dist) | |
458 | { // aka lambda | |
459 | return dist.scale(); | |
460 | } | |
461 | ||
462 | template <class RealType, class Policy> | |
463 | inline RealType shape(const inverse_gaussian_distribution<RealType, Policy>& dist) | |
464 | { // aka phi | |
465 | return dist.shape(); | |
466 | } | |
467 | ||
468 | template <class RealType, class Policy> | |
469 | inline RealType standard_deviation(const inverse_gaussian_distribution<RealType, Policy>& dist) | |
470 | { | |
471 | BOOST_MATH_STD_USING | |
472 | RealType scale = dist.scale(); | |
473 | RealType mean = dist.mean(); | |
474 | RealType result = sqrt(mean * mean * mean / scale); | |
475 | return result; | |
476 | } | |
477 | ||
478 | template <class RealType, class Policy> | |
479 | inline RealType mode(const inverse_gaussian_distribution<RealType, Policy>& dist) | |
480 | { | |
481 | BOOST_MATH_STD_USING | |
482 | RealType scale = dist.scale(); | |
483 | RealType mean = dist.mean(); | |
484 | RealType result = mean * (sqrt(1 + (9 * mean * mean)/(4 * scale * scale)) | |
485 | - 3 * mean / (2 * scale)); | |
486 | return result; | |
487 | } | |
488 | ||
489 | template <class RealType, class Policy> | |
490 | inline RealType skewness(const inverse_gaussian_distribution<RealType, Policy>& dist) | |
491 | { | |
492 | BOOST_MATH_STD_USING | |
493 | RealType scale = dist.scale(); | |
494 | RealType mean = dist.mean(); | |
495 | RealType result = 3 * sqrt(mean/scale); | |
496 | return result; | |
497 | } | |
498 | ||
499 | template <class RealType, class Policy> | |
500 | inline RealType kurtosis(const inverse_gaussian_distribution<RealType, Policy>& dist) | |
501 | { | |
502 | RealType scale = dist.scale(); | |
503 | RealType mean = dist.mean(); | |
504 | RealType result = 15 * mean / scale -3; | |
505 | return result; | |
506 | } | |
507 | ||
508 | template <class RealType, class Policy> | |
509 | inline RealType kurtosis_excess(const inverse_gaussian_distribution<RealType, Policy>& dist) | |
510 | { | |
511 | RealType scale = dist.scale(); | |
512 | RealType mean = dist.mean(); | |
513 | RealType result = 15 * mean / scale; | |
514 | return result; | |
515 | } | |
516 | ||
517 | } // namespace math | |
518 | } // namespace boost | |
519 | ||
520 | // This include must be at the end, *after* the accessors | |
521 | // for this distribution have been defined, in order to | |
522 | // keep compilers that support two-phase lookup happy. | |
523 | #include <boost/math/distributions/detail/derived_accessors.hpp> | |
524 | ||
525 | #endif // BOOST_STATS_INVERSE_GAUSSIAN_HPP | |
526 | ||
527 |