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[section:saspoint5_dist S[alpha]S Point5 Distribution]
``#include <boost/math/distributions/saspoint5.hpp>``
template <class RealType = double,
class ``__Policy`` = ``__policy_class`` >
class saspoint5_distribution;
typedef saspoint5_distribution<> saspoint5;
template <class RealType, class ``__Policy``>
class saspoint5_distribution
{
public:
typedef RealType value_type;
typedef Policy policy_type;
BOOST_MATH_GPU_ENABLED saspoint5_distribution(RealType location = 0, RealType scale = 1);
BOOST_MATH_GPU_ENABLED RealType location()const;
BOOST_MATH_GPU_ENABLED RealType scale()const;
};
It is special case of a [@http://en.wikipedia.org/wiki/Stable_distribution stable distribution]
with shape parameter [alpha]=1/2, [beta]=0.
[@http://en.wikipedia.org/wiki/Probability_distribution probability distribution function PDF]
given by:
[equation saspoint5_ref1] [/f(x; \mu, c)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \exp( i t \mu - \sqrt{|c t|} ) e^{-i x t} dt]
The location parameter [mu] is the location of the distribution,
while the scale parameter [c] determines the width of the distribution.
If the location is
zero, and the scale 1, then the result is a standard S[alpha]S Point5
distribution.
This distribution has heavier tails than the Cauchy distribution.
The following graph shows how the distributions moves as the
location parameter changes:
[graph saspoint5_pdf1]
While the following graph shows how the shape (scale) parameter alters
the distribution:
[graph saspoint5_pdf2]
[h4 Member Functions]
BOOST_MATH_GPU_ENABLED saspoint5_distribution(RealType location = 0, RealType scale = 1);
Constructs a S[alpha]S Point5 distribution, with location parameter /location/
and scale parameter /scale/. When these parameters take their default
values (location = 0, scale = 1)
then the result is a Standard S[alpha]S Point5 Distribution.
Requires scale > 0, otherwise calls __domain_error.
BOOST_MATH_GPU_ENABLED RealType location()const;
Returns the location parameter of the distribution.
BOOST_MATH_GPU_ENABLED RealType scale()const;
Returns the scale parameter of the 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.
Note however that the S[alpha]S Point5 distribution does not have a mean,
standard deviation, etc. See __math_undefined
[/link math_toolkit.pol_ref.assert_undefined mathematically undefined function]
to control whether these should fail to compile with a BOOST_STATIC_ASSERTION_FAILURE,
which is the default.
Alternately, the functions __mean, __sd,
__variance, __skewness, __kurtosis and __kurtosis_excess will all
return a __domain_error if called.
The domain of the random variable is \[-[max_value], +[min_value]\].
[h4 Accuracy]
The error is within 4 epsilon.
Errors in the PDF at 64-bit double precision:
[$../graphs/saspoint5_pdf_accuracy_64.png]
Errors in the CDF-complement at 64-bit double precision:
[$../graphs/saspoint5_ccdf_accuracy_64.png]
[h4 Implementation]
See references.
[h4 References]
* T. Yoshimura, Numerical Evaluation and High Precision Approximation Formula for S[alpha]S Point5 Distribution,
DOI: 10.36227/techrxiv.172055253.37208198/v1, 2024.
[endsect][/section:saspoint5_dist saspoint5]
[/ saspoint5.qbk
Copyright Takuma Yoshimura 2024.
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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