path: root/libs/math/test/test_nc_t.cpp

// test_nc_t.cpp

// Copyright John Maddock 2008, 2012.
// Copyright Paul A. Bristow 2012.

// 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)

#include <pch.hpp> // Need to include lib/math/test in path.

#ifdef _MSC_VER
#pragma warning (disable:4127 4512)
#endif

#if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT)
#  define TEST_FLOAT
#  define TEST_DOUBLE
#  define TEST_LDOUBLE
#  define TEST_REAL_CONCEPT
#endif

#include <boost/math/concepts/real_concept.hpp> // for real_concept
#include <boost/math/distributions/non_central_t.hpp> // for chi_squared_distribution.
#include <boost/math/distributions/normal.hpp> // for normal distribution (for comparison).

#define BOOST_TEST_MAIN
#include <boost/test/unit_test.hpp> // for test_main
#include <boost/test/results_collector.hpp>
#include <boost/test/unit_test.hpp>
#include <boost/test/floating_point_comparison.hpp> // for BOOST_CHECK_CLOSE

#include "functor.hpp"
#include "handle_test_result.hpp"
#include "table_type.hpp"
#include "test_out_of_range.hpp"

#include <iostream>
#include <iomanip>
using std::cout;
using std::endl;
#include <limits>
using std::numeric_limits;

#define BOOST_CHECK_CLOSE_EX(a, b, prec, i) \
   {\
      unsigned int failures = boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed;\
      BOOST_CHECK_CLOSE(a, b, prec); \
      if(failures != boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed)\
      {\
         std::cerr << "Failure was at row " << i << std::endl;\
         std::cerr << std::setprecision(35); \
         std::cerr << "{ " << data[i][0] << " , " << data[i][1] << " , " << data[i][2];\
         std::cerr << " , " << data[i][3] << " , " << data[i][4] << " } " << std::endl;\
      }\
   }

#define BOOST_CHECK_EX(a, i) \
   {\
      unsigned int failures = boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed;\
      BOOST_CHECK(a); \
      if(failures != boost::unit_test::results_collector.results( boost::unit_test::framework::current_test_case().p_id ).p_assertions_failed)\
      {\
         std::cerr << "Failure was at row " << i << std::endl;\
         std::cerr << std::setprecision(35); \
         std::cerr << "{ " << data[i][0] << " , " << data[i][1] << " , " << data[i][2];\
         std::cerr << " , " << data[i][3] << " , " << data[i][4] << " } " << std::endl;\
      }\
   }

void expected_results()
{
   //
   // Define the max and mean errors expected for
   // various compilers and platforms.
   //
   const char* largest_type;
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
   if(boost::math::policies::digits<double, boost::math::policies::policy<> >() == boost::math::policies::digits<long double, boost::math::policies::policy<> >())
   {
      largest_type = "(long\\s+)?double|real_concept";
   }
   else
   {
      largest_type = "long double|real_concept";
   }
#else
   largest_type = "(long\\s+)?double|real_concept";
#endif

   //
   // Catch all cases come last:
   //
   if(std::numeric_limits<long double>::digits > 54)
   {
      add_expected_result(
         "[^|]*",                          // compiler
         "[^|]*",                          // stdlib
         "[^|]*",                          // platform
         largest_type,                     // test type(s)
         "[^|]*large[^|]*",                // test data group
         "[^|]*", 2000000, 200000);        // test function
      add_expected_result(
         "[^|]*",                          // compiler
         "[^|]*",                          // stdlib
         "[^|]*",                          // platform
         "double",                         // test type(s)
         "[^|]*large[^|]*",                // test data group
         "[^|]*", 500, 100);               // test function
   }
   add_expected_result(
      "[^|]*",                          // compiler
      "[^|]*",                          // stdlib
      "[^|]*",                          // platform
      "real_concept",                   // test type(s)
      "[^|]*",                          // test data group
      "[^|]*", 300000, 100000);                // test function
   add_expected_result(
      "[^|]*",                          // compiler
      "[^|]*",                          // stdlib
      "[^|]*",                          // platform
      largest_type,                     // test type(s)
      "[^|]*large[^|]*",                // test data group
      "[^|]*", 1500, 300);              // test function
   add_expected_result(
      "[^|]*",                          // compiler
      "[^|]*",                          // stdlib
      "[^|]*",                          // platform
      largest_type,                     // test type(s)
      "[^|]*small[^|]*",                // test data group
      "[^|]*", 400, 100);              // test function
   add_expected_result(
      "[^|]*",                          // compiler
      "[^|]*",                          // stdlib
      "[^|]*",                          // platform
      largest_type,                     // test type(s)
      "[^|]*",                          // test data group
      "[^|]*", 250, 50);                // test function

   //
   // Finish off by printing out the compiler/stdlib/platform names,
   // we do this to make it easier to mark up expected error rates.
   //
   std::cout << "Tests run with " << BOOST_COMPILER << ", " 
      << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl;
}

template <class RealType>
RealType naive_pdf(RealType v, RealType delta, RealType x)
{
}

template <class RealType>
RealType naive_mean(RealType v, RealType delta)
{
   using boost::math::tgamma;
   return delta * sqrt(v / 2) * tgamma((v-1)/2) / tgamma(v/2);
}

float naive_mean(float v, float delta)
{
   return (float)naive_mean((double)v, (double)delta);
}

template <class RealType>
RealType naive_variance(RealType v, RealType delta)
{
   using boost::math::tgamma;
   RealType r = tgamma((v-1)/2) / tgamma(v/2);
   r *= r;
   r *= -delta * delta * v / 2;
   r += (1 + delta * delta) * v / (v - 2);
   return r;
}

float naive_variance(float v, float delta)
{
   return (float)naive_variance((double)v, (double)delta);
}

template <class RealType>
RealType naive_skewness(RealType v, RealType delta)
{
   using boost::math::tgamma;
   RealType tgr = tgamma((v-1)/2) / tgamma(v / 2);
   RealType r = delta * sqrt(v) * tgamma((v-1)/2)
      * (v * (-3 + delta * delta + 2 * v) / ((-3 + v) * (-2 + v)) 
         - 2 * ((1 + delta * delta) * v / (-2 + v) - delta * delta * v * tgr * tgr / 2));
   r /= boost::math::constants::root_two<RealType>()
      * pow(((1+delta*delta) * v / (-2+v) - delta*delta*v*tgr*tgr/2), RealType(1.5f))
      * tgamma(v/2);
   return r;
}

float naive_skewness(float v, float delta)
{
   return (float)naive_skewness((double)v, (double)delta);
}

template <class RealType>
RealType naive_kurtosis_excess(RealType v, RealType delta)
{
   using boost::math::tgamma;
   RealType tgr = tgamma((v-1)/2) / tgamma(v / 2);
   RealType r = -delta * delta * v * tgr * tgr / 2;
   r *= v * (delta * delta * (1 + v) + 3 * (-5 + 3 * v)) / ((-3 + v)*(-2+v))
      - 3 * ((1 + delta * delta) * v / (-2 + v) - delta * delta * v * tgr * tgr / 2);
   r += (3 + 6 * delta * delta + delta * delta * delta * delta)* v * v 
      / ((-4+v) * (-2+v));
   r /= (1+delta*delta)*v / (-2+v) - delta*delta*v *tgr*tgr/2;
   r /= (1+delta*delta)*v / (-2+v) - delta*delta*v *tgr*tgr/2;
   return r;
}

float naive_kurtosis_excess(float v, float delta)
{
   return (float)naive_kurtosis_excess((double)v, (double)delta);
}

template <class RealType>
void test_spot(
     RealType df,    // Degrees of freedom
     RealType ncp,   // non-centrality param
     RealType t,     // T statistic
     RealType P,     // CDF
     RealType Q,     // Complement of CDF
     RealType tol)   // Test tolerance
{
   // An extra fudge factor for real_concept which has a less accurate tgamma:
   RealType tolerance_tgamma_extra = std::numeric_limits<RealType>::is_specialized ? 1 : 5;

   boost::math::non_central_t_distribution<RealType> dist(df, ncp);
   BOOST_CHECK_CLOSE(
      cdf(dist, t), P, tol);
   try{
      BOOST_CHECK_CLOSE(
         mean(dist), naive_mean(df, ncp), tol);
      BOOST_CHECK_CLOSE(
         variance(dist), naive_variance(df, ncp), tol);
      BOOST_CHECK_CLOSE(
         skewness(dist), naive_skewness(df, ncp), tol * 10 * tolerance_tgamma_extra);
      BOOST_CHECK_CLOSE(
         kurtosis_excess(dist), naive_kurtosis_excess(df, ncp), tol * 50 * tolerance_tgamma_extra);
      BOOST_CHECK_CLOSE(
         kurtosis(dist), 3 + naive_kurtosis_excess(df, ncp), tol * 50 * tolerance_tgamma_extra);
   }
   catch(const std::domain_error&)
   {
   }
   /*
   BOOST_CHECK_CLOSE(
      pdf(dist, t), naive_pdf(dist.degrees_of_freedom(), ncp, t), tol * 50);
   */
   if((P < 0.99) && (Q < 0.99))
   {
      //
      // We can only check this if P is not too close to 1,
      // so that we can guarantee Q is reasonably free of error:
      //
      BOOST_CHECK_CLOSE(
         cdf(complement(dist, t)), Q, tol);
      BOOST_CHECK_CLOSE(
            quantile(dist, P), t, tol * 10);
      BOOST_CHECK_CLOSE(
            quantile(complement(dist, Q)), t, tol * 10);
      /*  Removed because can give more than one solution.
      BOOST_CHECK_CLOSE(
         dist.find_degrees_of_freedom(ncp, t, P), df, tol * 10);
      BOOST_CHECK_CLOSE(
         dist.find_degrees_of_freedom(boost::math::complement(ncp, t, Q)), df, tol * 10);
      BOOST_CHECK_CLOSE(
         dist.find_non_centrality(df, t, P), ncp, tol * 10);
      BOOST_CHECK_CLOSE(
         dist.find_non_centrality(boost::math::complement(df, t, Q)), ncp, tol * 10);
         */
   }
}

template <class RealType> // Any floating-point type RealType.
void test_spots(RealType)
{
   //
   // Approx limit of test data is 12 digits expressed here as a percentage:
   //
   RealType tolerance = (std::max)(
      boost::math::tools::epsilon<RealType>(),
      (RealType)5e-12f) * 100;
   //
   // At float precision we need to up the tolerance, since 
   // the input values are rounded off to inexact quantities
   // the results get thrown off by a noticeable amount.
   //
   if(boost::math::tools::digits<RealType>() < 50)
      tolerance *= 50;
   if(boost::is_floating_point<RealType>::value != 1)
      tolerance *= 20; // real_concept special functions are less accurate

   cout << "Tolerance = " << tolerance << "%." << endl;

   //
   // Test data is taken from:
   //
   // Computing discrete mixtures of continuous
   // distributions: noncentral chisquare, noncentral t
   // and the distribution of the square of the sample
   // multiple correlation coeficient.
   // Denise Benton, K. Krishnamoorthy.
   // Computational Statistics & Data Analysis 43 (2003) 249 - 267
   //
   test_spot(
      static_cast<RealType>(3),   // degrees of freedom
      static_cast<RealType>(1),   // non centrality
      static_cast<RealType>(2.34),   // T
      static_cast<RealType>(0.801888999613917),       // Probability of result (CDF), P
      static_cast<RealType>(1-0.801888999613917),           // Q = 1 - P
      tolerance);
   test_spot(
      static_cast<RealType>(126),   // degrees of freedom
      static_cast<RealType>(-2),   // non centrality
      static_cast<RealType>(-4.33),   // T
      static_cast<RealType>(1.252846196792878e-2),       // Probability of result (CDF), P
      static_cast<RealType>(1-1.252846196792878e-2),           // Q = 1 - P
      tolerance);
   test_spot(
      static_cast<RealType>(20),   // degrees of freedom
      static_cast<RealType>(23),   // non centrality
      static_cast<RealType>(23),   // T
      static_cast<RealType>(0.460134400391924),       // Probability of result (CDF), P
      static_cast<RealType>(1-0.460134400391924),           // Q = 1 - P
      tolerance);
   test_spot(
      static_cast<RealType>(20),   // degrees of freedom
      static_cast<RealType>(33),   // non centrality
      static_cast<RealType>(34),   // T
      static_cast<RealType>(0.532008386378725),       // Probability of result (CDF), P
      static_cast<RealType>(1-0.532008386378725),           // Q = 1 - P
      tolerance);
   test_spot(
      static_cast<RealType>(12),   // degrees of freedom
      static_cast<RealType>(38),   // non centrality
      static_cast<RealType>(39),   // T
      static_cast<RealType>(0.495868184917805),       // Probability of result (CDF), P
      static_cast<RealType>(1-0.495868184917805),           // Q = 1 - P
      tolerance);
   test_spot(
      static_cast<RealType>(12),   // degrees of freedom
      static_cast<RealType>(39),   // non centrality
      static_cast<RealType>(39),   // T
      static_cast<RealType>(0.446304024668836),       // Probability of result (CDF), P
      static_cast<RealType>(1-0.446304024668836),           // Q = 1 - P
      tolerance);
   test_spot(
      static_cast<RealType>(200),   // degrees of freedom
      static_cast<RealType>(38),   // non centrality
      static_cast<RealType>(39),   // T
      static_cast<RealType>(0.666194209961795),       // Probability of result (CDF), P
      static_cast<RealType>(1-0.666194209961795),           // Q = 1 - P
      tolerance);
   test_spot(
      static_cast<RealType>(200),   // degrees of freedom
      static_cast<RealType>(42),   // non centrality
      static_cast<RealType>(40),   // T
      static_cast<RealType>(0.179292265426085),       // Probability of result (CDF), P
      static_cast<RealType>(1-0.179292265426085),           // Q = 1 - P
      tolerance);

   // From https://svn.boost.org/trac/boost/ticket/10480.
   // Test value from Mathematica N[CDF[NoncentralStudentTDistribution[2, 4], 5], 35]:
   test_spot(
      static_cast<RealType>(2),   // degrees of freedom
      static_cast<RealType>(4),   // non centrality
      static_cast<RealType>(5),   // T
      static_cast<RealType>(0.53202069866995310466912357978934321L),       // Probability of result (CDF), P
      static_cast<RealType>(1 - 0.53202069866995310466912357978934321L),           // Q = 1 - P
      tolerance);

   /* This test fails
   "Result of tgamma is too large to represent" at naive_mean check for max and infinity.
   if (std::numeric_limits<RealType>::has_infinity)
   {
      test_spot(
      //static_cast<RealType>(std::numeric_limits<RealType>::infinity()),   // degrees of freedom
      static_cast<RealType>((std::numeric_limits<RealType>::max)()),   // degrees of freedom
      static_cast<RealType>(10),   // non centrality
      static_cast<RealType>(11),   // T
      static_cast<RealType>(0.84134474606854293),       // Probability of result (CDF), P
      static_cast<RealType>(0.15865525393145707),           // Q = 1 - P
      tolerance);
   }
   */

   boost::math::non_central_t_distribution<RealType> dist(static_cast<RealType>(8), static_cast<RealType>(12));
   BOOST_CHECK_CLOSE(pdf(dist, 12), static_cast<RealType>(1.235329715425894935157684607751972713457e-1L), tolerance);
   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(126, -2), -4), static_cast<RealType>(5.797932289365814702402873546466798025787e-2L), tolerance);
   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(126, 2), 4), static_cast<RealType>(5.797932289365814702402873546466798025787e-2L), tolerance);
   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(126, 2), 0), static_cast<RealType>(5.388394890639957139696546086044839573749e-2L), tolerance);

   // Error handling checks:
   //check_out_of_range<boost::math::non_central_t_distribution<RealType> >(1, 1);  // Fails one check because df for this distribution *can* be infinity.
   BOOST_CHECK_THROW(pdf(boost::math::non_central_t_distribution<RealType>(0, 1), 0), std::domain_error);
   BOOST_CHECK_THROW(pdf(boost::math::non_central_t_distribution<RealType>(-1, 1), 0), std::domain_error);
   BOOST_CHECK_THROW(quantile(boost::math::non_central_t_distribution<RealType>(1, 1), -1), std::domain_error);
   BOOST_CHECK_THROW(quantile(boost::math::non_central_t_distribution<RealType>(1, 1), 2), std::domain_error);
} // template <class RealType>void test_spots(RealType)

template <class T>
T nct_cdf(T df, T nc, T x)
{
   return cdf(boost::math::non_central_t_distribution<T>(df, nc), x);
}

template <class T>
T nct_ccdf(T df, T nc, T x)
{
   return cdf(complement(boost::math::non_central_t_distribution<T>(df, nc), x));
}

template <typename Real, typename T>
void do_test_nc_t(T& data, const char* type_name, const char* test)
{
   typedef typename T::value_type row_type;
   typedef Real                   value_type;

   std::cout << "Testing: " << test << std::endl;

   value_type (*fp1)(value_type, value_type, value_type) = nct_cdf;
   boost::math::tools::test_result<value_type> result;

   result = boost::math::tools::test_hetero<Real>(
      data,
      bind_func<Real>(fp1, 0, 1, 2),
      extract_result<Real>(3));
   handle_test_result(result, data[result.worst()], result.worst(),
      type_name, "CDF", test);

   fp1 = nct_ccdf;
   result = boost::math::tools::test_hetero<Real>(
      data,
      bind_func<Real>(fp1, 0, 1, 2),
      extract_result<Real>(4));
   handle_test_result(result, data[result.worst()], result.worst(),
      type_name, "CCDF", test);

   std::cout << std::endl;

}

template <typename Real, typename T>
void quantile_sanity_check(T& data, const char* type_name, const char* test)
{
   typedef typename T::value_type row_type;
   typedef Real                   value_type;

   //
   // Tests with type real_concept take rather too long to run, so
   // for now we'll disable them:
   //
   if(!boost::is_floating_point<value_type>::value)
      return;

   std::cout << "Testing: " << type_name << " quantile sanity check, with tests " << test << std::endl;

   //
   // These sanity checks test for a round trip accuracy of one half
   // of the bits in T, unless T is type float, in which case we check
   // for just one decimal digit.  The problem here is the sensitivity
   // of the functions, not their accuracy.  This test data was generated
   // for the forward functions, which means that when it is used as
   // the input to the inverses then it is necessarily inexact.  This rounding
   // of the input is what makes the data unsuitable for use as an accuracy check,
   // and also demonstrates that you can't in general round-trip these functions.
   // It is however a useful sanity check.
   //
   value_type precision = static_cast<value_type>(ldexp(1.0, 1-boost::math::policies::digits<value_type, boost::math::policies::policy<> >()/2)) * 100;
   if(boost::math::policies::digits<value_type, boost::math::policies::policy<> >() < 50)
      precision = 1;   // 1% or two decimal digits, all we can hope for when the input is truncated to float

   for(unsigned i = 0; i < data.size(); ++i)
   {
      if(data[i][3] == 0)
      {
         BOOST_CHECK(0 == quantile(boost::math::non_central_t_distribution<value_type>(data[i][0], data[i][1]), data[i][3]));
      }
      else if(data[i][3] < 0.9999f)
      {
         value_type p = quantile(boost::math::non_central_t_distribution<value_type>(data[i][0], data[i][1]), data[i][3]);
         value_type pt = data[i][2];
         BOOST_CHECK_CLOSE_EX(pt, p, precision, i);
      }
      if(data[i][4] == 0)
      {
         BOOST_CHECK(0 == quantile(complement(boost::math::non_central_t_distribution<value_type>(data[i][0], data[i][1]), data[i][3])));
      }
      else if(data[i][4] < 0.9999f)
      {
         value_type p = quantile(complement(boost::math::non_central_t_distribution<value_type>(data[i][0], data[i][1]), data[i][4]));
         value_type pt = data[i][2];
         BOOST_CHECK_CLOSE_EX(pt, p, precision, i);
      }
      if(boost::math::tools::digits<value_type>() > 50)
      {
         //
         // Sanity check mode, the accuracy of
         // the mode is at *best* the square root of the accuracy of the PDF:
         //
         try{
            value_type m = mode(boost::math::non_central_t_distribution<value_type>(data[i][0], data[i][1]));
            value_type p = pdf(boost::math::non_central_t_distribution<value_type>(data[i][0], data[i][1]), m);
            value_type delta = (std::max)(fabs(m * sqrt(precision) * 50), sqrt(precision) * 50);
            BOOST_CHECK_EX(pdf(boost::math::non_central_t_distribution<value_type>(data[i][0], data[i][1]), m + delta) <= p, i);
            BOOST_CHECK_EX(pdf(boost::math::non_central_t_distribution<value_type>(data[i][0], data[i][1]), m - delta) <= p, i);
         }
         catch(const boost::math::evaluation_error& ) {}
#if 0
         //
         // Sanity check degrees-of-freedom finder, don't bother at float
         // precision though as there's not enough data in the probability
         // values to get back to the correct degrees of freedom or 
         // non-centrality parameter:
         //
         try{
            if((data[i][3] < 0.99) && (data[i][3] != 0))
            {
               BOOST_CHECK_CLOSE_EX(
                  boost::math::non_central_t_distribution<value_type>::find_degrees_of_freedom(data[i][1], data[i][2], data[i][3]),
                  data[i][0], precision, i);
               BOOST_CHECK_CLOSE_EX(
                  boost::math::non_central_t_distribution<value_type>::find_non_centrality(data[i][0], data[i][2], data[i][3]),
                  data[i][1], precision, i);
            }
            if((data[i][4] < 0.99) && (data[i][4] != 0))
            {
               BOOST_CHECK_CLOSE_EX(
                  boost::math::non_central_t_distribution<value_type>::find_degrees_of_freedom(boost::math::complement(data[i][1], data[i][2], data[i][4])),
                  data[i][0], precision, i);
               BOOST_CHECK_CLOSE_EX(
                  boost::math::non_central_t_distribution<value_type>::find_non_centrality(boost::math::complement(data[i][0], data[i][2], data[i][4])),
                  data[i][1], precision, i);
            }
         }
         catch(const std::exception& e)
         {
            BOOST_ERROR(e.what());
         }
#endif
      }
   }
}

template <typename T>
void test_accuracy(T, const char* type_name)
{
#include "nct.ipp"
    do_test_nc_t<T>(nct, type_name, "Non Central T");
    quantile_sanity_check<T>(nct, type_name, "Non Central T");
    if(std::numeric_limits<T>::is_specialized)
    {
       //
       // Don't run these tests for real_concept: they take too long and don't converge
       // without numeric_limits and lanczos support:
       //
#include "nct_small_delta.ipp"
       do_test_nc_t<T>(nct_small_delta, type_name, "Non Central T (small non-centrality)");
       quantile_sanity_check<T>(nct_small_delta, type_name, "Non Central T (small non-centrality)");
#include "nct_asym.ipp"
       do_test_nc_t<T>(nct_asym, type_name, "Non Central T (large parameters)");
       quantile_sanity_check<T>(nct_asym, type_name, "Non Central T (large parameters)");
    }
}


template <class RealType>
void test_big_df(RealType)
{
  using namespace boost::math;

  if (typeid(RealType) != typeid(boost::math::concepts::real_concept))
  { // Ordinary floats only.
    // Could also test if (std::numeric_limits<RealType>::is_specialized);
    
    RealType tolerance = 10 * boost::math::tools::epsilon<RealType>(); // static_cast<RealType>(1e-14); // 
    std::cout.precision(17); // Note: need to reset after calling BOOST_CHECK_s
    // due to buglet in Boost.test that fails to restore precision corrrectly.

    // Test for large degrees of freedom when should be same as normal.
    RealType inf = 
      (std::numeric_limits<RealType>::has_infinity) ?
      std::numeric_limits<RealType>::infinity()
      :
      boost::math::tools::max_value<RealType>();
    RealType nan = std::numeric_limits<RealType>::quiet_NaN();

    // Tests for df = max_value and infinity.
    RealType max_val = boost::math::tools::max_value<RealType>();
    non_central_t_distribution<RealType> maxdf(max_val, 0);
    BOOST_CHECK_EQUAL(maxdf.degrees_of_freedom(), max_val);

    non_central_t_distribution<RealType> infdf(inf, 0);
    BOOST_CHECK_EQUAL(infdf.degrees_of_freedom(), inf);
    BOOST_CHECK_EQUAL(mean(infdf), 0); 
    BOOST_CHECK_EQUAL(mean(maxdf), 0); 
    BOOST_CHECK_EQUAL(variance(infdf), 1); 
    BOOST_CHECK_EQUAL(variance(maxdf), 1); 
    BOOST_CHECK_EQUAL(skewness(infdf), 0); 
    BOOST_CHECK_EQUAL(skewness(maxdf), 0); 
    BOOST_CHECK_EQUAL(kurtosis_excess(infdf), 3); 
    BOOST_CHECK_CLOSE_FRACTION(kurtosis_excess(maxdf), static_cast<RealType>(3), tolerance); 

    // Bad df examples.
    BOOST_CHECK_THROW(non_central_t_distribution<RealType> minfdf(-inf, 0), std::domain_error);
    BOOST_CHECK_THROW(non_central_t_distribution<RealType> minfdf(nan, 0), std::domain_error);
    BOOST_CHECK_THROW(non_central_t_distribution<RealType> minfdf(-nan, 0), std::domain_error);


    // BOOST_CHECK_CLOSE_FRACTION(pdf(infdf, 0), static_cast<RealType>(0.3989422804014326779399460599343818684759L), tolerance);
    BOOST_CHECK_CLOSE_FRACTION(pdf(maxdf, 0), boost::math::constants::one_div_root_two_pi<RealType>() , tolerance);
    BOOST_CHECK_CLOSE_FRACTION(pdf(infdf, 0), boost::math::constants::one_div_root_two_pi<RealType>() , tolerance);
    BOOST_CHECK_CLOSE_FRACTION(cdf(infdf, 0), boost::math::constants::half<RealType>() , tolerance);
    BOOST_CHECK_CLOSE_FRACTION(cdf(maxdf, 0), boost::math::constants::half<RealType>() , tolerance);

    // non-centrality delta = 10
    // Degrees of freedom = Max value and  = infinity should be very close.
    non_central_t_distribution<RealType> maxdf10(max_val, 10);
    non_central_t_distribution<RealType> infdf10(inf, 10);
    BOOST_CHECK_EQUAL(infdf10.degrees_of_freedom(), inf);
    BOOST_CHECK_EQUAL(infdf10.non_centrality(), 10);
    BOOST_CHECK_EQUAL(mean(infdf10), 10); 
    BOOST_CHECK_CLOSE_FRACTION(mean(maxdf10), static_cast<RealType>(10), tolerance); 

    BOOST_CHECK_CLOSE_FRACTION(pdf(infdf10, 11), pdf(maxdf10, 11), tolerance); // 

    BOOST_CHECK_CLOSE_FRACTION(cdf(complement(infdf10, 11)), 1 - cdf(infdf10, 11), tolerance); // 
    BOOST_CHECK_CLOSE_FRACTION(cdf(complement(maxdf10, 11)), 1 - cdf(maxdf10, 11), tolerance); // 
    BOOST_CHECK_CLOSE_FRACTION(cdf(complement(infdf10, 11)), 1 - cdf(maxdf10, 11), tolerance); // 
    std::cout.precision(17);
    //std::cout  << "cdf(maxdf10, 11)  = " << cdf(maxdf10, 11) << ' ' << cdf(complement(maxdf10, 11)) << endl;
    //std::cout  << "cdf(infdf10, 11)  = " << cdf(infdf10, 11) << ' ' << cdf(complement(infdf10, 11)) << endl;
    //std::cout  << "quantile(maxdf10, 0.5)  = " << quantile(maxdf10, 0.5) << std::endl; // quantile(maxdf10, 0.5)  = 10.000000000000004
    //std::cout  << "quantile(infdf10, 0.5) = " << ' ' << quantile(infdf10, 0.5) << std::endl; // quantile(infdf10, 0.5) =  10

    BOOST_CHECK_CLOSE_FRACTION(quantile(infdf10, 0.5), static_cast<RealType>(10), tolerance);
    BOOST_CHECK_CLOSE_FRACTION(quantile(maxdf10, 0.5), static_cast<RealType>(10), tolerance);

    BOOST_TEST_MESSAGE("non_central_t_distribution<RealType> infdf100(inf, 100);");
    non_central_t_distribution<RealType> infdf100(inf, 100);
    BOOST_TEST_MESSAGE("non_central_t_distribution<RealType> maxdf100(max_val, 100);");
    non_central_t_distribution<RealType> maxdf100(max_val, 100);
    BOOST_TEST_MESSAGE("BOOST_CHECK_CLOSE_FRACTION(quantile(infdf100, 0.5), static_cast<RealType>(100), tolerance);");
    BOOST_CHECK_CLOSE_FRACTION(quantile(infdf100, 0.5), static_cast<RealType>(100), tolerance);
    BOOST_TEST_MESSAGE("BOOST_CHECK_CLOSE_FRACTION(quantile(maxdf100, 0.5), static_cast<RealType>(100), tolerance);");
    BOOST_CHECK_CLOSE_FRACTION(quantile(maxdf100, 0.5), static_cast<RealType>(100), tolerance);
    { // Loop back.
      RealType p = static_cast<RealType>(0.01);
      RealType x = quantile(infdf10, p);
      RealType c = cdf(infdf10, x);
      BOOST_CHECK_CLOSE_FRACTION(c, p, tolerance);
    }
    {
      RealType q = static_cast<RealType>(0.99);
      RealType x = quantile(complement(infdf10, q));
      RealType c = cdf(complement(infdf10, x));
      BOOST_CHECK_CLOSE_FRACTION(c, q, tolerance);
    }
    { // Loop back.
      RealType p = static_cast<RealType>(0.99);
      RealType x = quantile(infdf10, p);
      RealType c = cdf(infdf10, x);
      BOOST_CHECK_CLOSE_FRACTION(c, p, tolerance);
    }
    {
      RealType q = static_cast<RealType>(0.01);
      RealType x = quantile(complement(infdf10, q));
      RealType c = cdf(complement(infdf10, x));
      BOOST_CHECK_CLOSE_FRACTION(c, q, tolerance *2); // c{0.0100000128} and q{0.00999999978}
    }

    //RealType cinf = quantile(infdf10, 0.25);
    //std::cout << cinf << ' ' << cdf(infdf10, cinf) << std::endl; // 9.32551 0.25

    //RealType cmax = quantile(maxdf10, 0.25);
    //std::cout << cmax << ' ' << cdf(maxdf10, cmax) << std::endl; //  9.32551 0.25

    //RealType cinfc = quantile(complement(infdf10, 0.75));
    //std::cout << cinfc << ' ' << cdf(infdf10, cinfc) << std::endl; // 9.32551 0.25

    //RealType cmaxc = quantile(complement(maxdf10, 0.75));
    //std::cout << cmaxc << ' ' << cdf(maxdf10, cmaxc) << std::endl; // 9.32551 0.25

    BOOST_CHECK_CLOSE_FRACTION(quantile(infdf10, 0.5), quantile(maxdf10, 0.5), tolerance); // 
    BOOST_CHECK_CLOSE_FRACTION(quantile(infdf10, 0.2), quantile(maxdf10, 0.2), tolerance); // 
    BOOST_CHECK_CLOSE_FRACTION(quantile(infdf10, 0.8), quantile(maxdf10, 0.8), tolerance); // 

    BOOST_CHECK_CLOSE_FRACTION(quantile(infdf10, 0.25), quantile(complement(infdf10, 0.75)), tolerance); // 
    BOOST_CHECK_CLOSE_FRACTION(quantile(complement(infdf10, 0.5)), quantile(complement(maxdf10, 0.5)), tolerance); // 

    BOOST_CHECK_CLOSE_FRACTION(quantile(maxdf10, 0.25), quantile(complement(maxdf10, 0.75)), tolerance); // 

    BOOST_CHECK_CLOSE_FRACTION(quantile(infdf10, 0.99), quantile(complement(infdf10, 0.01)), tolerance); // 
    BOOST_CHECK_CLOSE_FRACTION(quantile(infdf10, 0.4), quantile(complement(infdf10, 0.6)), tolerance); // 
    BOOST_CHECK_CLOSE_FRACTION(quantile(infdf10, 0.01), quantile(complement(infdf10, 1 - 0.01)), tolerance); // 
  }
} // void test_big_df(RealType)

template <class RealType>
void test_ignore_policy(RealType)
{
// Check on returns when errors are ignored.
   if ((typeid(RealType) != typeid(boost::math::concepts::real_concept))
     && std::numeric_limits<RealType>::has_infinity
     && std::numeric_limits<RealType>::has_quiet_NaN
     )
   { // Ordinary floats only.

   using namespace boost::math;
//   RealType inf = std::numeric_limits<RealType>::infinity();
   RealType nan = std::numeric_limits<RealType>::quiet_NaN();

     using boost::math::policies::policy;
  // Types of error whose action can be altered by policies:.
  //using boost::math::policies::evaluation_error;
  //using boost::math::policies::domain_error;
  //using boost::math::policies::overflow_error;
  //using boost::math::policies::underflow_error;
  //using boost::math::policies::domain_error;
  //using boost::math::policies::pole_error;

  //// Actions on error (in enum error_policy_type):
  //using boost::math::policies::errno_on_error;
  //using boost::math::policies::ignore_error;
  //using boost::math::policies::throw_on_error;
  //using boost::math::policies::denorm_error;
  //using boost::math::policies::pole_error;
  //using boost::math::policies::user_error;

  typedef policy<
    boost::math::policies::domain_error<boost::math::policies::ignore_error>,
    boost::math::policies::overflow_error<boost::math::policies::ignore_error>,
    boost::math::policies::underflow_error<boost::math::policies::ignore_error>,
    boost::math::policies::denorm_error<boost::math::policies::ignore_error>,
    boost::math::policies::pole_error<boost::math::policies::ignore_error>,
    boost::math::policies::evaluation_error<boost::math::policies::ignore_error>
              > ignore_all_policy;

   typedef non_central_t_distribution<RealType, ignore_all_policy> ignore_error_non_central_t;

  // Only test NaN and infinity if type has these features (realconcept returns zero).
  // Integers are always converted to RealType,
  // others requires static cast to RealType from long double.

  if(std::numeric_limits<RealType>::has_quiet_NaN)
  {
  // Mean
    BOOST_CHECK((boost::math::isnan)(mean(ignore_error_non_central_t(-nan, 0))));
    BOOST_CHECK((boost::math::isnan)(mean(ignore_error_non_central_t(+nan, 0))));
    BOOST_CHECK((boost::math::isnan)(mean(ignore_error_non_central_t(-1, 0))));
    BOOST_CHECK((boost::math::isnan)(mean(ignore_error_non_central_t(0, 0))));
    BOOST_CHECK((boost::math::isnan)(mean(ignore_error_non_central_t(1, 0))));
    BOOST_CHECK((boost::math::isnan)(mean(ignore_error_non_central_t(2, nan))));
    BOOST_CHECK((boost::math::isnan)(mean(ignore_error_non_central_t(nan, nan))));
    BOOST_CHECK(boost::math::isfinite(mean(ignore_error_non_central_t(2, 0)))); // OK

    // Variance
    BOOST_CHECK((boost::math::isnan)(variance(ignore_error_non_central_t(nan, 0))));
    BOOST_CHECK((boost::math::isnan)(variance(ignore_error_non_central_t(1, nan))));
    BOOST_CHECK((boost::math::isnan)(variance(ignore_error_non_central_t(2, nan))));
    BOOST_CHECK((boost::math::isnan)(variance(ignore_error_non_central_t(-1, 0))));
    BOOST_CHECK((boost::math::isnan)(variance(ignore_error_non_central_t(0, 0))));
    BOOST_CHECK((boost::math::isnan)(variance(ignore_error_non_central_t(1, 0))));
    BOOST_CHECK((boost::math::isnan)(variance(ignore_error_non_central_t(static_cast<RealType>(1.7L), 0))));
    BOOST_CHECK((boost::math::isnan)(variance(ignore_error_non_central_t(2, 0))));

  // Skewness
    BOOST_CHECK((boost::math::isnan)(skewness(ignore_error_non_central_t(std::numeric_limits<RealType>::quiet_NaN(), 0))));
    BOOST_CHECK((boost::math::isnan)(skewness(ignore_error_non_central_t(-1, 0))));
    BOOST_CHECK((boost::math::isnan)(skewness(ignore_error_non_central_t(0, 0))));
    BOOST_CHECK((boost::math::isnan)(skewness(ignore_error_non_central_t(1, 0))));
    BOOST_CHECK((boost::math::isnan)(skewness(ignore_error_non_central_t(2, 0))));
    BOOST_CHECK((boost::math::isnan)(skewness(ignore_error_non_central_t(3, 0))));

  // Kurtosis 
    BOOST_CHECK((boost::math::isnan)(kurtosis(ignore_error_non_central_t(std::numeric_limits<RealType>::quiet_NaN(), 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis(ignore_error_non_central_t(-1, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis(ignore_error_non_central_t(0, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis(ignore_error_non_central_t(1, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis(ignore_error_non_central_t(2, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis(ignore_error_non_central_t(static_cast<RealType>(2.0001L), 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis(ignore_error_non_central_t(3, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis(ignore_error_non_central_t(4, 0))));
 
    // Kurtosis excess
    BOOST_CHECK((boost::math::isnan)(kurtosis_excess(ignore_error_non_central_t(std::numeric_limits<RealType>::quiet_NaN(), 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis_excess(ignore_error_non_central_t(-1, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis_excess(ignore_error_non_central_t(0, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis_excess(ignore_error_non_central_t(1, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis_excess(ignore_error_non_central_t(2, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis_excess(ignore_error_non_central_t(static_cast<RealType>(2.0001L), 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis_excess(ignore_error_non_central_t(3, 0))));
    BOOST_CHECK((boost::math::isnan)(kurtosis_excess(ignore_error_non_central_t(4, 0))));
  } // has_quiet_NaN
  BOOST_CHECK(boost::math::isfinite(mean(ignore_error_non_central_t(1 + std::numeric_limits<RealType>::epsilon(), 0))));
  BOOST_CHECK(boost::math::isfinite(variance(ignore_error_non_central_t(2 + 2 * std::numeric_limits<RealType>::epsilon(), 0))));
  BOOST_CHECK(boost::math::isfinite(variance(ignore_error_non_central_t(static_cast<RealType>(2.0001L), 0))));
  BOOST_CHECK(boost::math::isfinite(variance(ignore_error_non_central_t(2 + 2 * std::numeric_limits<RealType>::epsilon(), 0))));
  BOOST_CHECK(boost::math::isfinite(skewness(ignore_error_non_central_t(3 + 3 * std::numeric_limits<RealType>::epsilon(), 0))));
  BOOST_CHECK(boost::math::isfinite(kurtosis(ignore_error_non_central_t(4 + 4 * std::numeric_limits<RealType>::epsilon(), 0))));
  BOOST_CHECK(boost::math::isfinite(kurtosis(ignore_error_non_central_t(static_cast<RealType>(4.0001L), 0))));

  // check_out_of_range<non_central_t_distribution<RealType> >(1, 0); // Fails one check because allows df = infinity.
  check_support<non_central_t_distribution<RealType> >(non_central_t_distribution<RealType>(1, 0));
  } // ordinary floats.
} // template <class RealType> void test_ignore_policy(RealType)


BOOST_AUTO_TEST_CASE( test_main )
{
  BOOST_MATH_CONTROL_FP;
   // Basic sanity-check spot values.
   expected_results();

   // (Parameter value, arbitrarily zero, only communicates the floating point type).
#ifdef TEST_FLOAT
   test_spots(0.0F); // Test float.
#endif
#ifdef TEST_DOUBLE
   test_spots(0.0); // Test double.
#endif
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
#ifdef TEST_LDOUBLE
   test_spots(0.0L); // Test long double.
#endif
#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS
#ifdef TEST_REAL_CONCEPT
   test_spots(boost::math::concepts::real_concept(0.)); // Test real concept.
#endif
#endif
#endif
  
#ifdef TEST_FLOAT
   test_accuracy(0.0F, "float"); // Test float.
   test_big_df(0.F); // float
#endif
#ifdef TEST_DOUBLE
   test_accuracy(0.0, "double"); // Test double.
   test_big_df(0.); // double
   test_ignore_policy(0.0);
#endif
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
#ifdef TEST_LDOUBLE
   test_accuracy(0.0L, "long double"); // Test long double.
#endif
#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS
#ifdef TEST_REAL_CONCEPT
   test_accuracy(boost::math::concepts::real_concept(0.), "real_concept"); // Test real concept.
#endif
#endif
#endif
  /* */

   
} // BOOST_AUTO_TEST_CASE( test_main )

/*

Output:

  Description: Autorun "J:\Cpp\MathToolkit\test\Math_test\Debug\test_nc_t.exe"
  Running 1 test case...
  Tests run with Microsoft Visual C++ version 10.0, Dinkumware standard library version 520, Win32
  Tolerance = 0.000596046%.
  Tolerance = 5e-010%.
  Tolerance = 5e-010%.
  Tolerance = 1e-008%.
  Testing: Non Central T
  CDF<float> Max = 0 RMS Mean=0
  
  CCDF<float> Max = 0 RMS Mean=0
  
  
  Testing: float quantile sanity check, with tests Non Central T
  Testing: Non Central T (small non-centrality)
  CDF<float> Max = 0 RMS Mean=0
  
  CCDF<float> Max = 0 RMS Mean=0
  
  
  Testing: float quantile sanity check, with tests Non Central T (small non-centrality)
  Testing: Non Central T (large parameters)
  CDF<float> Max = 0 RMS Mean=0
  
  CCDF<float> Max = 0 RMS Mean=0
  
  
  Testing: float quantile sanity check, with tests Non Central T (large parameters)
  Testing: Non Central T
  CDF<double> Max = 137.7 RMS Mean=31.5
      worst case at row: 181
      { 188.01481628417969, -282.022216796875, -298.02532958984375, 0.1552789395983287, 0.84472106040167128 }
  
  CCDF<double> Max = 150.4 RMS Mean=32.32
      worst case at row: 184
      { 191.43339538574219, 765.73358154296875, 820.14422607421875, 0.89943076553533785, 0.10056923446466212 }
  
  
  Testing: double quantile sanity check, with tests Non Central T
  Testing: Non Central T (small non-centrality)
  CDF<double> Max = 3.605 RMS Mean=1.031
      worst case at row: 42
      { 7376104448, 7.3761043495323975e-007, -1.3614851236343384, 0.086680099352107118, 0.91331990064789292 }
  
  CCDF<double> Max = 5.207 RMS Mean=1.432
      worst case at row: 38
      { 1524088576, 1.5240885886669275e-007, 1.3784774541854858, 0.91597201432644526, 0.084027985673554725 }
  
  
  Testing: double quantile sanity check, with tests Non Central T (small non-centrality)
  Testing: Non Central T (large parameters)
  CDF<double> Max = 286.4 RMS Mean=62.79
      worst case at row: 24
      { 1.3091821180254421e+019, 1309.18212890625, 1308.01171875, 0.12091797523015677, 0.87908202476984321 }
  
  CCDF<double> Max = 226.9 RMS Mean=50.41
      worst case at row: 23
      { 7.9217674231144776e+018, 792.1767578125, 793.54827880859375, 0.91489369852628, 0.085106301473719961 }
  
  
  Testing: double quantile sanity check, with tests Non Central T (large parameters)
  Testing: Non Central T
  CDF<long double> Max = 137.7 RMS Mean=31.5
      worst case at row: 181
      { 188.01481628417969, -282.022216796875, -298.02532958984375, 0.1552789395983287, 0.84472106040167128 }
  
  CCDF<long double> Max = 150.4 RMS Mean=32.32
      worst case at row: 184
      { 191.43339538574219, 765.73358154296875, 820.14422607421875, 0.89943076553533785, 0.10056923446466212 }
  
  
  Testing: long double quantile sanity check, with tests Non Central T
  Testing: Non Central T (small non-centrality)
  CDF<long double> Max = 3.605 RMS Mean=1.031
      worst case at row: 42
      { 7376104448, 7.3761043495323975e-007, -1.3614851236343384, 0.086680099352107118, 0.91331990064789292 }
  
  CCDF<long double> Max = 5.207 RMS Mean=1.432
      worst case at row: 38
      { 1524088576, 1.5240885886669275e-007, 1.3784774541854858, 0.91597201432644526, 0.084027985673554725 }
  
  
  Testing: long double quantile sanity check, with tests Non Central T (small non-centrality)
  Testing: Non Central T (large parameters)
  CDF<long double> Max = 286.4 RMS Mean=62.79
      worst case at row: 24
      { 1.3091821180254421e+019, 1309.18212890625, 1308.01171875, 0.12091797523015677, 0.87908202476984321 }
  
  CCDF<long double> Max = 226.9 RMS Mean=50.41
      worst case at row: 23
      { 7.9217674231144776e+018, 792.1767578125, 793.54827880859375, 0.91489369852628, 0.085106301473719961 }
  
  
  Testing: long double quantile sanity check, with tests Non Central T (large parameters)
  Testing: Non Central T
  CDF<real_concept> Max = 2.816e+005 RMS Mean=2.029e+004
      worst case at row: 185
      { 191.50137329101562, -957.5068359375, -1035.4078369140625, 0.072545502958829097, 0.92745449704117089 }
  
  CCDF<real_concept> Max = 1.304e+005 RMS Mean=1.529e+004
      worst case at row: 184
      { 191.43339538574219, 765.73358154296875, 820.14422607421875, 0.89943076553533785, 0.10056923446466212 }
  
  
  cdf(n10, 11)  = 0.84134471416473389 0.15865525603294373
  cdf(n10, 9)  = 0.15865525603294373 0.84134471416473389
  cdf(maxdf10, 11)  = 0.84134477376937866 0.15865525603294373
  cdf(infdf10, 11)  = 0.84134477376937866 0.15865525603294373
  cdf(n10, 11)  = 0.84134474606854293 0.15865525393145707
  cdf(n10, 9)  = 0.15865525393145707 0.84134474606854293
  cdf(maxdf10, 11)  = 0.84134474606854293 0.15865525393145707
  cdf(infdf10, 11)  = 0.84134474606854293 0.15865525393145707
  
  *** No errors detected

    Description: Autorun "J:\Cpp\MathToolkit\test\Math_test\Debug\test_nc_t.exe"
  Running 1 test case...
  Tests run with Microsoft Visual C++ version 10.0, Dinkumware standard library version 520, Win32
  Tolerance = 0.000596046%.
  Tolerance = 5e-010%.
  Tolerance = 5e-010%.
  Tolerance = 1e-008%.
  Testing: Non Central T
  CDF<float> Max = 0 RMS Mean=0
  
  CCDF<float> Max = 0 RMS Mean=0
  
  
  Testing: float quantile sanity check, with tests Non Central T
  Testing: Non Central T (small non-centrality)
  CDF<float> Max = 0 RMS Mean=0
  
  CCDF<float> Max = 0 RMS Mean=0
  
  
  Testing: float quantile sanity check, with tests Non Central T (small non-centrality)
  Testing: Non Central T (large parameters)
  CDF<float> Max = 0 RMS Mean=0
  
  CCDF<float> Max = 0 RMS Mean=0
  
  
  Testing: float quantile sanity check, with tests Non Central T (large parameters)
  Testing: Non Central T
  CDF<double> Max = 137.7 RMS Mean=31.5
      worst case at row: 181
      { 188.01481628417969, -282.022216796875, -298.02532958984375, 0.1552789395983287, 0.84472106040167128 }
  
  CCDF<double> Max = 150.4 RMS Mean=32.32
      worst case at row: 184
      { 191.43339538574219, 765.73358154296875, 820.14422607421875, 0.89943076553533785, 0.10056923446466212 }
  
  
  Testing: double quantile sanity check, with tests Non Central T
  Testing: Non Central T (small non-centrality)
  CDF<double> Max = 3.605 RMS Mean=1.031
      worst case at row: 42
      { 7376104448, 7.3761043495323975e-007, -1.3614851236343384, 0.086680099352107118, 0.91331990064789292 }
  
  CCDF<double> Max = 5.207 RMS Mean=1.432
      worst case at row: 38
      { 1524088576, 1.5240885886669275e-007, 1.3784774541854858, 0.91597201432644526, 0.084027985673554725 }
  
  
  Testing: double quantile sanity check, with tests Non Central T (small non-centrality)
  Testing: Non Central T (large parameters)
  CDF<double> Max = 286.4 RMS Mean=62.79
      worst case at row: 24
      { 1.3091821180254421e+019, 1309.18212890625, 1308.01171875, 0.12091797523015677, 0.87908202476984321 }
  
  CCDF<double> Max = 226.9 RMS Mean=50.41
      worst case at row: 23
      { 7.9217674231144776e+018, 792.1767578125, 793.54827880859375, 0.91489369852628, 0.085106301473719961 }
  
  
  Testing: double quantile sanity check, with tests Non Central T (large parameters)
  Testing: Non Central T
  CDF<long double> Max = 137.7 RMS Mean=31.5
      worst case at row: 181
      { 188.01481628417969, -282.022216796875, -298.02532958984375, 0.1552789395983287, 0.84472106040167128 }
  
  CCDF<long double> Max = 150.4 RMS Mean=32.32
      worst case at row: 184
      { 191.43339538574219, 765.73358154296875, 820.14422607421875, 0.89943076553533785, 0.10056923446466212 }
  
  
  Testing: long double quantile sanity check, with tests Non Central T
  Testing: Non Central T (small non-centrality)
  CDF<long double> Max = 3.605 RMS Mean=1.031
      worst case at row: 42
      { 7376104448, 7.3761043495323975e-007, -1.3614851236343384, 0.086680099352107118, 0.91331990064789292 }
  
  CCDF<long double> Max = 5.207 RMS Mean=1.432
      worst case at row: 38
      { 1524088576, 1.5240885886669275e-007, 1.3784774541854858, 0.91597201432644526, 0.084027985673554725 }
  
  
  Testing: long double quantile sanity check, with tests Non Central T (small non-centrality)
  Testing: Non Central T (large parameters)
  CDF<long double> Max = 286.4 RMS Mean=62.79
      worst case at row: 24
      { 1.3091821180254421e+019, 1309.18212890625, 1308.01171875, 0.12091797523015677, 0.87908202476984321 }
  
  CCDF<long double> Max = 226.9 RMS Mean=50.41
      worst case at row: 23
      { 7.9217674231144776e+018, 792.1767578125, 793.54827880859375, 0.91489369852628, 0.085106301473719961 }
  
  
  Testing: long double quantile sanity check, with tests Non Central T (large parameters)
  Testing: Non Central T
  CDF<real_concept> Max = 2.816e+005 RMS Mean=2.029e+004
      worst case at row: 185
      { 191.50137329101562, -957.5068359375, -1035.4078369140625, 0.072545502958829097, 0.92745449704117089 }
  
  CCDF<real_concept> Max = 1.304e+005 RMS Mean=1.529e+004
      worst case at row: 184
      { 191.43339538574219, 765.73358154296875, 820.14422607421875, 0.89943076553533785, 0.10056923446466212 }
  
  
  
  *** No errors detected


*/



/*

Temporary stuff from student's t version.


   // Calculate 1 / eps, the point where student's t should change to normal distribution.
    RealType limit = 1 / boost::math::tools::epsilon<RealType>();

    using namespace boost::math::policies;
    typedef policy<digits10<17> > accurate_policy; // 17 = max_digits10 where available.
    limit = 1 / policies::get_epsilon<RealType, accurate_policy>();

    BOOST_CHECK_CLOSE_FRACTION(limit, static_cast<RealType>(1) / std::numeric_limits<RealType>::epsilon(), tolerance);
    // Default policy to get full accuracy.
    // std::cout << "Switch over to normal if df > " << limit << std::endl;
    // float Switch over to normal if df > 8.38861e+006
    // double Switch over to normal if df > 4.5036e+015
    // Can't test real_concept - doesn't converge.

    boost::math::normal_distribution<RealType> n01(0, 1); // 
    boost::math::normal_distribution<RealType> n10(10, 1); // 
    non_central_t_distribution<RealType> nct(boost::math::tools::max_value<RealType>(), 0); // Well over the switchover point,
    non_central_t_distribution<RealType> nct2(limit /5, 0); // Just below the switchover point,
    non_central_t_distribution<RealType> nct3(limit /100, 0); // Well below the switchover point,
    non_central_t_distribution<RealType> nct4(limit, 10); // Well below the switchover point, and 10 non-centrality.

    // PDF
    BOOST_CHECK_CLOSE_FRACTION(pdf(nct, 0), pdf(n01, 0.), tolerance); // normal and non-central t should be nearly equal.
    BOOST_CHECK_CLOSE_FRACTION(pdf(nct2, 0), pdf(n01, 0.), tolerance); // should be very close to normal.
    BOOST_CHECK_CLOSE_FRACTION(pdf(nct3, 0), pdf(n01, 0.), tolerance * 10); // should be close to normal.
 //   BOOST_CHECK_CLOSE_FRACTION(pdf(nct4, 10), pdf(n10, 0.), tolerance * 100); // should be fairly close to normal tolerance.

    RealType delta = 10; // non-centrality.
    RealType nu = static_cast<RealType>(limit); // df
    boost::math::normal_distribution<RealType> nl(delta, 1); // Normal distribution that nct tends to for big df. 
    non_central_t_distribution<RealType> nct5(nu, delta); //
    RealType x = delta;
  //  BOOST_CHECK_CLOSE_FRACTION(pdf(nct5, x), pdf(nl, x), tolerance * 10 ); // nu = 1e15
  //  BOOST_CHECK_CLOSE_FRACTION(pdf(nct5, x), pdf(nl, x), tolerance * 1000 ); // nu = 1e14
  //  BOOST_CHECK_CLOSE_FRACTION(pdf(nct5, x), pdf(nl, x), tolerance * 10000 ); // nu = 1e13
  //  BOOST_CHECK_CLOSE_FRACTION(pdf(nct5, x), pdf(nl, x), tolerance * 100000 ); // nu = 1e12
    BOOST_CHECK_CLOSE_FRACTION(pdf(nct5, x), pdf(nl, x), tolerance * 5  ); // nu = 1/eps

  // Increasing the non-centrality delta increases the difference too because increases asymmetry.
  // For example, with non-centrality = 100, need tolerance * 500 

      // CDF
    BOOST_CHECK_CLOSE_FRACTION(cdf(nct, 0), cdf(n01, 0.), tolerance); // should be exactly equal.
    BOOST_CHECK_CLOSE_FRACTION(cdf(nct2, 0), cdf(n01, 0.), tolerance); // should be very close to normal.

    BOOST_CHECK_CLOSE_FRACTION(cdf(complement(n10, 11)), 1 - cdf(n10, 11), tolerance); // 
    //   cdf(n10, 10)  = 0.841345 0.158655
    BOOST_CHECK_CLOSE_FRACTION(cdf(complement(n10, 9)), 1 - cdf(n10, 9), tolerance); // 
    std::cout.precision(17);
    std::cout  << "cdf(n10, 11)  = " << cdf(n10, 11) << ' ' << cdf(complement(n10, 11)) << endl;
    std::cout  << "cdf(n10, 9)  = " << cdf(n10, 9) << ' ' << cdf(complement(n10, 9)) << endl;

  std::cout << std::numeric_limits<double>::max_digits10 << std::endl;
   std::cout.precision(17);

   using boost::math::tools::max_value;

   double eps = std::numeric_limits<double>::epsilon();
   // Use policies so that if policy requests lower precision, 
   // then get the normal distribution approximation earlier.
   //limit = static_cast<double>(1) / limit; // 1/eps
   double delta = 1e2;
   double df = 
   delta / (4 * eps);

    std::cout << df << std::endl; // df = 1.125899906842624e+018
     
   {
     boost::math::non_central_t_distribution<double> dist(df, delta);

      std::cout <<"mean " << mean(dist) << std::endl; // mean 1000
      std::cout <<"variance " << variance(dist) << std::endl; // variance 1
      std::cout <<"skewness " << skewness(dist) << std::endl; //  skewness 8.8817841970012523e-010
      std::cout <<"kurtosis_excess " << kurtosis_excess(dist) << std::endl; // kurtosis_excess 3.0001220703125
  //1.125899906842624e+017
  //mean 100
  //variance 1
  //skewness 8.8817841970012523e-012
  //kurtosis_excess 3


   }



  */