negative_binomial.hpp

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    { // 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;
      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;
      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;
      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;
      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;
      }
      /*
      // 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, 1-p, P, 1-P, Policy());

      if(guess < 10)
      {
         //
         // Cornish-Fisher Negative binomial approximation not accurate in this area:
         //
         guess = (std::min)(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,
         1-P,
         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;
       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 <= boost::math::powm1(dist.success_fraction(), dist.successes(), Policy()))
       {  // q <= cdf(complement(dist, 0)) == pdf(dist, 0)
          return 0; //
       }
       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
       }
       //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, 1-p, 1-Q, Q, Policy());

       if(guess < 10)
       {
          //
          // Cornish-Fisher Negative binomial approximation not accurate in this area:
          //
          guess = (std::min)(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,
          1-Q,
          Q,
          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