test_students_t.cpp

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// Copyright Paul A. Bristow 2006.
// Copyright John Maddock 2006.

// 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)

// test_students_t.cpp

// http://en.wikipedia.org/wiki/Student%27s_t_distribution
// http://www.itl.nist.gov/div898/handbook/eda/section3/eda3664.htm

// Basic sanity test for Student's t probability (quantile) (0. < p < 1).
// and Student's t probability Quantile (0. < p < 1).

#include <boost/test/included/test_exec_monitor.hpp> // Boost.Test
#include <boost/test/floating_point_comparison.hpp>

#include <boost/math/concepts/real_concept.hpp> // for real_concept
#include <boost/math/distributions/students_t.hpp>
    using boost::math::students_t_distribution;
#include <boost/math/tools/test.hpp> // for real_concept

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

template <class RealType>
RealType naive_pdf(RealType v, RealType t)
{
   // Calculate the pdf of the students t in a deliberately
   // naive way, using equation (5) from
   // http://mathworld.wolfram.com/Studentst-Distribution.html
   // This is equivalent to, but a different method
   // to the one in the actual implementation, so can be used as
   // a very basic sanity check.  However some published values
   // would be nice....

   using namespace std;  // for ADL
   using boost::math::beta;

   //return pow(v / (v + t*t), (1+v) / 2) / (sqrt(v) * beta(v/2, RealType(0.5f)));
   RealType result = boost::math::tgamma_ratio((v+1)/2, v/2);
   result /= sqrt(v * boost::math::constants::pi<RealType>());
   result /= pow(1 + t*t/v, (v+1)/2);
   return result;
}

template <class RealType>
void test_spots(RealType)
{
  // Basic sanity checks
   RealType tolerance = 1e-4; // 1e-6 (as %)
   // Some tests only pass at 1e-5 because probability value is less accurate,
   // a digit in 6th decimal place, although calculated using
   // a t-distribution generator (claimed 6 decimal digits) at
  // http://faculty.vassar.edu/lowry/VassarStats.html
   // http://faculty.vassar.edu/lowry/tsamp.html
   // df = 5, +/-t = 2.0, 1-tailed = 0.050970, 2-tailed = 0.101939

   cout << "Tolerance for type " << typeid(RealType).name()  << " is " << tolerance << " %" << endl;

   // http://en.wikipedia.org/wiki/Student%27s_t_distribution#Table_of_selected_values
  // Using tabulated value of t = 3.182 for 0.975, 3 df, one-sided.

   // http://www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf refers to:

   // A lookup table of quantiles of the RealType distribution
  // for 1 to 25 in steps of 0.1 is provided in CSV form at:
  // www.mth.kcl.ac.uk/~shaww/web_page/papers/Tsupp/tquantiles.csv
   // gives accurate t of -3.1824463052837 and 3 degrees of freedom.
   // Values below are from this source, saved as tquantiles.xls.
   // DF are across the columns, probabilities down the rows
   // and the t- values (quantiles) are shown.
   // These values are probably accurate to nearly 64-bit double
  // (perhaps 14 decimal digits).

   BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(2),       // degrees_of_freedom
         static_cast<RealType>(-6.96455673428326)),  // t
         static_cast<RealType>(0.01),                // probability.
         tolerance); // %

   BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(5),       // degrees_of_freedom
         static_cast<RealType>(-3.36492999890721)),  // t
         static_cast<RealType>(0.01),                // probability.
         tolerance);

   BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(1),      // degrees_of_freedom
         static_cast<RealType>(-31830.988607907)),  // t
         static_cast<RealType>(0.00001),            // probability.
         tolerance);

   BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(25.),    // degrees_of_freedom
         static_cast<RealType>(-5.2410429995425)),  // t
         static_cast<RealType>(0.00001),            // probability.
         tolerance);

   BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(1),   // degrees_of_freedom
         static_cast<RealType>(-63661.97723)),   // t
         static_cast<RealType>(0.000005),        // probability.
         tolerance);

    BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(5.),  // degrees_of_freedom
         static_cast<RealType>(-17.89686614)),   // t
         static_cast<RealType>(0.000005),        // probability.
         tolerance);

    BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(25.),  // degrees_of_freedom
         static_cast<RealType>(-5.510848412)),    // t
         static_cast<RealType>(0.000005),         // probability.
         tolerance);

  BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(10.),  // degrees_of_freedom
         static_cast<RealType>(-1.812461123)),    // t
         static_cast<RealType>(0.05),             // probability.
         tolerance);

  BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(10),  // degrees_of_freedom
         static_cast<RealType>(1.812461123)),    // t
         static_cast<RealType>(0.95),            // probability.
         tolerance);

  BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         complement(
            students_t_distribution<RealType>(10),  // degrees_of_freedom
            static_cast<RealType>(1.812461123))),    // t
         static_cast<RealType>(0.05),            // probability.
         tolerance);

  BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(10),  // degrees_of_freedom
         static_cast<RealType>(9.751995491)),    // t
         static_cast<RealType>(0.999999),        // probability.
         tolerance);

  BOOST_CHECK_CLOSE(
      ::boost::math::cdf(
         students_t_distribution<RealType>(10.),  // degrees_of_freedom - for ALL degrees_of_freedom!
         static_cast<RealType>(0.)),              // t
         static_cast<RealType>(0.5),              // probability.
         tolerance);


   // Student's t Inverse function tests.
  // Special cases

  BOOST_CHECK_THROW(boost::math::quantile(
         students_t_distribution<RealType>(1.),  // degrees_of_freedom (ignored).
         static_cast<RealType>(0)), std::overflow_error); // t == -infinity.

  BOOST_CHECK_THROW(boost::math::quantile(
         students_t_distribution<RealType>(1.),  // degrees_of_freedom (ignored).
         static_cast<RealType>(1)), std::overflow_error); // t == +infinity.

  BOOST_CHECK_EQUAL(boost::math::quantile(
         students_t_distribution<RealType>(1.),  // degrees_of_freedom (ignored).
         static_cast<RealType>(0.5)),  //  probability == half - special case.
         static_cast<RealType>(0)); // t == zero.

  BOOST_CHECK_EQUAL(boost::math::quantile(
         complement(
            students_t_distribution<RealType>(1.),  // degrees_of_freedom (ignored).
            static_cast<RealType>(0.5))),  //  probability == half - special case.
         static_cast<RealType>(0)); // t == zero.

  BOOST_CHECK_CLOSE(boost::math::quantile(
         students_t_distribution<RealType>(1.),  // degrees_of_freedom (ignored).
         static_cast<RealType>(0.5)),  //  probability == half - special case.
         static_cast<RealType>(0), // t == zero.
         tolerance);

   BOOST_CHECK_CLOSE( // Tests of p middling.
      ::boost::math::cdf(
         students_t_distribution<RealType>(5.),  // degrees_of_freedom
         static_cast<RealType>(-0.559429644)),  // t
         static_cast<RealType>(0.3), // probability.
         tolerance);
   BOOST_CHECK_CLOSE(
      ::boost::math::quantile(
         students_t_distribution<RealType>(5.),  // degrees_of_freedom
         static_cast<RealType>(0.3)),  // probability.
         static_cast<RealType>(-0.559429644), // t
         tolerance);

   BOOST_CHECK_CLOSE(