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// boost\math\distributions\non_central_beta.hpp
// Copyright John Maddock 2008.
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt
// or copy at http://www.boost.org/LICENSE_1_0.txt)
#ifndef BOOST_MATH_SPECIAL_NON_CENTRAL_BETA_HPP
#define BOOST_MATH_SPECIAL_NON_CENTRAL_BETA_HPP
#include <boost/math/distributions/fwd.hpp>
#include <boost/math/special_functions/beta.hpp> // for incomplete gamma. gamma_q
#include <boost/math/distributions/complement.hpp> // complements
#include <boost/math/distributions/beta.hpp> // central distribution
#include <boost/math/distributions/detail/generic_mode.hpp>
#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks
#include <boost/math/special_functions/fpclassify.hpp> // isnan.
#include <boost/math/tools/roots.hpp> // for root finding.
#include <boost/math/tools/series.hpp>
namespace boost
{
namespace math
{
template <class RealType, class Policy>
class non_central_beta_distribution;
namespace detail{
template <class T, class Policy>
T non_central_beta_p(T a, T b, T lam, T x, T y, const Policy& pol, T init_val = 0)
{
BOOST_MATH_STD_USING
using namespace boost::math;
//
// Variables come first:
//
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
T errtol = boost::math::policies::get_epsilon<T, Policy>();
T l2 = lam / 2;
//
// k is the starting point for iteration, and is the
// maximum of the poisson weighting term,
// note that unlike other similar code, we do not set
// k to zero, when l2 is small, as forward iteration
// is unstable:
//
int k = itrunc(l2);
if(k == 0)
k = 1;
// Starting Poisson weight:
T pois = gamma_p_derivative(T(k+1), l2, pol);
if(pois == 0)
return init_val;
// recurance term:
T xterm;
// Starting beta term:
T beta = x < y
? detail::ibeta_imp(T(a + k), b, x, pol, false, true, &xterm)
: detail::ibeta_imp(b, T(a + k), y, pol, true, true, &xterm);
xterm *= y / (a + b + k - 1);
T poisf(pois), betaf(beta), xtermf(xterm);
T sum = init_val;
if((beta == 0) && (xterm == 0))
return init_val;
//
// Backwards recursion first, this is the stable
// direction for recursion:
//
T last_term = 0;
boost::uintmax_t count = k;
for(int i = k; i >= 0; --i)
{
T term = beta * pois;
sum += term;
if(((fabs(term/sum) < errtol) && (last_term >= term)) || (term == 0))
{
count = k - i;
break;
}
pois *= i / l2;
beta += xterm;
xterm *= (a + i - 1) / (x * (a + b + i - 2));
last_term = term;
}
for(int i = k + 1; ; ++i)
{
poisf *= l2 / i;
xtermf *= (x * (a + b + i - 2)) / (a + i - 1);
betaf -= xtermf;
T term = poisf * betaf;
sum += term;
if((fabs(term/sum) < errtol) || (term == 0))
{
break;
}
if(static_cast<boost::uintmax_t>(count + i - k) > max_iter)
{
return policies::raise_evaluation_error(
"cdf(non_central_beta_distribution<%1%>, %1%)",
"Series did not converge, closest value was %1%", sum, pol);
}
}
return sum;
}
template <class T, class Policy>
T non_central_beta_q(T a, T b, T lam, T x, T y, const Policy& pol, T init_val = 0)
{
BOOST_MATH_STD_USING
using namespace boost::math;
//
// Variables come first:
//
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
T errtol = boost::math::policies::get_epsilon<T, Policy>();
T l2 = lam / 2;
//
// k is the starting point for iteration, and is the
// maximum of the poisson weighting term:
//
int k = itrunc(l2);
T pois;
if(k <= 30)
{
//
// Might as well start at 0 since we'll likely have this number of terms anyway:
//
if(a + b > 1)
k = 0;
else if(k == 0)
k = 1;
}
if(k == 0)
{
// Starting Poisson weight:
pois = exp(-l2);
}
else
{
// Starting Poisson weight:
pois = gamma_p_derivative(T(k+1), l2, pol);
}
if(pois == 0)
return init_val;
// recurance term:
T xterm;
// Starting beta term:
T beta = x < y
? detail::ibeta_imp(T(a + k), b, x, pol, true, true, &xterm)
: detail::ibeta_imp(b, T(a + k), y, pol, false, true, &xterm);
xterm *= y / (a + b + k - 1);
T poisf(pois), betaf(beta), xtermf(xterm);
T sum = init_val;
if((beta == 0) && (xterm == 0))
return init_val;
//
// Forwards recursion first, this is the stable
// direction for recursion, and the location
// of the bulk of the sum:
//
T last_term = 0;
boost::uintmax_t count = 0;
for(int i = k + 1; ; ++i)
{
poisf *= l2 / i;
xtermf *= (x * (a + b + i - 2)) / (a + i - 1);
betaf += xtermf;
T term = poisf * betaf;
sum += term;
if((fabs(term/sum) < errtol) && (last_term >= term))
{
count = i - k;
break;
}
if(static_cast<boost::uintmax_t>(i - k) > max_iter)
{
return policies::raise_evaluation_error(
"cdf(non_central_beta_distribution<%1%>, %1%)",
"Series did not converge, closest value was %1%", sum, pol);
}
last_term = term;
}
for(int i = k; i >= 0; --i)
{
T term = beta * pois;
sum += term;
if(fabs(term/sum) < errtol)
{
break;
}
if(static_cast<boost::uintmax_t>(count + k - i) > max_iter)
{
return policies::raise_evaluation_error(
"cdf(non_central_beta_distribution<%1%>, %1%)",
"Series did not converge, closest value was %1%", sum, pol);
}
pois *= i / l2;
beta -= xterm;
xterm *= (a + i - 1) / (x * (a + b + i - 2));
}
return sum;
}
template <class RealType, class Policy>
inline RealType non_central_beta_cdf(RealType x, RealType y, RealType a, RealType b, RealType l, bool invert, const Policy&)
{
typedef typename policies::evaluation<RealType, Policy>::type value_type;
typedef typename policies::normalise<
Policy,
policies::promote_float<false>,
policies::promote_double<false>,
policies::discrete_quantile<>,
policies::assert_undefined<> >::type forwarding_policy;
BOOST_MATH_STD_USING
if(x == 0)
return invert ? 1.0f : 0.0f;
if(y == 0)
return invert ? 0.0f : 1.0f;
value_type result;
value_type c = a + b + l / 2;
value_type cross = 1 - (b / c) * (1 + l / (2 * c * c));
if(l == 0)
result = cdf(boost::math::beta_distribution<RealType, Policy>(a, b), x);
else if(x > cross)
{
// Complement is the smaller of the two:
result = detail::non_central_beta_q(
static_cast<value_type>(a),
static_cast<value_type>(b),
static_cast<value_type>(l),
static_cast<value_type>(x),
static_cast<value_type>(y),
forwarding_policy(),
static_cast<value_type>(invert ? 0 : -1));
invert = !invert;
}
else
{
result = detail::non_central_beta_p(
static_cast<value_type>(a),
static_cast<value_type>(b),
static_cast<value_type>(l),
static_cast<value_type>(x),
static_cast<value_type>(y),
forwarding_policy(),
static_cast<value_type>(invert ? -1 : 0));
}
if(invert)
result = -result;
return policies::checked_narrowing_cast<RealType, forwarding_policy>(
result,
"boost::math::non_central_beta_cdf<%1%>(%1%, %1%, %1%)");
}
template <class T, class Policy>
struct nc_beta_quantile_functor
{
nc_beta_quantile_functor(const non_central_beta_distribution<T,Policy>& d, T t, bool c)
: dist(d), target(t), comp(c) {}
T operator()(const T& x)
{
return comp ?
T(target - cdf(complement(dist, x)))
: T(cdf(dist, x) - target);
}
private:
non_central_beta_distribution<T,Policy> dist;
T target;
bool comp;
};
//
// This is more or less a copy of bracket_and_solve_root, but
// modified to search only the interval [0,1] using similar
// heuristics.
//
template <class F, class T, class Tol, class Policy>
std::pair<T, T> bracket_and_solve_root_01(F f, const T& guess, T factor, bool rising, Tol tol, boost::uintmax_t& max_iter, const Policy& pol)
{
BOOST_MATH_STD_USING
static const char* function = "boost::math::tools::bracket_and_solve_root_01<%1%>";
//
// Set up inital brackets:
//
T a = guess;
T b = a;
T fa = f(a);
T fb = fa;
//
// Set up invocation count:
//
boost::uintmax_t count = max_iter - 1;
if((fa < 0) == (guess < 0 ? !rising : rising))
{
//
// Zero is to the right of b, so walk upwards
// until we find it:
//
while((boost::math::sign)(fb) == (boost::math::sign)(fa))
{
if(count == 0)
{
b = policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", b, pol);
return std::make_pair(a, b);
}
//
// Heuristic: every 20 iterations we double the growth factor in case the
// initial guess was *really* bad !
//
if((max_iter - count) % 20 == 0)
factor *= 2;
//
// Now go ahead and move are guess by "factor",
// we do this by reducing 1-guess by factor:
//
a = b;
fa = fb;
b = 1 - ((1 - b) / factor);
fb = f(b);
--count;
BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
}
}
else
{
//
// Zero is to the left of a, so walk downwards
// until we find it:
//
while((boost::math::sign)(fb) == (boost::math::sign)(fa))
{
if(fabs(a) < tools::min_value<T>())
{
// Escape route just in case the answer is zero!
max_iter -= count;
max_iter += 1;
return a > 0 ? std::make_pair(T(0), T(a)) : std::make_pair(T(a), T(0));
}
if(count == 0)
{
a = policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", a, pol);
return std::make_pair(a, b);
}
//
// Heuristic: every 20 iterations we double the growth factor in case the
// initial guess was *really* bad !
//
if((max_iter - count) % 20 == 0)
factor *= 2;
//
// Now go ahead and move are guess by "factor":
//
b = a;
fb = fa;
a /= factor;
fa = f(a);
--count;
BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
}
}
max_iter -= count;
max_iter += 1;
std::pair<T, T> r = toms748_solve(
f,
(a < 0 ? b : a),
(a < 0 ? a : b),
(a < 0 ? fb : fa),
(a < 0 ? fa : fb),
tol,
count,
pol);
max_iter += count;
BOOST_MATH_INSTRUMENT_CODE("max_iter = " << max_iter << " count = " << count);
return r;
}
template <class RealType, class Policy>
RealType nc_beta_quantile(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& p, bool comp)
{
static const char* function = "quantile(non_central_beta_distribution<%1%>, %1%)";
typedef typename policies::evaluation<RealType, Policy>::type value_type;
typedef typename policies::normalise<
Policy,
policies::promote_float<false>,
policies::promote_double<false>,
policies::discrete_quantile<>,
policies::assert_undefined<> >::type forwarding_policy;
value_type a = dist.alpha();
value_type b = dist.beta();
value_type l = dist.non_centrality();
value_type r;
if(!beta_detail::check_alpha(
function,
a, &r, Policy())
||
!beta_detail::check_beta(
function,
b, &r, Policy())
||
!detail::check_non_centrality(
function,
l,
&r,
Policy())
||
!detail::check_probability(
function,
static_cast<value_type>(p),
&r,
Policy()))
return (RealType)r;
//
// Special cases first:
//
if(p == 0)
return comp
? 1.0f
: 0.0f;
if(p == 1)
return !comp
? 1.0f
: 0.0f;
value_type c = a + b + l / 2;
value_type mean = 1 - (b / c) * (1 + l / (2 * c * c));
/*
//
// Calculate a normal approximation to the quantile,
// uses mean and variance approximations from:
// Algorithm AS 310:
// Computing the Non-Central Beta Distribution Function
// R. Chattamvelli; R. Shanmugam
// Applied Statistics, Vol. 46, No. 1. (1997), pp. 146-156.
//
// Unfortunately, when this is wrong it tends to be *very*
// wrong, so it's disabled for now, even though it often
// gets the initial guess quite close. Probably we could
// do much better by factoring in the skewness if only
// we could calculate it....
//
value_type delta = l / 2;
value_type delta2 = delta * delta;
value_type delta3 = delta * delta2;
value_type delta4 = delta2 * delta2;
value_type G = c * (c + 1) + delta;
value_type alpha = a + b;
value_type alpha2 = alpha * alpha;
value_type eta = (2 * alpha + 1) * (2 * alpha + 1) + 1;
value_type H = 3 * alpha2 + 5 * alpha + 2;
value_type F = alpha2 * (alpha + 1) + H * delta
+ (2 * alpha + 4) * delta2 + delta3;
value_type P = (3 * alpha + 1) * (9 * alpha + 17)
+ 2 * alpha * (3 * alpha + 2) * (3 * alpha + 4) + 15;
value_type Q = 54 * alpha2 + 162 * alpha + 130;
value_type R = 6 * (6 * alpha + 11);
value_type D = delta
* (H * H + 2 * P * delta + Q * delta2 + R * delta3 + 9 * delta4);
value_type variance = (b / G)
* (1 + delta * (l * l + 3 * l + eta) / (G * G))
- (b * b / F) * (1 + D / (F * F));
value_type sd = sqrt(variance);
value_type guess = comp
? quantile(complement(normal_distribution<RealType, Policy>(static_cast<RealType>(mean), static_cast<RealType>(sd)), p))
: quantile(normal_distribution<RealType, Policy>(static_cast<RealType>(mean), static_cast<RealType>(sd)), p);
if(guess >= 1)
guess = mean;
if(guess <= tools::min_value<value_type>())
guess = mean;
*/
value_type guess = mean;
detail::nc_beta_quantile_functor<value_type, Policy>
f(non_central_beta_distribution<value_type, Policy>(a, b, l), p, comp);
tools::eps_tolerance<value_type> tol(policies::digits<RealType, Policy>());
boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
std::pair<value_type, value_type> ir
= bracket_and_solve_root_01(
f, guess, value_type(2.5), true, tol,
max_iter, Policy());
value_type result = ir.first + (ir.second - ir.first) / 2;
if(max_iter >= policies::get_max_root_iterations<Policy>())
{
return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:"
" either there is no answer to quantile of the non central beta distribution"
" or the answer is infinite. Current best guess is %1%",
policies::checked_narrowing_cast<RealType, forwarding_policy>(
result,
function), Policy());
}
return policies::checked_narrowing_cast<RealType, forwarding_policy>(
result,
function);
}
template <class T, class Policy>
T non_central_beta_pdf(T a, T b, T lam, T x, T y, const Policy& pol)
{
BOOST_MATH_STD_USING
//
// Special cases:
//
if((x == 0) || (y == 0))
return 0;
//
// Variables come first:
//
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
T errtol = boost::math::policies::get_epsilon<T, Policy>();
T l2 = lam / 2;
//
// k is the starting point for iteration, and is the
// maximum of the poisson weighting term:
//
int k = itrunc(l2);
// Starting Poisson weight:
T pois = gamma_p_derivative(T(k+1), l2, pol);
// Starting beta term:
T beta = x < y ?
ibeta_derivative(a + k, b, x, pol)
: ibeta_derivative(b, a + k, y, pol);
T sum = 0;
T poisf(pois);
T betaf(beta);
//
// Stable backwards recursion first:
//
boost::uintmax_t count = k;
for(int i = k; i >= 0; --i)
{
T term = beta * pois;
sum += term;
if((fabs(term/sum) < errtol) || (term == 0))
{
count = k - i;
break;
}
pois *= i / l2;
beta *= (a + i - 1) / (x * (a + i + b - 1));
}
for(int i = k + 1; ; ++i)
{
poisf *= l2 / i;
betaf *= x * (a + b + i - 1) / (a + i - 1);
T term = poisf * betaf;
sum += term;
if((fabs(term/sum) < errtol) || (term == 0))
{
break;
}
if(static_cast<boost::uintmax_t>(count + i - k) > max_iter)
{
return policies::raise_evaluation_error(
"pdf(non_central_beta_distribution<%1%>, %1%)",
"Series did not converge, closest value was %1%", sum, pol);
}
}
return sum;
}
template <class RealType, class Policy>
RealType nc_beta_pdf(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& x)
{
BOOST_MATH_STD_USING
static const char* function = "pdf(non_central_beta_distribution<%1%>, %1%)";
typedef typename policies::evaluation<RealType, Policy>::type value_type;
typedef typename policies::normalise<
Policy,
policies::promote_float<false>,
policies::promote_double<false>,
policies::discrete_quantile<>,
policies::assert_undefined<> >::type forwarding_policy;
value_type a = dist.alpha();
value_type b = dist.beta();
value_type l = dist.non_centrality();
value_type r;
if(!beta_detail::check_alpha(
function,
a, &r, Policy())
||
!beta_detail::check_beta(
function,
b, &r, Policy())
||
!detail::check_non_centrality(
function,
l,
&r,
Policy())
||
!beta_detail::check_x(
function,
static_cast<value_type>(x),
&r,
Policy()))
return (RealType)r;
if(l == 0)
return pdf(boost::math::beta_distribution<RealType, Policy>(dist.alpha(), dist.beta()), x);
return policies::checked_narrowing_cast<RealType, forwarding_policy>(
non_central_beta_pdf(a, b, l, static_cast<value_type>(x), value_type(1 - static_cast<value_type>(x)), forwarding_policy()),
"function");
}
template <class T>
struct hypergeometric_2F2_sum
{
typedef T result_type;
hypergeometric_2F2_sum(T a1_, T a2_, T b1_, T b2_, T z_) : a1(a1_), a2(a2_), b1(b1_), b2(b2_), z(z_), term(1), k(0) {}
T operator()()
{
T result = term;
term *= a1 * a2 / (b1 * b2);
a1 += 1;
a2 += 1;
b1 += 1;
b2 += 1;
k += 1;
term /= k;
term *= z;
return result;
}
T a1, a2, b1, b2, z, term, k;
};
template <class T, class Policy>
T hypergeometric_2F2(T a1, T a2, T b1, T b2, T z, const Policy& pol)
{
typedef typename policies::evaluation<T, Policy>::type value_type;
const char* function = "boost::math::detail::hypergeometric_2F2<%1%>(%1%,%1%,%1%,%1%,%1%)";
hypergeometric_2F2_sum<value_type> s(a1, a2, b1, b2, z);
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
#if BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582))
value_type zero = 0;
value_type result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<value_type, Policy>(), max_iter, zero);
#else
value_type result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<value_type, Policy>(), max_iter);
#endif
policies::check_series_iterations<T>(function, max_iter, pol);
return policies::checked_narrowing_cast<T, Policy>(result, function);
}
} // namespace detail
template <class RealType = double, class Policy = policies::policy<> >
class non_central_beta_distribution
{
public:
typedef RealType value_type;
typedef Policy policy_type;
non_central_beta_distribution(RealType a_, RealType b_, RealType lambda) : a(a_), b(b_), ncp(lambda)
{
const char* function = "boost::math::non_central_beta_distribution<%1%>::non_central_beta_distribution(%1%,%1%)";
RealType r;
beta_detail::check_alpha(
function,
a, &r, Policy());
beta_detail::check_beta(
function,
b, &r, Policy());
detail::check_non_centrality(
function,
lambda,
&r,
Policy());
} // non_central_beta_distribution constructor.
RealType alpha() const
{ // Private data getter function.
return a;
}
RealType beta() const
{ // Private data getter function.
return b;
}
RealType non_centrality() const
{ // Private data getter function.
return ncp;
}
private:
// Data member, initialized by constructor.
RealType a; // alpha.
RealType b; // beta.
RealType ncp; // non-centrality parameter
}; // template <class RealType, class Policy> class non_central_beta_distribution
typedef non_central_beta_distribution<double> non_central_beta; // Reserved name of type double.
// Non-member functions to give properties of the distribution.
template <class RealType, class Policy>
inline const std::pair<RealType, RealType> range(const non_central_beta_distribution<RealType, Policy>& /* dist */)
{ // Range of permissible values for random variable k.
using boost::math::tools::max_value;
return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
}
template <class RealType, class Policy>
inline const std::pair<RealType, RealType> support(const non_central_beta_distribution<RealType, Policy>& /* dist */)
{ // Range of supported values for random variable k.
// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
using boost::math::tools::max_value;
return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
}
template <class RealType, class Policy>
inline RealType mode(const non_central_beta_distribution<RealType, Policy>& dist)
{ // mode.
static const char* function = "mode(non_central_beta_distribution<%1%> const&)";
RealType a = dist.alpha();
RealType b = dist.beta();
RealType l = dist.non_centrality();
RealType r;
if(!beta_detail::check_alpha(
function,
a, &r, Policy())
||
!beta_detail::check_beta(
function,
b, &r, Policy())
||
!detail::check_non_centrality(
function,
l,
&r,
Policy()))
return (RealType)r;
RealType c = a + b + l / 2;
RealType mean = 1 - (b / c) * (1 + l / (2 * c * c));
return detail::generic_find_mode_01(
dist,
mean,
function);
}
//
// We don't have the necessary information to implement
// these at present. These are just disabled for now,
// prototypes retained so we can fill in the blanks
// later:
//
template <class RealType, class Policy>
inline RealType mean(const non_central_beta_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING
RealType a = dist.alpha();
RealType b = dist.beta();
RealType d = dist.non_centrality();
RealType apb = a + b;
return exp(-d / 2) * a * detail::hypergeometric_2F2<RealType, Policy>(1 + a, apb, a, 1 + apb, d / 2, Policy()) / apb;
} // mean
template <class RealType, class Policy>
inline RealType variance(const non_central_beta_distribution<RealType, Policy>& dist)
{
//
// Relative error of this function may be arbitarily large... absolute
// error will be small however... that's the best we can do for now.
//
BOOST_MATH_STD_USING
RealType a = dist.alpha();
RealType b = dist.beta();
RealType d = dist.non_centrality();
RealType apb = a + b;
RealType result = detail::hypergeometric_2F2(RealType(1 + a), apb, a, RealType(1 + apb), RealType(d / 2), Policy());
result *= result * -exp(-d) * a * a / (apb * apb);
result += exp(-d / 2) * a * (1 + a) * detail::hypergeometric_2F2(RealType(2 + a), apb, a, RealType(2 + apb), RealType(d / 2), Policy()) / (apb * (1 + apb));
return result;
}
// RealType standard_deviation(const non_central_beta_distribution<RealType, Policy>& dist)
// standard_deviation provided by derived accessors.
template <class RealType, class Policy>
inline RealType skewness(const non_central_beta_distribution<RealType, Policy>& /*dist*/)
{ // skewness = sqrt(l).
const char* function = "boost::math::non_central_beta_distribution<%1%>::skewness()";
typedef typename Policy::assert_undefined_type assert_type;
BOOST_STATIC_ASSERT(assert_type::value == 0);
return policies::raise_evaluation_error<RealType>(
function,
"This function is not yet implemented, the only sensible result is %1%.",
std::numeric_limits<RealType>::quiet_NaN(), Policy()); // infinity?
}
template <class RealType, class Policy>
inline RealType kurtosis_excess(const non_central_beta_distribution<RealType, Policy>& /*dist*/)
{
const char* function = "boost::math::non_central_beta_distribution<%1%>::kurtosis_excess()";
typedef typename Policy::assert_undefined_type assert_type;
BOOST_STATIC_ASSERT(assert_type::value == 0);
return policies::raise_evaluation_error<RealType>(
function,
"This function is not yet implemented, the only sensible result is %1%.",
std::numeric_limits<RealType>::quiet_NaN(), Policy()); // infinity?
} // kurtosis_excess
template <class RealType, class Policy>
inline RealType kurtosis(const non_central_beta_distribution<RealType, Policy>& dist)
{
return kurtosis_excess(dist) + 3;
}
template <class RealType, class Policy>
inline RealType pdf(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& x)
{ // Probability Density/Mass Function.
return detail::nc_beta_pdf(dist, x);
} // pdf
template <class RealType, class Policy>
RealType cdf(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& x)
{
const char* function = "boost::math::non_central_beta_distribution<%1%>::cdf(%1%)";
RealType a = dist.alpha();
RealType b = dist.beta();
RealType l = dist.non_centrality();
RealType r;
if(!beta_detail::check_alpha(
function,
a, &r, Policy())
||
!beta_detail::check_beta(
function,
b, &r, Policy())
||
!detail::check_non_centrality(
function,
l,
&r,
Policy())
||
!beta_detail::check_x(
function,
x,
&r,
Policy()))
return (RealType)r;
if(l == 0)
return cdf(beta_distribution<RealType, Policy>(a, b), x);
return detail::non_central_beta_cdf(x, RealType(1 - x), a, b, l, false, Policy());
} // cdf
template <class RealType, class Policy>
RealType cdf(const complemented2_type<non_central_beta_distribution<RealType, Policy>, RealType>& c)
{ // Complemented Cumulative Distribution Function
const char* function = "boost::math::non_central_beta_distribution<%1%>::cdf(%1%)";
non_central_beta_distribution<RealType, Policy> const& dist = c.dist;
RealType a = dist.alpha();
RealType b = dist.beta();
RealType l = dist.non_centrality();
RealType x = c.param;
RealType r;
if(!beta_detail::check_alpha(
function,
a, &r, Policy())
||
!beta_detail::check_beta(
function,
b, &r, Policy())
||
!detail::check_non_centrality(
function,
l,
&r,
Policy())
||
!beta_detail::check_x(
function,
x,
&r,
Policy()))
return (RealType)r;
if(l == 0)
return cdf(complement(beta_distribution<RealType, Policy>(a, b), x));
return detail::non_central_beta_cdf(x, RealType(1 - x), a, b, l, true, Policy());
} // ccdf
template <class RealType, class Policy>
inline RealType quantile(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& p)
{ // Quantile (or Percent Point) function.
return detail::nc_beta_quantile(dist, p, false);
} // quantile
template <class RealType, class Policy>
inline RealType quantile(const complemented2_type<non_central_beta_distribution<RealType, Policy>, RealType>& c)
{ // Quantile (or Percent Point) function.
return detail::nc_beta_quantile(c.dist, c.param, true);
} // quantile complement.
} // namespace math
} // namespace boost
// This include must be at the end, *after* the accessors
// for this distribution have been defined, in order to
// keep compilers that support two-phase lookup happy.
#include <boost/math/distributions/detail/derived_accessors.hpp>
#endif // BOOST_MATH_SPECIAL_NON_CENTRAL_BETA_HPP
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