# Boost C++ Libraries

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### boost/math/distributions/negative_binomial.hpp

```// boost\math\special_functions\negative_binomial.hpp

// Copyright Paul A. Bristow 2007.

// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.

// 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,
// 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,
// 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:
//
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
```