1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
|
[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;
BOOST_MATH_GPU_ENABLED inverse_gamma_distribution(RealType shape, RealType scale = 1)
BOOST_MATH_GPU_ENABLED RealType shape()const;
BOOST_MATH_GPU_ENABLED 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]
BOOST_MATH_GPU_ENABLED 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.
BOOST_MATH_GPU_ENABLED RealType shape()const;
Returns the [alpha] shape parameter of this inverse gamma distribution.
BOOST_MATH_GPU_ENABLED 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.
For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
be run on both host and device.
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.]
In this distribution the implementation of `logpdf` is specialized
to improve numerical accuracy.
[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,
[beta] 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 ]]
[[logpdf][log(pdf) = [alpha] * log([beta]) + (-[alpha]-1)*log(x) - lgamma([alpha]) - ([beta]/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).
]
|
