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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
 // precede 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 continuous 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 <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*>(nullptr), 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*>(nullptr), 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.
 
     #ifdef __cpp_deduction_guides
     template <class RealType>
     negative_binomial_distribution(RealType,RealType)->negative_binomial_distribution<typename boost::math::tools::promote_args<RealType>::type>;
     #endif
 
     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:
       //
       std::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:
        //
        std::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

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