Chris@16: // boost\math\distributions\non_central_t.hpp
Chris@16: 
Chris@16: // Copyright John Maddock 2008.
Chris@16: 
Chris@16: // Use, modification and distribution are subject to the
Chris@16: // Boost Software License, Version 1.0.
Chris@16: // (See accompanying file LICENSE_1_0.txt
Chris@16: // or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_SPECIAL_NON_CENTRAL_T_HPP
Chris@16: #define BOOST_MATH_SPECIAL_NON_CENTRAL_T_HPP
Chris@16: 
Chris@16: #include <boost/math/distributions/fwd.hpp>
Chris@16: #include <boost/math/distributions/non_central_beta.hpp> // for nc beta
Chris@16: #include <boost/math/distributions/normal.hpp> // for normal CDF and quantile
Chris@16: #include <boost/math/distributions/students_t.hpp>
Chris@16: #include <boost/math/distributions/detail/generic_quantile.hpp> // quantile
Chris@16: 
Chris@16: namespace boost
Chris@16: {
Chris@16:    namespace math
Chris@16:    {
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       class non_central_t_distribution;
Chris@16: 
Chris@16:       namespace detail{
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T non_central_t2_p(T v, T delta, T x, T y, const Policy& pol, T init_val)
Chris@16:          {
Chris@16:             BOOST_MATH_STD_USING
Chris@16:             //
Chris@16:             // Variables come first:
Chris@16:             //
Chris@16:             boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
Chris@16:             T errtol = policies::get_epsilon<T, Policy>();
Chris@16:             T d2 = delta * delta / 2;
Chris@16:             //
Chris@16:             // k is the starting point for iteration, and is the
Chris@16:             // maximum of the poisson weighting term, we don't
Chris@16:             // ever allow k == 0 as this can lead to catastrophic
Chris@16:             // cancellation errors later (test case is v = 1621286869049072.3
Chris@16:             // delta = 0.16212868690490723, x = 0.86987415482475994).
Chris@16:             //
Chris@16:             int k = itrunc(d2);
Chris@16:             T pois;
Chris@16:             if(k == 0) k = 1;
Chris@16:             // Starting Poisson weight:
Chris@16:             pois = gamma_p_derivative(T(k+1), d2, pol) 
Chris@16:                * tgamma_delta_ratio(T(k + 1), T(0.5f))
Chris@16:                * delta / constants::root_two<T>();
Chris@16:             if(pois == 0)
Chris@16:                return init_val;
Chris@16:             T xterm, beta;
Chris@16:             // Recurrance & starting beta terms:
Chris@16:             beta = x < y
Chris@16:                ? detail::ibeta_imp(T(k + 1), T(v / 2), x, pol, false, true, &xterm)
Chris@16:                : detail::ibeta_imp(T(v / 2), T(k + 1), y, pol, true, true, &xterm);
Chris@16:             xterm *= y / (v / 2 + k);
Chris@16:             T poisf(pois), betaf(beta), xtermf(xterm);
Chris@16:             T sum = init_val;
Chris@16:             if((xterm == 0) && (beta == 0))
Chris@16:                return init_val;
Chris@16: 
Chris@16:             //
Chris@16:             // Backwards recursion first, this is the stable
Chris@16:             // direction for recursion:
Chris@16:             //
Chris@16:             boost::uintmax_t count = 0;
Chris@16:             T last_term = 0;
Chris@16:             for(int i = k; i >= 0; --i)
Chris@16:             {
Chris@16:                T term = beta * pois;
Chris@16:                sum += term;
Chris@16:                // Don't terminate on first term in case we "fixed" k above:
Chris@16:                if((fabs(last_term) > fabs(term)) && fabs(term/sum) < errtol)
Chris@16:                   break;
Chris@16:                last_term = term;
Chris@16:                pois *= (i + 0.5f) / d2;
Chris@16:                beta += xterm;
Chris@16:                xterm *= (i) / (x * (v / 2 + i - 1));
Chris@16:                ++count;
Chris@16:             }
Chris@16:             last_term = 0;
Chris@16:             for(int i = k + 1; ; ++i)
Chris@16:             {
Chris@16:                poisf *= d2 / (i + 0.5f);
Chris@16:                xtermf *= (x * (v / 2 + i - 1)) / (i);
Chris@16:                betaf -= xtermf;
Chris@16:                T term = poisf * betaf;
Chris@16:                sum += term;
Chris@16:                if((fabs(last_term) > fabs(term)) && (fabs(term/sum) < errtol))
Chris@16:                   break;
Chris@16:                last_term = term;
Chris@16:                ++count;
Chris@16:                if(count > max_iter)
Chris@16:                {
Chris@16:                   return policies::raise_evaluation_error(
Chris@16:                      "cdf(non_central_t_distribution<%1%>, %1%)", 
Chris@16:                      "Series did not converge, closest value was %1%", sum, pol);
Chris@16:                }
Chris@16:             }
Chris@16:             return sum;
Chris@16:          }
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T non_central_t2_q(T v, T delta, T x, T y, const Policy& pol, T init_val)
Chris@16:          {
Chris@16:             BOOST_MATH_STD_USING
Chris@16:             //
Chris@16:             // Variables come first:
Chris@16:             //
Chris@16:             boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
Chris@16:             T errtol = boost::math::policies::get_epsilon<T, Policy>();
Chris@16:             T d2 = delta * delta / 2;
Chris@16:             //
Chris@16:             // k is the starting point for iteration, and is the
Chris@16:             // maximum of the poisson weighting term, we don't allow
Chris@16:             // k == 0 as this can cause catastrophic cancellation errors
Chris@16:             // (test case is v = 561908036470413.25, delta = 0.056190803647041321,
Chris@16:             // x = 1.6155232703966216):
Chris@16:             //
Chris@16:             int k = itrunc(d2);
Chris@16:             if(k == 0) k = 1;
Chris@16:             // Starting Poisson weight:
Chris@16:             T pois;
Chris@16:             if((k < (int)(max_factorial<T>::value)) && (d2 < tools::log_max_value<T>()) && (log(d2) * k < tools::log_max_value<T>()))
Chris@16:             {
Chris@16:                //
Chris@16:                // For small k we can optimise this calculation by using
Chris@16:                // a simpler reduced formula:
Chris@16:                //
Chris@16:                pois = exp(-d2);
Chris@16:                pois *= pow(d2, static_cast<T>(k));
Chris@16:                pois /= boost::math::tgamma(T(k + 1 + 0.5), pol);
Chris@16:                pois *= delta / constants::root_two<T>();
Chris@16:             }
Chris@16:             else
Chris@16:             {
Chris@16:                pois = gamma_p_derivative(T(k+1), d2, pol) 
Chris@16:                   * tgamma_delta_ratio(T(k + 1), T(0.5f))
Chris@16:                   * delta / constants::root_two<T>();
Chris@16:             }
Chris@16:             if(pois == 0)
Chris@16:                return init_val;
Chris@16:             // Recurance term:
Chris@16:             T xterm;
Chris@16:             T beta;
Chris@16:             // Starting beta term:
Chris@16:             if(k != 0)
Chris@16:             {
Chris@16:                beta = x < y 
Chris@16:                   ? detail::ibeta_imp(T(k + 1), T(v / 2), x, pol, true, true, &xterm) 
Chris@16:                   : detail::ibeta_imp(T(v / 2), T(k + 1), y, pol, false, true, &xterm);
Chris@16: 
Chris@16:                xterm *= y / (v / 2 + k);
Chris@16:             }
Chris@16:             else
Chris@16:             {
Chris@16:                beta = pow(y, v / 2);
Chris@16:                xterm = beta;
Chris@16:             }
Chris@16:             T poisf(pois), betaf(beta), xtermf(xterm);
Chris@16:             T sum = init_val;
Chris@16:             if((xterm == 0) && (beta == 0))
Chris@16:                return init_val;
Chris@16: 
Chris@16:             //
Chris@16:             // Fused forward and backwards recursion:
Chris@16:             //
Chris@16:             boost::uintmax_t count = 0;
Chris@16:             T last_term = 0;
Chris@16:             for(int i = k + 1, j = k; ; ++i, --j)
Chris@16:             {
Chris@16:                poisf *= d2 / (i + 0.5f);
Chris@16:                xtermf *= (x * (v / 2 + i - 1)) / (i);
Chris@16:                betaf += xtermf;
Chris@16:                T term = poisf * betaf;
Chris@16: 
Chris@16:                if(j >= 0)
Chris@16:                {
Chris@16:                   term += beta * pois;
Chris@16:                   pois *= (j + 0.5f) / d2;
Chris@16:                   beta -= xterm;
Chris@16:                   xterm *= (j) / (x * (v / 2 + j - 1));
Chris@16:                }
Chris@16: 
Chris@16:                sum += term;
Chris@16:                // Don't terminate on first term in case we "fixed" the value of k above:
Chris@16:                if((fabs(last_term) > fabs(term)) && fabs(term/sum) < errtol)
Chris@16:                   break;
Chris@16:                last_term = term;
Chris@16:                if(count > max_iter)
Chris@16:                {
Chris@16:                   return policies::raise_evaluation_error(
Chris@16:                      "cdf(non_central_t_distribution<%1%>, %1%)", 
Chris@16:                      "Series did not converge, closest value was %1%", sum, pol);
Chris@16:                }
Chris@16:                ++count;
Chris@16:             }
Chris@16:             return sum;
Chris@16:          }
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T non_central_t_cdf(T v, T delta, T t, bool invert, const Policy& pol)
Chris@16:          {
Chris@16:             BOOST_MATH_STD_USING
Chris@16:             if ((boost::math::isinf)(v))
Chris@16:             { // Infinite degrees of freedom, so use normal distribution located at delta.
Chris@16:                normal_distribution<T, Policy> n(delta, 1); 
Chris@16:                return cdf(n, t);
Chris@16:             }
Chris@16:             //
Chris@16:             // Otherwise, for t < 0 we have to use the reflection formula:
Chris@16:             if(t < 0)
Chris@16:             {
Chris@16:                t = -t;
Chris@16:                delta = -delta;
Chris@16:                invert = !invert;
Chris@16:             }
Chris@16:             if(fabs(delta / (4 * v)) < policies::get_epsilon<T, Policy>())
Chris@16:             {
Chris@16:                // Approximate with a Student's T centred on delta,
Chris@16:                // the crossover point is based on eq 2.6 from
Chris@16:                // "A Comparison of Approximations To Percentiles of the
Chris@16:                // Noncentral t-Distribution".  H. Sahai and M. M. Ojeda,
Chris@16:                // Revista Investigacion Operacional Vol 21, No 2, 2000.
Chris@16:                // Original sources referenced in the above are:
Chris@16:                // "Some Approximations to the Percentage Points of the Noncentral
Chris@16:                // t-Distribution". C. van Eeden. International Statistical Review, 29, 4-31.
Chris@16:                // "Continuous Univariate Distributions".  N.L. Johnson, S. Kotz and
Chris@16:                // N. Balkrishnan. 1995. John Wiley and Sons New York.
Chris@16:                T result = cdf(students_t_distribution<T, Policy>(v), t - delta);
Chris@16:                return invert ? 1 - result : result;
Chris@16:             }
Chris@16:             //
Chris@16:             // x and y are the corresponding random
Chris@16:             // variables for the noncentral beta distribution,
Chris@16:             // with y = 1 - x:
Chris@16:             //
Chris@16:             T x = t * t / (v + t * t);
Chris@16:             T y = v / (v + t * t);
Chris@16:             T d2 = delta * delta;
Chris@16:             T a = 0.5f;
Chris@16:             T b = v / 2;
Chris@16:             T c = a + b + d2 / 2;
Chris@16:             //
Chris@16:             // Crossover point for calculating p or q is the same
Chris@16:             // as for the noncentral beta:
Chris@16:             //
Chris@16:             T cross = 1 - (b / c) * (1 + d2 / (2 * c * c));
Chris@16:             T result;
Chris@16:             if(x < cross)
Chris@16:             {
Chris@16:                //
Chris@16:                // Calculate p:
Chris@16:                //
Chris@16:                if(x != 0)
Chris@16:                {
Chris@16:                   result = non_central_beta_p(a, b, d2, x, y, pol);
Chris@16:                   result = non_central_t2_p(v, delta, x, y, pol, result);
Chris@16:                   result /= 2;
Chris@16:                }
Chris@16:                else
Chris@16:                   result = 0;
Chris@16:                result += cdf(boost::math::normal_distribution<T, Policy>(), -delta);
Chris@16:             }
Chris@16:             else
Chris@16:             {
Chris@16:                //
Chris@16:                // Calculate q:
Chris@16:                //
Chris@16:                invert = !invert;
Chris@16:                if(x != 0)
Chris@16:                {
Chris@16:                   result = non_central_beta_q(a, b, d2, x, y, pol);
Chris@16:                   result = non_central_t2_q(v, delta, x, y, pol, result);
Chris@16:                   result /= 2;
Chris@16:                }
Chris@16:                else // x == 0
Chris@16:                   result = cdf(complement(boost::math::normal_distribution<T, Policy>(), -delta));
Chris@16:             }
Chris@16:             if(invert)
Chris@16:                result = 1 - result;
Chris@16:             return result;
Chris@16:          }
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T non_central_t_quantile(const char* function, T v, T delta, T p, T q, const Policy&)
Chris@16:          {
Chris@16:             BOOST_MATH_STD_USING
Chris@16:      //       static const char* function = "quantile(non_central_t_distribution<%1%>, %1%)";
Chris@16:      // now passed as function
Chris@16:             typedef typename policies::evaluation<T, Policy>::type value_type;
Chris@16:             typedef typename policies::normalise<
Chris@16:                Policy, 
Chris@16:                policies::promote_float<false>, 
Chris@16:                policies::promote_double<false>, 
Chris@16:                policies::discrete_quantile<>,
Chris@16:                policies::assert_undefined<> >::type forwarding_policy;
Chris@16: 
Chris@16:                T r;
Chris@16:                if(!detail::check_df_gt0_to_inf(
Chris@16:                   function,
Chris@16:                   v, &r, Policy())
Chris@16:                   ||
Chris@16:                !detail::check_finite(
Chris@16:                   function,
Chris@16:                   delta,
Chris@16:                   &r,
Chris@16:                   Policy())
Chris@16:                   ||
Chris@16:                !detail::check_probability(
Chris@16:                   function,
Chris@16:                   p,
Chris@16:                   &r,
Chris@16:                   Policy()))
Chris@16:                      return r;
Chris@16: 
Chris@16: 
Chris@16:             value_type guess = 0;
Chris@16:             if ( ((boost::math::isinf)(v)) || (v > 1 / boost::math::tools::epsilon<T>()) )
Chris@16:             { // Infinite or very large degrees of freedom, so use normal distribution located at delta.
Chris@16:                normal_distribution<T, Policy> n(delta, 1);
Chris@16:                if (p < q)
Chris@16:                {
Chris@16:                  return quantile(n, p);
Chris@16:                }
Chris@16:                else
Chris@16:                {
Chris@16:                  return quantile(complement(n, q));
Chris@16:                }
Chris@16:             }
Chris@16:             else if(v > 3)
Chris@16:             { // Use normal distribution to calculate guess.
Chris@16:                value_type mean = (v > 1 / policies::get_epsilon<T, Policy>()) ? delta : delta * sqrt(v / 2) * tgamma_delta_ratio((v - 1) * 0.5f, T(0.5f));
Chris@16:                value_type var = (v > 1 / policies::get_epsilon<T, Policy>()) ? value_type(1) : (((delta * delta + 1) * v) / (v - 2) - mean * mean);
Chris@16:                if(p < q)
Chris@16:                   guess = quantile(normal_distribution<value_type, forwarding_policy>(mean, var), p);
Chris@16:                else
Chris@16:                   guess = quantile(complement(normal_distribution<value_type, forwarding_policy>(mean, var), q));
Chris@16:             }
Chris@16:             //
Chris@16:             // We *must* get the sign of the initial guess correct, 
Chris@16:             // or our root-finder will fail, so double check it now:
Chris@16:             //
Chris@16:             value_type pzero = non_central_t_cdf(
Chris@16:                static_cast<value_type>(v), 
Chris@16:                static_cast<value_type>(delta), 
Chris@16:                static_cast<value_type>(0), 
Chris@16:                !(p < q), 
Chris@16:                forwarding_policy());
Chris@16:             int s;
Chris@16:             if(p < q)
Chris@16:                s = boost::math::sign(p - pzero);
Chris@16:             else
Chris@16:                s = boost::math::sign(pzero - q);
Chris@16:             if(s != boost::math::sign(guess))
Chris@16:             {
Chris@16:                guess = s;
Chris@16:             }
Chris@16: 
Chris@16:             value_type result = detail::generic_quantile(
Chris@16:                non_central_t_distribution<value_type, forwarding_policy>(v, delta), 
Chris@16:                (p < q ? p : q), 
Chris@16:                guess, 
Chris@16:                (p >= q), 
Chris@16:                function);
Chris@16:             return policies::checked_narrowing_cast<T, forwarding_policy>(
Chris@16:                result, 
Chris@16:                function);
Chris@16:          }
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T non_central_t2_pdf(T n, T delta, T x, T y, const Policy& pol, T init_val)
Chris@16:          {
Chris@16:             BOOST_MATH_STD_USING
Chris@16:             //
Chris@16:             // Variables come first:
Chris@16:             //
Chris@16:             boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
Chris@16:             T errtol = boost::math::policies::get_epsilon<T, Policy>();
Chris@16:             T d2 = delta * delta / 2;
Chris@16:             //
Chris@16:             // k is the starting point for iteration, and is the
Chris@16:             // maximum of the poisson weighting term:
Chris@16:             //
Chris@16:             int k = itrunc(d2);
Chris@16:             T pois, xterm;
Chris@16:             if(k == 0)
Chris@16:                k = 1;
Chris@16:             // Starting Poisson weight:
Chris@16:             pois = gamma_p_derivative(T(k+1), d2, pol) 
Chris@16:                * tgamma_delta_ratio(T(k + 1), T(0.5f))
Chris@16:                * delta / constants::root_two<T>();
Chris@16:             // Starting beta term:
Chris@16:             xterm = x < y
Chris@16:                ? ibeta_derivative(T(k + 1), n / 2, x, pol)
Chris@16:                : ibeta_derivative(n / 2, T(k + 1), y, pol);
Chris@16:             T poisf(pois), xtermf(xterm);
Chris@16:             T sum = init_val;
Chris@16:             if((pois == 0) || (xterm == 0))
Chris@16:                return init_val;
Chris@16: 
Chris@16:             //
Chris@16:             // Backwards recursion first, this is the stable
Chris@16:             // direction for recursion:
Chris@16:             //
Chris@16:             boost::uintmax_t count = 0;
Chris@16:             for(int i = k; i >= 0; --i)
Chris@16:             {
Chris@16:                T term = xterm * pois;
Chris@16:                sum += term;
Chris@16:                if(((fabs(term/sum) < errtol) && (i != k)) || (term == 0))
Chris@16:                   break;
Chris@16:                pois *= (i + 0.5f) / d2;
Chris@16:                xterm *= (i) / (x * (n / 2 + i));
Chris@16:                ++count;
Chris@16:                if(count > max_iter)
Chris@16:                {
Chris@16:                   return policies::raise_evaluation_error(
Chris@16:                      "pdf(non_central_t_distribution<%1%>, %1%)", 
Chris@16:                      "Series did not converge, closest value was %1%", sum, pol);
Chris@16:                }
Chris@16:             }
Chris@16:             for(int i = k + 1; ; ++i)
Chris@16:             {
Chris@16:                poisf *= d2 / (i + 0.5f);
Chris@16:                xtermf *= (x * (n / 2 + i)) / (i);
Chris@16:                T term = poisf * xtermf;
Chris@16:                sum += term;
Chris@16:                if((fabs(term/sum) < errtol) || (term == 0))
Chris@16:                   break;
Chris@16:                ++count;
Chris@16:                if(count > max_iter)
Chris@16:                {
Chris@16:                   return policies::raise_evaluation_error(
Chris@16:                      "pdf(non_central_t_distribution<%1%>, %1%)", 
Chris@16:                      "Series did not converge, closest value was %1%", sum, pol);
Chris@16:                }
Chris@16:             }
Chris@16:             return sum;
Chris@16:          }
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T non_central_t_pdf(T n, T delta, T t, const Policy& pol)
Chris@16:          {
Chris@16:             BOOST_MATH_STD_USING
Chris@16:             if ((boost::math::isinf)(n))
Chris@16:             { // Infinite degrees of freedom, so use normal distribution located at delta.
Chris@16:                normal_distribution<T, Policy> norm(delta, 1); 
Chris@16:                return pdf(norm, t);
Chris@16:             }
Chris@16:             //
Chris@16:             // Otherwise, for t < 0 we have to use the reflection formula:
Chris@16:             if(t < 0)
Chris@16:             {
Chris@16:                t = -t;
Chris@16:                delta = -delta;
Chris@16:             }
Chris@16:             if(t == 0)
Chris@16:             {
Chris@16:                //
Chris@16:                // Handle this as a special case, using the formula
Chris@16:                // from Weisstein, Eric W. 
Chris@16:                // "Noncentral Student's t-Distribution." 
Chris@16:                // From MathWorld--A Wolfram Web Resource. 
Chris@16:                // http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html 
Chris@16:                // 
Chris@16:                // The formula is simplified thanks to the relation
Chris@16:                // 1F1(a,b,0) = 1.
Chris@16:                //
Chris@16:                return tgamma_delta_ratio(n / 2 + 0.5f, T(0.5f))
Chris@16:                   * sqrt(n / constants::pi<T>()) 
Chris@16:                   * exp(-delta * delta / 2) / 2;
Chris@16:             }
Chris@16:             if(fabs(delta / (4 * n)) < policies::get_epsilon<T, Policy>())
Chris@16:             {
Chris@16:                // Approximate with a Student's T centred on delta,
Chris@16:                // the crossover point is based on eq 2.6 from
Chris@16:                // "A Comparison of Approximations To Percentiles of the
Chris@16:                // Noncentral t-Distribution".  H. Sahai and M. M. Ojeda,
Chris@16:                // Revista Investigacion Operacional Vol 21, No 2, 2000.
Chris@16:                // Original sources referenced in the above are:
Chris@16:                // "Some Approximations to the Percentage Points of the Noncentral
Chris@16:                // t-Distribution". C. van Eeden. International Statistical Review, 29, 4-31.
Chris@16:                // "Continuous Univariate Distributions".  N.L. Johnson, S. Kotz and
Chris@16:                // N. Balkrishnan. 1995. John Wiley and Sons New York.
Chris@16:                return pdf(students_t_distribution<T, Policy>(n), t - delta);
Chris@16:             }
Chris@16:             //
Chris@16:             // x and y are the corresponding random
Chris@16:             // variables for the noncentral beta distribution,
Chris@16:             // with y = 1 - x:
Chris@16:             //
Chris@16:             T x = t * t / (n + t * t);
Chris@16:             T y = n / (n + t * t);
Chris@16:             T a = 0.5f;
Chris@16:             T b = n / 2;
Chris@16:             T d2 = delta * delta;
Chris@16:             //
Chris@16:             // Calculate pdf:
Chris@16:             //
Chris@16:             T dt = n * t / (n * n + 2 * n * t * t + t * t * t * t);
Chris@16:             T result = non_central_beta_pdf(a, b, d2, x, y, pol);
Chris@16:             T tol = tools::epsilon<T>() * result * 500;
Chris@16:             result = non_central_t2_pdf(n, delta, x, y, pol, result);
Chris@16:             if(result <= tol)
Chris@16:                result = 0;
Chris@16:             result *= dt;
Chris@16:             return result;
Chris@16:          }
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T mean(T v, T delta, const Policy& pol)
Chris@16:          {
Chris@16:             if ((boost::math::isinf)(v))
Chris@16:             {
Chris@16:                return delta;
Chris@16:             }
Chris@16:             BOOST_MATH_STD_USING
Chris@16:             if (v > 1 / boost::math::tools::epsilon<T>() )
Chris@16:             {
Chris@16:               //normal_distribution<T, Policy> n(delta, 1);
Chris@16:               //return boost::math::mean(n); 
Chris@16:               return delta;
Chris@16:             }
Chris@16:             else
Chris@16:             {
Chris@16:              return delta * sqrt(v / 2) * tgamma_delta_ratio((v - 1) * 0.5f, T(0.5f), pol);
Chris@16:             }
Chris@16:             // Other moments use mean so using normal distribution is propagated.
Chris@16:          }
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T variance(T v, T delta, const Policy& pol)
Chris@16:          {
Chris@16:             if ((boost::math::isinf)(v))
Chris@16:             {
Chris@16:                return 1;
Chris@16:             }
Chris@16:             if (delta == 0)
Chris@16:             {  // == Student's t
Chris@16:               return v / (v - 2);
Chris@16:             }
Chris@16:             T result = ((delta * delta + 1) * v) / (v - 2);
Chris@16:             T m = mean(v, delta, pol);
Chris@16:             result -= m * m;
Chris@16:             return result;
Chris@16:          }
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T skewness(T v, T delta, const Policy& pol)
Chris@16:          {
Chris@16:             BOOST_MATH_STD_USING
Chris@16:             if ((boost::math::isinf)(v))
Chris@16:             {
Chris@16:                return 0;
Chris@16:             }
Chris@16:             if(delta == 0)
Chris@16:             { // == Student's t
Chris@16:               return 0;
Chris@16:             }
Chris@16:             T mean = boost::math::detail::mean(v, delta, pol);
Chris@16:             T l2 = delta * delta;
Chris@16:             T var = ((l2 + 1) * v) / (v - 2) - mean * mean;
Chris@16:             T result = -2 * var;
Chris@16:             result += v * (l2 + 2 * v - 3) / ((v - 3) * (v - 2));
Chris@16:             result *= mean;
Chris@16:             result /= pow(var, T(1.5f));
Chris@16:             return result;
Chris@16:          }
Chris@16: 
Chris@16:          template <class T, class Policy>
Chris@16:          T kurtosis_excess(T v, T delta, const Policy& pol)
Chris@16:          {
Chris@16:             BOOST_MATH_STD_USING
Chris@16:             if ((boost::math::isinf)(v))
Chris@16:             {
Chris@16:                return 3;
Chris@16:             }
Chris@16:             if (delta == 0)
Chris@16:             { // == Student's t
Chris@16:               return 3;
Chris@16:             }
Chris@16:             T mean = boost::math::detail::mean(v, delta, pol);
Chris@16:             T l2 = delta * delta;
Chris@16:             T var = ((l2 + 1) * v) / (v - 2) - mean * mean;
Chris@16:             T result = -3 * var;
Chris@16:             result += v * (l2 * (v + 1) + 3 * (3 * v - 5)) / ((v - 3) * (v - 2));
Chris@16:             result *= -mean * mean;
Chris@16:             result += v * v * (l2 * l2 + 6 * l2 + 3) / ((v - 4) * (v - 2));
Chris@16:             result /= var * var;
Chris@16:             return result;
Chris@16:          }
Chris@16: 
Chris@16: #if 0
Chris@16:          // 
Chris@16:          // This code is disabled, since there can be multiple answers to the
Chris@16:          // question, and it's not clear how to find the "right" one.
Chris@16:          //
Chris@16:          template <class RealType, class Policy>
Chris@16:          struct t_degrees_of_freedom_finder
Chris@16:          {
Chris@16:             t_degrees_of_freedom_finder(
Chris@16:                RealType delta_, RealType x_, RealType p_, bool c)
Chris@16:                : delta(delta_), x(x_), p(p_), comp(c) {}
Chris@16: 
Chris@16:             RealType operator()(const RealType& v)
Chris@16:             {
Chris@16:                non_central_t_distribution<RealType, Policy> d(v, delta);
Chris@16:                return comp ?
Chris@16:                   p - cdf(complement(d, x))
Chris@16:                   : cdf(d, x) - p;
Chris@16:             }
Chris@16:          private:
Chris@16:             RealType delta;
Chris@16:             RealType x;
Chris@16:             RealType p;
Chris@16:             bool comp;
Chris@16:          };
Chris@16: 
Chris@16:          template <class RealType, class Policy>
Chris@16:          inline RealType find_t_degrees_of_freedom(
Chris@16:             RealType delta, RealType x, RealType p, RealType q, const Policy& pol)
Chris@16:          {
Chris@16:             const char* function = "non_central_t<%1%>::find_degrees_of_freedom";
Chris@16:             if((p == 0) || (q == 0))
Chris@16:             {
Chris@16:                //
Chris@16:                // Can't a thing if one of p and q is zero:
Chris@16:                //
Chris@16:                return policies::raise_evaluation_error<RealType>(function, 
Chris@16:                   "Can't find degrees of freedom when the probability is 0 or 1, only possible answer is %1%", 
Chris@16:                   RealType(std::numeric_limits<RealType>::quiet_NaN()), Policy());
Chris@16:             }
Chris@16:             t_degrees_of_freedom_finder<RealType, Policy> f(delta, x, p < q ? p : q, p < q ? false : true);
Chris@16:             tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>());
Chris@16:             boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
Chris@16:             //
Chris@16:             // Pick an initial guess:
Chris@16:             //
Chris@16:             RealType guess = 200;
Chris@16:             std::pair<RealType, RealType> ir = tools::bracket_and_solve_root(
Chris@16:                f, guess, RealType(2), false, tol, max_iter, pol);
Chris@16:             RealType result = ir.first + (ir.second - ir.first) / 2;
Chris@16:             if(max_iter >= policies::get_max_root_iterations<Policy>())
Chris@16:             {
Chris@101:                return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:"
Chris@16:                   " or there is no answer to problem.  Current best guess is %1%", result, Policy());
Chris@16:             }
Chris@16:             return result;
Chris@16:          }
Chris@16: 
Chris@16:          template <class RealType, class Policy>
Chris@16:          struct t_non_centrality_finder
Chris@16:          {
Chris@16:             t_non_centrality_finder(
Chris@16:                RealType v_, RealType x_, RealType p_, bool c)
Chris@16:                : v(v_), x(x_), p(p_), comp(c) {}
Chris@16: 
Chris@16:             RealType operator()(const RealType& delta)
Chris@16:             {
Chris@16:                non_central_t_distribution<RealType, Policy> d(v, delta);
Chris@16:                return comp ?
Chris@16:                   p - cdf(complement(d, x))
Chris@16:                   : cdf(d, x) - p;
Chris@16:             }
Chris@16:          private:
Chris@16:             RealType v;
Chris@16:             RealType x;
Chris@16:             RealType p;
Chris@16:             bool comp;
Chris@16:          };
Chris@16: 
Chris@16:          template <class RealType, class Policy>
Chris@16:          inline RealType find_t_non_centrality(
Chris@16:             RealType v, RealType x, RealType p, RealType q, const Policy& pol)
Chris@16:          {
Chris@16:             const char* function = "non_central_t<%1%>::find_t_non_centrality";
Chris@16:             if((p == 0) || (q == 0))
Chris@16:             {
Chris@16:                //
Chris@16:                // Can't do a thing if one of p and q is zero:
Chris@16:                //
Chris@16:                return policies::raise_evaluation_error<RealType>(function, 
Chris@16:                   "Can't find non-centrality parameter when the probability is 0 or 1, only possible answer is %1%", 
Chris@16:                   RealType(std::numeric_limits<RealType>::quiet_NaN()), Policy());
Chris@16:             }
Chris@16:             t_non_centrality_finder<RealType, Policy> f(v, x, p < q ? p : q, p < q ? false : true);
Chris@16:             tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>());
Chris@16:             boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
Chris@16:             //
Chris@16:             // Pick an initial guess that we know is the right side of
Chris@16:             // zero:
Chris@16:             //
Chris@16:             RealType guess;
Chris@16:             if(f(0) < 0)
Chris@16:                guess = 1;
Chris@16:             else
Chris@16:                guess = -1;
Chris@16:             std::pair<RealType, RealType> ir = tools::bracket_and_solve_root(
Chris@16:                f, guess, RealType(2), false, tol, max_iter, pol);
Chris@16:             RealType result = ir.first + (ir.second - ir.first) / 2;
Chris@16:             if(max_iter >= policies::get_max_root_iterations<Policy>())
Chris@16:             {
Chris@101:                return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:"
Chris@16:                   " or there is no answer to problem.  Current best guess is %1%", result, Policy());
Chris@16:             }
Chris@16:             return result;
Chris@16:          }
Chris@16: #endif
Chris@16:       } // namespace detail ======================================================================
Chris@16: 
Chris@16:       template <class RealType = double, class Policy = policies::policy<> >
Chris@16:       class non_central_t_distribution
Chris@16:       {
Chris@16:       public:
Chris@16:          typedef RealType value_type;
Chris@16:          typedef Policy policy_type;
Chris@16: 
Chris@16:          non_central_t_distribution(RealType v_, RealType lambda) : v(v_), ncp(lambda)
Chris@16:          { 
Chris@16:             const char* function = "boost::math::non_central_t_distribution<%1%>::non_central_t_distribution(%1%,%1%)";
Chris@16:             RealType r;
Chris@16:             detail::check_df_gt0_to_inf(
Chris@16:                function,
Chris@16:                v, &r, Policy());
Chris@16:             detail::check_finite(
Chris@16:                function,
Chris@16:                lambda,
Chris@16:                &r,
Chris@16:                Policy());
Chris@16:          } // non_central_t_distribution constructor.
Chris@16: 
Chris@16:          RealType degrees_of_freedom() const
Chris@16:          { // Private data getter function.
Chris@16:             return v;
Chris@16:          }
Chris@16:          RealType non_centrality() const
Chris@16:          { // Private data getter function.
Chris@16:             return ncp;
Chris@16:          }
Chris@16: #if 0
Chris@16:          // 
Chris@16:          // This code is disabled, since there can be multiple answers to the
Chris@16:          // question, and it's not clear how to find the "right" one.
Chris@16:          //
Chris@16:          static RealType find_degrees_of_freedom(RealType delta, RealType x, RealType p)
Chris@16:          {
Chris@16:             const char* function = "non_central_t<%1%>::find_degrees_of_freedom";
Chris@16:             typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:             typedef typename policies::normalise<
Chris@16:                Policy, 
Chris@16:                policies::promote_float<false>, 
Chris@16:                policies::promote_double<false>, 
Chris@16:                policies::discrete_quantile<>,
Chris@16:                policies::assert_undefined<> >::type forwarding_policy;
Chris@16:             value_type result = detail::find_t_degrees_of_freedom(
Chris@16:                static_cast<value_type>(delta), 
Chris@16:                static_cast<value_type>(x), 
Chris@16:                static_cast<value_type>(p), 
Chris@16:                static_cast<value_type>(1-p), 
Chris@16:                forwarding_policy());
Chris@16:             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:                result, 
Chris@16:                function);
Chris@16:          }
Chris@16:          template <class A, class B, class C>
Chris@16:          static RealType find_degrees_of_freedom(const complemented3_type<A,B,C>& c)
Chris@16:          {
Chris@16:             const char* function = "non_central_t<%1%>::find_degrees_of_freedom";
Chris@16:             typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:             typedef typename policies::normalise<
Chris@16:                Policy, 
Chris@16:                policies::promote_float<false>, 
Chris@16:                policies::promote_double<false>, 
Chris@16:                policies::discrete_quantile<>,
Chris@16:                policies::assert_undefined<> >::type forwarding_policy;
Chris@16:             value_type result = detail::find_t_degrees_of_freedom(
Chris@16:                static_cast<value_type>(c.dist), 
Chris@16:                static_cast<value_type>(c.param1), 
Chris@16:                static_cast<value_type>(1-c.param2), 
Chris@16:                static_cast<value_type>(c.param2), 
Chris@16:                forwarding_policy());
Chris@16:             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:                result, 
Chris@16:                function);
Chris@16:          }
Chris@16:          static RealType find_non_centrality(RealType v, RealType x, RealType p)
Chris@16:          {
Chris@16:             const char* function = "non_central_t<%1%>::find_t_non_centrality";
Chris@16:             typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:             typedef typename policies::normalise<
Chris@16:                Policy, 
Chris@16:                policies::promote_float<false>, 
Chris@16:                policies::promote_double<false>, 
Chris@16:                policies::discrete_quantile<>,
Chris@16:                policies::assert_undefined<> >::type forwarding_policy;
Chris@16:             value_type result = detail::find_t_non_centrality(
Chris@16:                static_cast<value_type>(v), 
Chris@16:                static_cast<value_type>(x), 
Chris@16:                static_cast<value_type>(p), 
Chris@16:                static_cast<value_type>(1-p), 
Chris@16:                forwarding_policy());
Chris@16:             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:                result, 
Chris@16:                function);
Chris@16:          }
Chris@16:          template <class A, class B, class C>
Chris@16:          static RealType find_non_centrality(const complemented3_type<A,B,C>& c)
Chris@16:          {
Chris@16:             const char* function = "non_central_t<%1%>::find_t_non_centrality";
Chris@16:             typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:             typedef typename policies::normalise<
Chris@16:                Policy, 
Chris@16:                policies::promote_float<false>, 
Chris@16:                policies::promote_double<false>, 
Chris@16:                policies::discrete_quantile<>,
Chris@16:                policies::assert_undefined<> >::type forwarding_policy;
Chris@16:             value_type result = detail::find_t_non_centrality(
Chris@16:                static_cast<value_type>(c.dist), 
Chris@16:                static_cast<value_type>(c.param1), 
Chris@16:                static_cast<value_type>(1-c.param2), 
Chris@16:                static_cast<value_type>(c.param2), 
Chris@16:                forwarding_policy());
Chris@16:             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:                result, 
Chris@16:                function);
Chris@16:          }
Chris@16: #endif
Chris@16:       private:
Chris@16:          // Data member, initialized by constructor.
Chris@16:          RealType v;   // degrees of freedom
Chris@16:          RealType ncp; // non-centrality parameter
Chris@16:       }; // template <class RealType, class Policy> class non_central_t_distribution
Chris@16: 
Chris@16:       typedef non_central_t_distribution<double> non_central_t; // Reserved name of type double.
Chris@16: 
Chris@16:       // Non-member functions to give properties of the distribution.
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline const std::pair<RealType, RealType> range(const non_central_t_distribution<RealType, Policy>& /* dist */)
Chris@16:       { // Range of permissible values for random variable k.
Chris@16:          using boost::math::tools::max_value;
Chris@16:          return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>());
Chris@16:       }
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline const std::pair<RealType, RealType> support(const non_central_t_distribution<RealType, Policy>& /* dist */)
Chris@16:       { // Range of supported values for random variable k.
Chris@16:          // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
Chris@16:          using boost::math::tools::max_value;
Chris@16:          return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>());
Chris@16:       }
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline RealType mode(const non_central_t_distribution<RealType, Policy>& dist)
Chris@16:       { // mode.
Chris@16:          static const char* function = "mode(non_central_t_distribution<%1%> const&)";
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality();
Chris@16:          RealType r;
Chris@16:          if(!detail::check_df_gt0_to_inf(
Chris@16:             function,
Chris@16:             v, &r, Policy())
Chris@16:             ||
Chris@16:          !detail::check_finite(
Chris@16:             function,
Chris@16:             l,
Chris@16:             &r,
Chris@16:             Policy()))
Chris@16:                return (RealType)r;
Chris@16: 
Chris@16:          BOOST_MATH_STD_USING
Chris@16: 
Chris@16:          RealType m = v < 3 ? 0 : detail::mean(v, l, Policy());
Chris@16:          RealType var = v < 4 ? 1 : detail::variance(v, l, Policy());
Chris@16: 
Chris@16:          return detail::generic_find_mode(
Chris@16:             dist, 
Chris@16:             m,
Chris@16:             function,
Chris@16:             sqrt(var));
Chris@16:       }
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline RealType mean(const non_central_t_distribution<RealType, Policy>& dist)
Chris@16:       { 
Chris@16:          BOOST_MATH_STD_USING
Chris@16:          const char* function = "mean(const non_central_t_distribution<%1%>&)";
Chris@16:          typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:          typedef typename policies::normalise<
Chris@16:             Policy, 
Chris@16:             policies::promote_float<false>, 
Chris@16:             policies::promote_double<false>, 
Chris@16:             policies::discrete_quantile<>,
Chris@16:             policies::assert_undefined<> >::type forwarding_policy;
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality();
Chris@16:          RealType r;
Chris@16:          if(!detail::check_df_gt0_to_inf(
Chris@16:             function,
Chris@16:             v, &r, Policy())
Chris@16:             ||
Chris@16:          !detail::check_finite(
Chris@16:             function,
Chris@16:             l,
Chris@16:             &r,
Chris@16:             Policy()))
Chris@16:                return (RealType)r;
Chris@16:          if(v <= 1)
Chris@16:             return policies::raise_domain_error<RealType>(
Chris@16:                function, 
Chris@16:                "The non-central t distribution has no defined mean for degrees of freedom <= 1: got v=%1%.", v, Policy());
Chris@16:          // return l * sqrt(v / 2) * tgamma_delta_ratio((v - 1) * 0.5f, RealType(0.5f));
Chris@16:          return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:             detail::mean(static_cast<value_type>(v), static_cast<value_type>(l), forwarding_policy()), function);
Chris@16: 
Chris@16:       } // mean
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline RealType variance(const non_central_t_distribution<RealType, Policy>& dist)
Chris@16:       { // variance.
Chris@16:          const char* function = "variance(const non_central_t_distribution<%1%>&)";
Chris@16:          typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:          typedef typename policies::normalise<
Chris@16:             Policy, 
Chris@16:             policies::promote_float<false>, 
Chris@16:             policies::promote_double<false>, 
Chris@16:             policies::discrete_quantile<>,
Chris@16:             policies::assert_undefined<> >::type forwarding_policy;
Chris@16:          BOOST_MATH_STD_USING
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality();
Chris@16:          RealType r;
Chris@16:          if(!detail::check_df_gt0_to_inf(
Chris@16:             function,
Chris@16:             v, &r, Policy())
Chris@16:             ||
Chris@16:          !detail::check_finite(
Chris@16:             function,
Chris@16:             l,
Chris@16:             &r,
Chris@16:             Policy()))
Chris@16:                return (RealType)r;
Chris@16:          if(v <= 2)
Chris@16:             return policies::raise_domain_error<RealType>(
Chris@16:                function, 
Chris@16:                "The non-central t distribution has no defined variance for degrees of freedom <= 2: got v=%1%.", v, Policy());
Chris@16:          return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:             detail::variance(static_cast<value_type>(v), static_cast<value_type>(l), forwarding_policy()), function);
Chris@16:       }
Chris@16: 
Chris@16:       // RealType standard_deviation(const non_central_t_distribution<RealType, Policy>& dist)
Chris@16:       // standard_deviation provided by derived accessors.
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline RealType skewness(const non_central_t_distribution<RealType, Policy>& dist)
Chris@16:       { // skewness = sqrt(l).
Chris@16:          const char* function = "skewness(const non_central_t_distribution<%1%>&)";
Chris@16:          typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:          typedef typename policies::normalise<
Chris@16:             Policy, 
Chris@16:             policies::promote_float<false>, 
Chris@16:             policies::promote_double<false>, 
Chris@16:             policies::discrete_quantile<>,
Chris@16:             policies::assert_undefined<> >::type forwarding_policy;
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality();
Chris@16:          RealType r;
Chris@16:          if(!detail::check_df_gt0_to_inf(
Chris@16:             function,
Chris@16:             v, &r, Policy())
Chris@16:             ||
Chris@16:          !detail::check_finite(
Chris@16:             function,
Chris@16:             l,
Chris@16:             &r,
Chris@16:             Policy()))
Chris@16:                return (RealType)r;
Chris@16:          if(v <= 3)
Chris@16:             return policies::raise_domain_error<RealType>(
Chris@16:                function, 
Chris@16:                "The non-central t distribution has no defined skewness for degrees of freedom <= 3: got v=%1%.", v, Policy());;
Chris@16:          return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:             detail::skewness(static_cast<value_type>(v), static_cast<value_type>(l), forwarding_policy()), function);
Chris@16:       }
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline RealType kurtosis_excess(const non_central_t_distribution<RealType, Policy>& dist)
Chris@16:       { 
Chris@16:          const char* function = "kurtosis_excess(const non_central_t_distribution<%1%>&)";
Chris@16:          typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:          typedef typename policies::normalise<
Chris@16:             Policy, 
Chris@16:             policies::promote_float<false>, 
Chris@16:             policies::promote_double<false>, 
Chris@16:             policies::discrete_quantile<>,
Chris@16:             policies::assert_undefined<> >::type forwarding_policy;
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality();
Chris@16:          RealType r;
Chris@16:          if(!detail::check_df_gt0_to_inf(
Chris@16:             function,
Chris@16:             v, &r, Policy())
Chris@16:             ||
Chris@16:          !detail::check_finite(
Chris@16:             function,
Chris@16:             l,
Chris@16:             &r,
Chris@16:             Policy()))
Chris@16:                return (RealType)r;
Chris@16:          if(v <= 4)
Chris@16:             return policies::raise_domain_error<RealType>(
Chris@16:                function, 
Chris@16:                "The non-central t distribution has no defined kurtosis for degrees of freedom <= 4: got v=%1%.", v, Policy());;
Chris@16:          return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:             detail::kurtosis_excess(static_cast<value_type>(v), static_cast<value_type>(l), forwarding_policy()), function);
Chris@16:       } // kurtosis_excess
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline RealType kurtosis(const non_central_t_distribution<RealType, Policy>& dist)
Chris@16:       {
Chris@16:          return kurtosis_excess(dist) + 3;
Chris@16:       }
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline RealType pdf(const non_central_t_distribution<RealType, Policy>& dist, const RealType& t)
Chris@16:       { // Probability Density/Mass Function.
Chris@16:          const char* function = "pdf(non_central_t_distribution<%1%>, %1%)";
Chris@16:          typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:          typedef typename policies::normalise<
Chris@16:             Policy, 
Chris@16:             policies::promote_float<false>, 
Chris@16:             policies::promote_double<false>, 
Chris@16:             policies::discrete_quantile<>,
Chris@16:             policies::assert_undefined<> >::type forwarding_policy;
Chris@16: 
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality();
Chris@16:          RealType r;
Chris@16:          if(!detail::check_df_gt0_to_inf(
Chris@16:             function,
Chris@16:             v, &r, Policy())
Chris@16:             ||
Chris@16:          !detail::check_finite(
Chris@16:             function,
Chris@16:             l,
Chris@16:             &r,
Chris@16:             Policy())
Chris@16:             ||
Chris@16:          !detail::check_x(
Chris@16:             function,
Chris@16:             t,
Chris@16:             &r,
Chris@16:             Policy()))
Chris@16:                return (RealType)r;
Chris@16:          return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:             detail::non_central_t_pdf(static_cast<value_type>(v), 
Chris@16:                static_cast<value_type>(l), 
Chris@16:                static_cast<value_type>(t), 
Chris@16:                Policy()),
Chris@16:             function);
Chris@16:       } // pdf
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       RealType cdf(const non_central_t_distribution<RealType, Policy>& dist, const RealType& x)
Chris@16:       { 
Chris@16:          const char* function = "boost::math::cdf(non_central_t_distribution<%1%>&, %1%)";
Chris@16: //   was const char* function = "boost::math::non_central_t_distribution<%1%>::cdf(%1%)";
Chris@16:          typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:          typedef typename policies::normalise<
Chris@16:             Policy, 
Chris@16:             policies::promote_float<false>, 
Chris@16:             policies::promote_double<false>, 
Chris@16:             policies::discrete_quantile<>,
Chris@16:             policies::assert_undefined<> >::type forwarding_policy;
Chris@16: 
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality();
Chris@16:          RealType r;
Chris@16:          if(!detail::check_df_gt0_to_inf(
Chris@16:             function,
Chris@16:             v, &r, Policy())
Chris@16:             ||
Chris@16:          !detail::check_finite(
Chris@16:             function,
Chris@16:             l,
Chris@16:             &r,
Chris@16:             Policy())
Chris@16:             ||
Chris@16:          !detail::check_x(
Chris@16:             function,
Chris@16:             x,
Chris@16:             &r,
Chris@16:             Policy()))
Chris@16:                return (RealType)r;
Chris@16:           if ((boost::math::isinf)(v))
Chris@16:           { // Infinite degrees of freedom, so use normal distribution located at delta.
Chris@16:              normal_distribution<RealType, Policy> n(l, 1); 
Chris@16:              cdf(n, x);
Chris@16:               //return cdf(normal_distribution<RealType, Policy>(l, 1), x);
Chris@16:           }
Chris@16: 
Chris@16:          if(l == 0)
Chris@16:          { // NO non-centrality, so use Student's t instead.
Chris@16:             return cdf(students_t_distribution<RealType, Policy>(v), x);
Chris@16:          }
Chris@16:          return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:             detail::non_central_t_cdf(
Chris@16:                static_cast<value_type>(v), 
Chris@16:                static_cast<value_type>(l), 
Chris@16:                static_cast<value_type>(x), 
Chris@16:                false, Policy()),
Chris@16:             function);
Chris@16:       } // cdf
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       RealType cdf(const complemented2_type<non_central_t_distribution<RealType, Policy>, RealType>& c)
Chris@16:       { // Complemented Cumulative Distribution Function
Chris@16:   // was       const char* function = "boost::math::non_central_t_distribution<%1%>::cdf(%1%)";
Chris@16:          const char* function = "boost::math::cdf(const complement(non_central_t_distribution<%1%>&), %1%)";
Chris@16:          typedef typename policies::evaluation<RealType, Policy>::type value_type;
Chris@16:          typedef typename policies::normalise<
Chris@16:             Policy, 
Chris@16:             policies::promote_float<false>, 
Chris@16:             policies::promote_double<false>, 
Chris@16:             policies::discrete_quantile<>,
Chris@16:             policies::assert_undefined<> >::type forwarding_policy;
Chris@16: 
Chris@16:          non_central_t_distribution<RealType, Policy> const& dist = c.dist;
Chris@16:          RealType x = c.param;
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality(); // aka delta
Chris@16:          RealType r;
Chris@16:          if(!detail::check_df_gt0_to_inf(
Chris@16:             function,
Chris@16:             v, &r, Policy())
Chris@16:             ||
Chris@16:          !detail::check_finite(
Chris@16:             function,
Chris@16:             l,
Chris@16:             &r,
Chris@16:             Policy())
Chris@16:             ||
Chris@16:          !detail::check_x(
Chris@16:             function,
Chris@16:             x,
Chris@16:             &r,
Chris@16:             Policy()))
Chris@16:                return (RealType)r;
Chris@16: 
Chris@16:          if ((boost::math::isinf)(v))
Chris@16:          { // Infinite degrees of freedom, so use normal distribution located at delta.
Chris@16:              normal_distribution<RealType, Policy> n(l, 1); 
Chris@16:              return cdf(complement(n, x));
Chris@16:          }
Chris@16:          if(l == 0)
Chris@16:          { // zero non-centrality so use Student's t distribution.
Chris@16:             return cdf(complement(students_t_distribution<RealType, Policy>(v), x));
Chris@16:          }
Chris@16:          return policies::checked_narrowing_cast<RealType, forwarding_policy>(
Chris@16:             detail::non_central_t_cdf(
Chris@16:                static_cast<value_type>(v), 
Chris@16:                static_cast<value_type>(l), 
Chris@16:                static_cast<value_type>(x), 
Chris@16:                true, Policy()),
Chris@16:             function);
Chris@16:       } // ccdf
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline RealType quantile(const non_central_t_distribution<RealType, Policy>& dist, const RealType& p)
Chris@16:       { // Quantile (or Percent Point) function.
Chris@16:          static const char* function = "quantile(const non_central_t_distribution<%1%>, %1%)";
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality();
Chris@16:          return detail::non_central_t_quantile(function, v, l, p, RealType(1-p), Policy());
Chris@16:       } // quantile
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline RealType quantile(const complemented2_type<non_central_t_distribution<RealType, Policy>, RealType>& c)
Chris@16:       { // Quantile (or Percent Point) function.
Chris@16:          static const char* function = "quantile(const complement(non_central_t_distribution<%1%>, %1%))";
Chris@16:          non_central_t_distribution<RealType, Policy> const& dist = c.dist;
Chris@16:          RealType q = c.param;
Chris@16:          RealType v = dist.degrees_of_freedom();
Chris@16:          RealType l = dist.non_centrality();
Chris@16:          return detail::non_central_t_quantile(function, v, l, RealType(1-q), q, Policy());
Chris@16:       } // quantile complement.
Chris@16: 
Chris@16:    } // namespace math
Chris@16: } // namespace boost
Chris@16: 
Chris@16: // This include must be at the end, *after* the accessors
Chris@16: // for this distribution have been defined, in order to
Chris@16: // keep compilers that support two-phase lookup happy.
Chris@16: #include <boost/math/distributions/detail/derived_accessors.hpp>
Chris@16: 
Chris@16: #endif // BOOST_MATH_SPECIAL_NON_CENTRAL_T_HPP
Chris@16: