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- [section:inverse_gamma_dist Inverse Gamma Distribution]
- ``#include <boost/math/distributions/inverse_gamma.hpp>``
- namespace boost{ namespace math{
-
- template <class RealType = double,
- class ``__Policy`` = ``__policy_class`` >
- class inverse_gamma_distribution
- {
- public:
- typedef RealType value_type;
- typedef Policy policy_type;
- inverse_gamma_distribution(RealType shape, RealType scale = 1)
- RealType shape()const;
- RealType scale()const;
- };
-
- }} // namespaces
-
- The inverse_gamma distribution is a continuous probability distribution
- of the reciprocal of a variable distributed according to the gamma distribution.
- The inverse_gamma distribution is used in Bayesian statistics.
- See [@http://en.wikipedia.org/wiki/Inverse-gamma_distribution inverse gamma distribution].
- [@http://rss.acs.unt.edu/Rdoc/library/pscl/html/igamma.html R inverse gamma distribution functions].
- [@http://reference.wolfram.com/mathematica/ref/InverseGammaDistribution.html Wolfram inverse gamma distribution].
- See also __gamma_distrib.
- [note
- In spite of potential confusion with the inverse gamma function, this
- distribution *does* provide the typedef:
- ``typedef inverse_gamma_distribution<double> gamma;``
- If you want a `double` precision gamma distribution you can use
- ``boost::math::inverse_gamma_distribution<>``
- or you can write `inverse_gamma my_ig(2, 3);`]
- For shape parameter [alpha] and scale parameter [beta], it is defined
- by the probability density function (PDF):
- [expression f(x;[alpha], [beta]) = [beta][super [alpha]] * (1/x) [super [alpha]+1] exp(-[beta]/x) / [Gamma]([alpha])]
- and cumulative density function (CDF)
- [expression F(x;[alpha], [beta]) = [Gamma]([alpha], [beta]/x) / [Gamma]([alpha])]
- The following graphs illustrate how the PDF and CDF of the inverse gamma distribution
- varies as the parameters vary:
- [graph inverse_gamma_pdf] [/png or svg]
- [graph inverse_gamma_cdf]
- [h4 Member Functions]
- inverse_gamma_distribution(RealType shape = 1, RealType scale = 1);
-
- Constructs an inverse gamma distribution with shape [alpha] and scale [beta].
- Requires that the shape and scale parameters are greater than zero, otherwise calls
- __domain_error.
- RealType shape()const;
-
- Returns the [alpha] shape parameter of this inverse gamma distribution.
-
- RealType scale()const;
-
- Returns the [beta] scale parameter of this inverse gamma distribution.
- [h4 Non-member Accessors]
- All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
- distributions are supported: __usual_accessors.
- The domain of the random variate is \[0,+[infin]\].
- [note Unlike some definitions, this implementation supports a random variate
- equal to zero as a special case, returning zero for pdf and cdf.]
- [h4 Accuracy]
- The inverse gamma distribution is implemented in terms of the
- incomplete gamma functions __gamma_p and __gamma_q and their
- inverses __gamma_p_inv and __gamma_q_inv: refer to the accuracy
- data for those functions for more information.
- But in general, inverse_gamma results are accurate to a few epsilon,
- >14 decimal digits accuracy for 64-bit double.
- [h4 Implementation]
- In the following table [alpha] is the shape parameter of the distribution,
- [alpha] is its scale parameter, /x/ is the random variate, /p/ is the probability
- and /q = 1-p/.
- [table
- [[Function][Implementation Notes]]
- [[pdf][Using the relation: pdf = __gamma_p_derivative([alpha], [beta]/ x, [beta]) / x * x ]]
- [[cdf][Using the relation: p = __gamma_q([alpha], [beta] / x) ]]
- [[cdf complement][Using the relation: q = __gamma_p([alpha], [beta] / x) ]]
- [[quantile][Using the relation: x = [beta]/ __gamma_q_inv([alpha], p) ]]
- [[quantile from the complement][Using the relation: x = [alpha]/ __gamma_p_inv([alpha], q) ]]
- [[mode][[beta] / ([alpha] + 1) ]]
- [[median][no analytic equation is known, but is evaluated as quantile(0.5)]]
- [[mean][[beta] / ([alpha] - 1) for [alpha] > 1, else a __domain_error]]
- [[variance][([beta] * [beta]) / (([alpha] - 1) * ([alpha] - 1) * ([alpha] - 2)) for [alpha] >2, else a __domain_error]]
- [[skewness][4 * sqrt ([alpha] -2) / ([alpha] -3) for [alpha] >3, else a __domain_error]]
- [[kurtosis_excess][(30 * [alpha] - 66) / (([alpha]-3)*([alpha] - 4)) for [alpha] >4, else a __domain_error]]
- ] [/table]
- [endsect] [/section:inverse_gamma_dist Inverse Gamma Distribution]
- [/
- Copyright 2010 John Maddock and Paul A. Bristow.
- Distributed under 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).
- ]
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