...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#include <boost/math/distributions/inverse_gamma.hpp>
namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > 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 inverse gamma distribution.
R inverse gamma distribution functions.
Wolfram inverse gamma distribution.
See also Gamma Distribution.
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 boost::math::inverse_gamma_distribution<>
or you can write |
For shape parameter α and scale parameter β, it is defined by the probability density function (PDF):
f(x;α, β) = βα * (1/x) α+1 exp(-β/x) / Γ(α)
and cumulative density function (CDF)
F(x;α, β) = Γ(α, β/x) / Γ(α)
The following graphs illustrate how the PDF and CDF of the inverse gamma distribution varies as the parameters vary:
inverse_gamma_distribution(RealType shape = 1, RealType scale = 1);
Constructs an inverse gamma distribution with shape α and scale β.
Requires that the shape and scale parameters are greater than zero, otherwise calls domain_error.
RealType shape()const;
Returns the α shape parameter of this inverse gamma distribution.
RealType scale()const;
Returns the β scale parameter of this inverse gamma distribution.
All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.
The domain of the random variate is [0,+∞].
Note | |
---|---|
Unlike some definitions, this implementation supports a random variate equal to zero as a special case, returning zero for pdf and cdf. |
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.
In the following table α is the shape parameter of the distribution, α is its scale parameter, x is the random variate, p is the probability and q = 1-p.
Function |
Implementation Notes |
---|---|
|
Using the relation: pdf = gamma_p_derivative(α, β/ x, β) / x * x |
cdf |
Using the relation: p = gamma_q(α, β / x) |
cdf complement |
Using the relation: q = gamma_p(α, β / x) |
quantile |
Using the relation: x = β/ gamma_q_inv(α, p) |
quantile from the complement |
Using the relation: x = α/ gamma_p_inv(α, q) |
mode |
β / (α + 1) |
median |
no analytic equation is known, but is evaluated as quantile(0.5) |
mean |
β / (α - 1) for α > 1, else a domain_error |
variance |
(β * β) / ((α - 1) * (α - 1) * (α - 2)) for α >2, else a domain_error |
skewness |
4 * sqrt (α -2) / (α -3) for α >3, else a domain_error |
kurtosis_excess |
(30 * α - 66) / ((α-3)*(α - 4)) for α >4, else a domain_error |