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// boost\math\special_functions\negative_binomial.hpp

// Copyright Paul A. Bristow 2007.

// Copyright John Maddock 2007.

// 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)

// http://en.wikipedia.org/wiki/negative_binomial_distribution

// http://mathworld.wolfram.com/NegativeBinomialDistribution.html

// http://documents.wolfram.com/teachersedition/Teacher/Statistics/DiscreteDistributions.html

// The negative binomial distribution NegativeBinomialDistribution[n, p]

// is the distribution of the number (k) of failures that occur in a sequence of trials before

// r successes have occurred, where the probability of success in each trial is p.

// In a sequence of Bernoulli trials or events

// (independent, yes or no, succeed or fail) with success_fraction probability p,

// negative_binomial is the probability that k or fewer failures

// preceed the r th trial's success.

// random variable k is the number of failures (NOT the probability).

// Negative_binomial distribution is a discrete probability distribution.

// But note that the negative binomial distribution

// (like others including the binomial, Poisson & Bernoulli)

// is strictly defined as a discrete function: only integral values of k are envisaged.

// However because of the method of calculation using a continuous gamma function,

// it is convenient to treat it as if a continous function,

// and permit non-integral values of k.

// However, by default the policy is to use discrete_quantile_policy.

// To enforce the strict mathematical model, users should use conversion

// on k outside this function to ensure that k is integral.

// MATHCAD cumulative negative binomial pnbinom(k, n, p)

// Implementation note: much greater speed, and perhaps greater accuracy,

// might be achieved for extreme values by using a normal approximation.

// This is NOT been tested or implemented.

#ifndef BOOST_MATH_SPECIAL_NEGATIVE_BINOMIAL_HPP

#define BOOST_MATH_SPECIAL_NEGATIVE_BINOMIAL_HPP

#include <boost/math/distributions/fwd.hpp>

#include <boost/math/special_functions/beta.hpp> // for ibeta(a, b, x) == Ix(a, b).

#include <boost/math/distributions/complement.hpp> // complement.

#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks domain_error & logic_error.

#include <boost/math/special_functions/fpclassify.hpp> // isnan.

#include <boost/math/tools/roots.hpp> // for root finding.

#include <boost/math/distributions/detail/inv_discrete_quantile.hpp>

#include <boost/type_traits/is_floating_point.hpp>

#include <boost/type_traits/is_integral.hpp>

#include <boost/type_traits/is_same.hpp>

#include <boost/mpl/if.hpp>

#include <limits> // using std::numeric_limits;

#include <utility>

#if defined (BOOST_MSVC)

#  pragma warning(push)

// This believed not now necessary, so commented out.

//#  pragma warning(disable: 4702) // unreachable code.

// in domain_error_imp in error_handling.

#endif

namespace boost

{

  namespace math

  {

    namespace negative_binomial_detail

    {

      // Common error checking routines for negative binomial distribution functions:

      template <class RealType, class Policy>

      inline bool check_successes(const char* function, const RealType& r, RealType* result, const Policy& pol)

      {

        if( !(boost::math::isfinite)(r) || (r <= 0) )

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "Number of successes argument is %1%, but must be > 0 !", r, pol);

          return false;

        }

        return true;

      }

      template <class RealType, class Policy>

      inline bool check_success_fraction(const char* function, const RealType& p, RealType* result, const Policy& pol)

      {

        if( !(boost::math::isfinite)(p) || (p < 0) || (p > 1) )

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "Success fraction argument is %1%, but must be >= 0 and <= 1 !", p, pol);

          return false;

        }

        return true;

      }

      template <class RealType, class Policy>

      inline bool check_dist(const char* function, const RealType& r, const RealType& p, RealType* result, const Policy& pol)

      {

        return check_success_fraction(function, p, result, pol)

          && check_successes(function, r, result, pol);

      }

      template <class RealType, class Policy>

      inline bool check_dist_and_k(const char* function, const RealType& r, const RealType& p, RealType k, RealType* result, const Policy& pol)

      {

        if(check_dist(function, r, p, result, pol) == false)

        {

          return false;

        }

        if( !(boost::math::isfinite)(k) || (k < 0) )

        { // Check k failures.

          *result = policies::raise_domain_error<RealType>(

            function,

            "Number of failures argument is %1%, but must be >= 0 !", k, pol);

          return false;

        }

        return true;

      } // Check_dist_and_k

      template <class RealType, class Policy>

      inline bool check_dist_and_prob(const char* function, const RealType& r, RealType p, RealType prob, RealType* result, const Policy& pol)

      {

        if((check_dist(function, r, p, result, pol) && detail::check_probability(function, prob, result, pol)) == false)

        {

          return false;

        }

        return true;

      } // check_dist_and_prob

    } //  namespace negative_binomial_detail

    template <class RealType = double, class Policy = policies::policy<> >

    class negative_binomial_distribution

    {

    public:

      typedef RealType value_type;

      typedef Policy policy_type;

      negative_binomial_distribution(RealType r, RealType p) : m_r(r), m_p(p)

      { // Constructor.

        RealType result;

        negative_binomial_detail::check_dist(

          "negative_binomial_distribution<%1%>::negative_binomial_distribution",

          m_r, // Check successes r > 0.

          m_p, // Check success_fraction 0 <= p <= 1.

          &result, Policy());

      } // negative_binomial_distribution constructor.

      // Private data getter class member functions.

      RealType success_fraction() const

      { // Probability of success as fraction in range 0 to 1.

        return m_p;

      }

      RealType successes() const

      { // Total number of successes r.

        return m_r;

      }

      static RealType find_lower_bound_on_p(

        RealType trials,

        RealType successes,

        RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.

      {

        static const char* function = "boost::math::negative_binomial<%1%>::find_lower_bound_on_p";

        RealType result = 0;  // of error checks.

        RealType failures = trials - successes;

        if(false == detail::check_probability(function, alpha, &result, Policy())

          && negative_binomial_detail::check_dist_and_k(

          function, successes, RealType(0), failures, &result, Policy()))

        {

          return result;

        }

        // Use complement ibeta_inv function for lower bound.

        // This is adapted from the corresponding binomial formula

        // here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm

        // This is a Clopper-Pearson interval, and may be overly conservative,

        // see also "A Simple Improved Inferential Method for Some

        // Discrete Distributions" Yong CAI and K. KRISHNAMOORTHY

        // http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf

        //

        return ibeta_inv(successes, failures + 1, alpha, static_cast<RealType*>(0), Policy());

      } // find_lower_bound_on_p

      static RealType find_upper_bound_on_p(

        RealType trials,

        RealType successes,

        RealType alpha) // alpha 0.05 equivalent to 95% for one-sided test.

      {

        static const char* function = "boost::math::negative_binomial<%1%>::find_upper_bound_on_p";

        RealType result = 0;  // of error checks.

        RealType failures = trials - successes;

        if(false == negative_binomial_detail::check_dist_and_k(

          function, successes, RealType(0), failures, &result, Policy())

          && detail::check_probability(function, alpha, &result, Policy()))

        {

          return result;

        }

        if(failures == 0)

           return 1;

        // Use complement ibetac_inv function for upper bound.

        // Note adjusted failures value: *not* failures+1 as usual.

        // This is adapted from the corresponding binomial formula

        // here: http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm

        // This is a Clopper-Pearson interval, and may be overly conservative,

        // see also "A Simple Improved Inferential Method for Some

        // Discrete Distributions" Yong CAI and K. KRISHNAMOORTHY

        // http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf

        //

        return ibetac_inv(successes, failures, alpha, static_cast<RealType*>(0), Policy());

      } // find_upper_bound_on_p

      // Estimate number of trials :

      // "How many trials do I need to be P% sure of seeing k or fewer failures?"

      static RealType find_minimum_number_of_trials(

        RealType k,     // number of failures (k >= 0).

        RealType p,     // success fraction 0 <= p <= 1.

        RealType alpha) // risk level threshold 0 <= alpha <= 1.

      {

        static const char* function = "boost::math::negative_binomial<%1%>::find_minimum_number_of_trials";

        // Error checks:

        RealType result = 0;

        if(false == negative_binomial_detail::check_dist_and_k(

          function, RealType(1), p, k, &result, Policy())

          && detail::check_probability(function, alpha, &result, Policy()))

        { return result; }

        result = ibeta_inva(k + 1, p, alpha, Policy());  // returns n - k

        return result + k;

      } // RealType find_number_of_failures

      static RealType find_maximum_number_of_trials(

        RealType k,     // number of failures (k >= 0).

        RealType p,     // success fraction 0 <= p <= 1.

        RealType alpha) // risk level threshold 0 <= alpha <= 1.

      {

        static const char* function = "boost::math::negative_binomial<%1%>::find_maximum_number_of_trials";

        // Error checks:

        RealType result = 0;

        if(false == negative_binomial_detail::check_dist_and_k(

          function, RealType(1), p, k, &result, Policy())

          &&  detail::check_probability(function, alpha, &result, Policy()))

        { return result; }

        result = ibetac_inva(k + 1, p, alpha, Policy());  // returns n - k

        return result + k;

      } // RealType find_number_of_trials complemented

    private:

      RealType m_r; // successes.

      RealType m_p; // success_fraction

    }; // template <class RealType, class Policy> class negative_binomial_distribution

    typedef negative_binomial_distribution<double> negative_binomial; // Reserved name of type double.

    template <class RealType, class Policy>

    inline const std::pair<RealType, RealType> range(const negative_binomial_distribution<RealType, Policy>& /* dist */)

    { // Range of permissible values for random variable k.

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

       return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // max_integer?

    }

    template <class RealType, class Policy>

    inline const std::pair<RealType, RealType> support(const negative_binomial_distribution<RealType, Policy>& /* dist */)

    { // Range of supported values for random variable k.

       // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.

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

       return std::pair<RealType, RealType>(static_cast<RealType>(0),  max_value<RealType>()); // max_integer?

    }

    template <class RealType, class Policy>

    inline RealType mean(const negative_binomial_distribution<RealType, Policy>& dist)

    { // Mean of Negative Binomial distribution = r(1-p)/p.

      return dist.successes() * (1 - dist.success_fraction() ) / dist.success_fraction();

    } // mean

    //template <class RealType, class Policy>

    //inline RealType median(const negative_binomial_distribution<RealType, Policy>& dist)

    //{ // Median of negative_binomial_distribution is not defined.

    //  return policies::raise_domain_error<RealType>(BOOST_CURRENT_FUNCTION, "Median is not implemented, result is %1%!", std::numeric_limits<RealType>::quiet_NaN());

    //} // median

    // Now implemented via quantile(half) in derived accessors.

    template <class RealType, class Policy>

    inline RealType mode(const negative_binomial_distribution<RealType, Policy>& dist)

    { // Mode of Negative Binomial distribution = floor[(r-1) * (1 - p)/p]

      BOOST_MATH_STD_USING // ADL of std functions.

      return floor((dist.successes() -1) * (1 - dist.success_fraction()) / dist.success_fraction());

    } // mode

    template <class RealType, class Policy>

    inline RealType skewness(const negative_binomial_distribution<RealType, Policy>& dist)

    { // skewness of Negative Binomial distribution = 2-p / (sqrt(r(1-p))

      BOOST_MATH_STD_USING // ADL of std functions.

      RealType p = dist.success_fraction();

      RealType r = dist.successes();

      return (2 - p) /

        sqrt(r * (1 - p));

    } // skewness

    template <class RealType, class Policy>

    inline RealType kurtosis(const negative_binomial_distribution<RealType, Policy>& dist)

    { // kurtosis of Negative Binomial distribution

      // http://en.wikipedia.org/wiki/Negative_binomial is kurtosis_excess so add 3

      RealType p = dist.success_fraction();

      RealType r = dist.successes();

      return 3 + (6 / r) + ((p * p) / (r * (1 - p)));

    } // kurtosis

     template <class RealType, class Policy>

    inline RealType kurtosis_excess(const negative_binomial_distribution<RealType, Policy>& dist)

    { // kurtosis excess of Negative Binomial distribution

      // http://mathworld.wolfram.com/Kurtosis.html table of kurtosis_excess

      RealType p = dist.success_fraction();

      RealType r = dist.successes();

      return (6 - p * (6-p)) / (r * (1-p));

    } // kurtosis_excess

    template <class RealType, class Policy>

    inline RealType variance(const negative_binomial_distribution<RealType, Policy>& dist)

    { // Variance of Binomial distribution = r (1-p) / p^2.

      return  dist.successes() * (1 - dist.success_fraction())

        / (dist.success_fraction() * dist.success_fraction());

    } // variance

    // RealType standard_deviation(const negative_binomial_distribution<RealType, Policy>& dist)

    // standard_deviation provided by derived accessors.

    // RealType hazard(const negative_binomial_distribution<RealType, Policy>& dist)

    // hazard of Negative Binomial distribution provided by derived accessors.

    // RealType chf(const negative_binomial_distribution<RealType, Policy>& dist)

    // chf of Negative Binomial distribution provided by derived accessors.

    template <class RealType, class Policy>

    inline RealType pdf(const negative_binomial_distribution<RealType, Policy>& dist, const RealType& k)

    { // Probability Density/Mass Function.

      BOOST_FPU_EXCEPTION_GUARD

      static const char* function = "boost::math::pdf(const negative_binomial_distribution<%1%>&, %1%)";

      RealType r = dist.successes();

      RealType p = dist.success_fraction();

      RealType result = 0;

      if(false == negative_binomial_detail::check_dist_and_k(

        function,

        r,

        dist.success_fraction(),

        k,

        &result, Policy()))

      {

        return result;

      }

      result = (p/(r + k)) * ibeta_derivative(r, static_cast<RealType>(k+1), p, Policy());

      // Equivalent to:

      // return exp(lgamma(r + k) - lgamma(r) - lgamma(k+1)) * pow(p, r) * pow((1-p), k);

      return result;

    } // negative_binomial_pdf

    template <class RealType, class Policy>

    inline RealType cdf(const negative_binomial_distribution<RealType, Policy>& dist, const RealType& k)

    { // Cumulative Distribution Function of Negative Binomial.

      static const char* function = "boost::math::cdf(const negative_binomial_distribution<%1%>&, %1%)";

      using boost::math::ibeta; // Regularized incomplete beta function.

      // k argument may be integral, signed, or unsigned, or floating point.

      // If necessary, it has already been promoted from an integral type.

      RealType p = dist.success_fraction();

      RealType r = dist.successes();

      // Error check:

      RealType result = 0;

      if(false == negative_binomial_detail::check_dist_and_k(

        function,

        r,

        dist.success_fraction(),

        k,

        &result, Policy()))

      {

        return result;

      }

      RealType probability = ibeta(r, static_cast<RealType>(k+1), p, Policy());

      // Ip(r, k+1) = ibeta(r, k+1, p)

      return probability;

    } // cdf Cumulative Distribution Function Negative Binomial.

      template <class RealType, class Policy>

      inline RealType cdf(const complemented2_type<negative_binomial_distribution<RealType, Policy>, RealType>& c)

      { // Complemented Cumulative Distribution Function Negative Binomial.

      static const char* function = "boost::math::cdf(const negative_binomial_distribution<%1%>&, %1%)";

      using boost::math::ibetac; // Regularized incomplete beta function complement.

      // k argument may be integral, signed, or unsigned, or floating point.

      // If necessary, it has already been promoted from an integral type.

      RealType const& k = c.param;

      negative_binomial_distribution<RealType, Policy> const& dist = c.dist;

      RealType p = dist.success_fraction();

      RealType r = dist.successes();

      // Error check:

      RealType result = 0;

      if(false == negative_binomial_detail::check_dist_and_k(

        function,

        r,

        p,

        k,

        &result, Policy()))

      {

        return result;

      }

      // Calculate cdf negative binomial using the incomplete beta function.

      // Use of ibeta here prevents cancellation errors in calculating

      // 1-p if p is very small, perhaps smaller than machine epsilon.

      // Ip(k+1, r) = ibetac(r, k+1, p)

      // constrain_probability here?

     RealType probability = ibetac(r, static_cast<RealType>(k+1), p, Policy());

      // Numerical errors might cause probability to be slightly outside the range < 0 or > 1.

      // This might cause trouble downstream, so warn, possibly throw exception, but constrain to the limits.

      return probability;

    } // cdf Cumulative Distribution Function Negative Binomial.

    template <class RealType, class Policy>

    inline RealType quantile(const negative_binomial_distribution<RealType, Policy>& dist, const RealType& P)

    { // Quantile, percentile/100 or Percent Point Negative Binomial function.

      // Return the number of expected failures k for a given probability p.

      // Inverse cumulative Distribution Function or Quantile (percentile / 100) of negative_binomial Probability.

      // MAthCAD pnbinom return smallest k such that negative_binomial(k, n, p) >= probability.

      // k argument may be integral, signed, or unsigned, or floating point.

      // BUT Cephes/CodeCogs says: finds argument p (0 to 1) such that cdf(k, n, p) = y

      static const char* function = "boost::math::quantile(const negative_binomial_distribution<%1%>&, %1%)";

      BOOST_MATH_STD_USING // ADL of std functions.

      RealType p = dist.success_fraction();

      RealType r = dist.successes();

      // Check dist and P.

      RealType result = 0;

      if(false == negative_binomial_detail::check_dist_and_prob

        (function, r, p, P, &result, Policy()))

      {

        return result;

      }

      // Special cases.

      if (P == 1)

      {  // Would need +infinity failures for total confidence.

        result = policies::raise_overflow_error<RealType>(

            function,

            "Probability argument is 1, which implies infinite failures !", Policy());

        return result;

       // usually means return +std::numeric_limits<RealType>::infinity();

       // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR

      }

      if (P == 0)

      { // No failures are expected if P = 0.

        return 0; // Total trials will be just dist.successes.

      }

      if (P <= pow(dist.success_fraction(), dist.successes()))

      { // p <= pdf(dist, 0) == cdf(dist, 0)

        return 0;

      }

      if(p == 0)

      {  // Would need +infinity failures for total confidence.

         result = policies::raise_overflow_error<RealType>(

            function,

            "Success fraction is 0, which implies infinite failures !", Policy());

         return result;

         // usually means return +std::numeric_limits<RealType>::infinity();

         // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR

      }

      /*

      // Calculate quantile of negative_binomial using the inverse incomplete beta function.

      using boost::math::ibeta_invb;

      return ibeta_invb(r, p, P, Policy()) - 1; //

      */

      RealType guess = 0;

      RealType factor = 5;

      if(r * r * r * P * p > 0.005)

         guess = detail::inverse_negative_binomial_cornish_fisher(r, p, RealType(1-p), P, RealType(1-P), Policy());

      if(guess < 10)

      {

         //

         // Cornish-Fisher Negative binomial approximation not accurate in this area:

         //

         guess = (std::min)(RealType(r * 2), RealType(10));

      }

      else

         factor = (1-P < sqrt(tools::epsilon<RealType>())) ? 2 : (guess < 20 ? 1.2f : 1.1f);

      BOOST_MATH_INSTRUMENT_CODE("guess = " << guess);

      //

      // Max iterations permitted:

      //

      boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();

      typedef typename Policy::discrete_quantile_type discrete_type;

      return detail::inverse_discrete_quantile(

         dist,

         P,

         false,

         guess,

         factor,

         RealType(1),

         discrete_type(),

         max_iter);

    } // RealType quantile(const negative_binomial_distribution dist, p)

    template <class RealType, class Policy>

    inline RealType quantile(const complemented2_type<negative_binomial_distribution<RealType, Policy>, RealType>& c)

    {  // Quantile or Percent Point Binomial function.

       // Return the number of expected failures k for a given

       // complement of the probability Q = 1 - P.

       static const char* function = "boost::math::quantile(const negative_binomial_distribution<%1%>&, %1%)";

       BOOST_MATH_STD_USING

       // Error checks:

       RealType Q = c.param;

       const negative_binomial_distribution<RealType, Policy>& dist = c.dist;

       RealType p = dist.success_fraction();

       RealType r = dist.successes();

       RealType result = 0;

       if(false == negative_binomial_detail::check_dist_and_prob(

          function,

          r,

          p,

          Q,

          &result, Policy()))

       {

          return result;

       }

       // Special cases:

       //

       if(Q == 1)

       {  // There may actually be no answer to this question,

          // since the probability of zero failures may be non-zero,

          return 0; // but zero is the best we can do:

       }

       if(Q == 0)

       {  // Probability 1 - Q  == 1 so infinite failures to achieve certainty.

          // Would need +infinity failures for total confidence.

          result = policies::raise_overflow_error<RealType>(

             function,

             "Probability argument complement is 0, which implies infinite failures !", Policy());

          return result;

          // usually means return +std::numeric_limits<RealType>::infinity();

          // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR

       }

       if (-Q <= boost::math::powm1(dist.success_fraction(), dist.successes(), Policy()))

       {  // q <= cdf(complement(dist, 0)) == pdf(dist, 0)

          return 0; //

       }

       if(p == 0)

       {  // Success fraction is 0 so infinite failures to achieve certainty.

          // Would need +infinity failures for total confidence.

          result = policies::raise_overflow_error<RealType>(

             function,

             "Success fraction is 0, which implies infinite failures !", Policy());

          return result;

          // usually means return +std::numeric_limits<RealType>::infinity();

          // unless #define BOOST_MATH_THROW_ON_OVERFLOW_ERROR

       }

       //return ibetac_invb(r, p, Q, Policy()) -1;

       RealType guess = 0;

       RealType factor = 5;

       if(r * r * r * (1-Q) * p > 0.005)

          guess = detail::inverse_negative_binomial_cornish_fisher(r, p, RealType(1-p), RealType(1-Q), Q, Policy());

       if(guess < 10)

       {

          //

          // Cornish-Fisher Negative binomial approximation not accurate in this area:

          //

          guess = (std::min)(RealType(r * 2), RealType(10));

       }

       else

          factor = (Q < sqrt(tools::epsilon<RealType>())) ? 2 : (guess < 20 ? 1.2f : 1.1f);

       BOOST_MATH_INSTRUMENT_CODE("guess = " << guess);

       //

       // Max iterations permitted:

       //

       boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();

       typedef typename Policy::discrete_quantile_type discrete_type;

       return detail::inverse_discrete_quantile(

          dist,

          Q,

          true,

          guess,

          factor,

          RealType(1),

          discrete_type(),

          max_iter);

    } // quantile complement

 } // namespace math

} // namespace boost

// This include must be at the end, *after* the accessors

// for this distribution have been defined, in order to

// keep compilers that support two-phase lookup happy.

#include <boost/math/distributions/detail/derived_accessors.hpp>

#if defined (BOOST_MSVC)

# pragma warning(pop)

#endif

#endif // BOOST_MATH_SPECIAL_NEGATIVE_BINOMIAL_HPP