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- [section:inverse_gaussian_dist Inverse Gaussian (or Inverse Normal) Distribution]
- ``#include <boost/math/distributions/inverse_gaussian.hpp>``
- namespace boost{ namespace math{
-
- template <class RealType = double,
- class ``__Policy`` = ``__policy_class`` >
- class inverse_gaussian_distribution
- {
- public:
- typedef RealType value_type;
- typedef Policy policy_type;
- inverse_gaussian_distribution(RealType mean = 1, RealType scale = 1);
- RealType mean()const; // mean default 1.
- RealType scale()const; // Optional scale, default 1 (unscaled).
- RealType shape()const; // Shape = scale/mean.
- };
- typedef inverse_gaussian_distribution<double> inverse_gaussian;
- }} // namespace boost // namespace math
-
- The Inverse Gaussian distribution distribution is a continuous probability distribution.
- The distribution is also called 'normal-inverse Gaussian distribution',
- and 'normal Inverse' distribution.
- It is also convenient to provide unity as default for both mean and scale.
- This is the Standard form for all distributions.
- The Inverse Gaussian distribution was first studied in relation to Brownian motion.
- In 1956 M.C.K. Tweedie used the name Inverse Gaussian because there is an inverse relationship
- between the time to cover a unit distance and distance covered in unit time.
- The inverse Gaussian is one of family of distributions that have been called the
- [@http://en.wikipedia.org/wiki/Tweedie_distributions Tweedie distributions].
- (So ['inverse] in the name may mislead: it does [*not] relate to the inverse of a distribution).
- The tails of the distribution decrease more slowly than the normal distribution.
- It is therefore suitable to model phenomena
- where numerically large values are more probable than is the case for the normal distribution.
- For stock market returns and prices, a key characteristic is that it models
- that extremely large variations from typical (crashes) can occur
- even when almost all (normal) variations are small.
- Examples are returns from financial assets and turbulent wind speeds.
- The normal-inverse Gaussian distributions form
- a subclass of the generalised hyperbolic distributions.
- See
- [@http://en.wikipedia.org/wiki/Normal-inverse_Gaussian_distribution distribution].
- [@http://mathworld.wolfram.com/InverseGaussianDistribution.html
- Weisstein, Eric W. "Inverse Gaussian Distribution." From MathWorld--A Wolfram Web Resource.]
-
- If you want a `double` precision inverse_gaussian distribution you can use
- ``boost::math::inverse_gaussian_distribution<>``
- or, more conveniently, you can write
- using boost::math::inverse_gaussian;
- inverse_gaussian my_ig(2, 3);
- For mean parameters [mu] and scale (also called precision) parameter [lambda],
- and random variate x,
- the inverse_gaussian distribution is defined by the probability density function (PDF):
- [expression f(x;[mu], [lambda]) = [sqrt]([lambda]/2[pi]x[super 3]) e[super -[lambda](x-[mu])[sup2]/2[mu][sup2]x] ]
- and Cumulative Density Function (CDF):
- [expression F(x;[mu], [lambda]) = [Phi]{[sqrt]([lambda]/x) (x/[mu]-1)} + e[super 2[mu]/[lambda]] [Phi]{-[sqrt]([lambda]/[mu]) (1+x/[mu])} ]
- where [Phi] is the standard normal distribution CDF.
- The following graphs illustrate how the PDF and CDF of the inverse_gaussian distribution
- varies for a few values of parameters [mu] and [lambda]:
- [graph inverse_gaussian_pdf] [/.png or .svg]
- [graph inverse_gaussian_cdf]
- Tweedie also provided 3 other parameterisations where ([mu] and [lambda])
- are replaced by their ratio [phi] = [lambda]/[mu] and by 1/[mu]:
- these forms may be more suitable for Bayesian applications.
- These can be found on Seshadri, page 2 and are also discussed by Chhikara and Folks on page 105.
- Another related parameterisation, the __wald_distrib (where mean [mu] is unity) is also provided.
- [h4 Member Functions]
- inverse_gaussian_distribution(RealType df = 1, RealType scale = 1); // optionally scaled.
- Constructs an inverse_gaussian distribution with [mu] mean,
- and scale [lambda], with both default values 1.
- Requires that both the mean [mu] parameter and scale [lambda] are greater than zero,
- otherwise calls __domain_error.
- RealType mean()const;
-
- Returns the mean [mu] parameter of this distribution.
- RealType scale()const;
-
- Returns the scale [lambda] parameter of this 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 both pdf and cdf.]
- [h4 Accuracy]
- The inverse_gaussian distribution is implemented in terms of the
- exponential function and standard normal distribution ['N]0,1 [Phi] :
- refer to the accuracy data for those functions for more information.
- But in general, gamma (and thus inverse gamma) results are often accurate to a few epsilon,
- >14 decimal digits accuracy for 64-bit double.
- [h4 Implementation]
- In the following table [mu] is the mean parameter and
- [lambda] is the scale parameter of the inverse_gaussian distribution,
- /x/ is the random variate, /p/ is the probability and /q = 1-p/ its complement.
- Parameters [mu] for shape and [lambda] for scale
- are used for the inverse gaussian function.
- [table
- [[Function] [Implementation Notes] ]
- [[pdf] [ [sqrt]([lambda]/ 2[pi]x[super 3]) e[super -[lambda](x - [mu])[sup2]/ 2[mu][sup2]x]]]
- [[cdf][ [Phi]{[sqrt]([lambda]/x) (x/[mu]-1)} + e[super 2[mu]/[lambda]] [Phi]{-[sqrt]([lambda]/[mu]) (1+x/[mu])} ]]
- [[cdf complement] [using complement of [Phi] above.] ]
- [[quantile][No closed form known. Estimated using a guess refined by Newton-Raphson iteration.]]
- [[quantile from the complement][No closed form known. Estimated using a guess refined by Newton-Raphson iteration.]]
- [[mode][[mu] {[sqrt](1+9[mu][sup2]/4[lambda][sup2])[sup2] - 3[mu]/2[lambda]} ]]
- [[median][No closed form analytic equation is known, but is evaluated as quantile(0.5)]]
- [[mean][[mu]] ]
- [[variance][[mu][cubed]/[lambda]] ]
- [[skewness][3 [sqrt] ([mu]/[lambda])] ]
- [[kurtosis_excess][15[mu]/[lambda]] ]
- [[kurtosis][12[mu]/[lambda]] ]
- ] [/table]
- [h4 References]
- #Wald, A. (1947). Sequential analysis. Wiley, NY.
- #The Inverse Gaussian distribution : theory, methodology, and applications, Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989).
- #The Inverse Gaussian distribution : statistical theory and applications, Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998).
- #[@http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.wald.html Numpy and Scipy Documentation].
- #[@http://bm2.genes.nig.ac.jp/RGM2/R_current/library/statmod/man/invgauss.html R statmod invgauss functions].
- #[@http://cran.r-project.org/web/packages/SuppDists/index.html R SuppDists invGauss functions].
- (Note that these R implementations names differ in case).
- #[@http://www.statsci.org/s/invgauss.html StatSci.org invgauss help].
- #[@http://www.statsci.org/s/invgauss.statSci.org invgauss R source].
- #[@http://www.biostat.wustl.edu/archives/html/s-news/2001-12/msg00144.html pwald, qwald].
- #[@http://www.brighton-webs.co.uk/distributions/wald.asp Brighton Webs wald].
- [endsect] [/section:inverse_gaussian_dist Inverse Gaussiann 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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