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191 lines
5.2 KiB
Plaintext
191 lines
5.2 KiB
Plaintext
[section:f_dist F Distribution]
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``#include <boost/math/distributions/fisher_f.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class fisher_f_distribution;
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typedef fisher_f_distribution<> fisher_f;
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template <class RealType, class ``__Policy``>
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class fisher_f_distribution
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{
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public:
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typedef RealType value_type;
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// Construct:
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fisher_f_distribution(const RealType& i, const RealType& j);
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// Accessors:
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RealType degrees_of_freedom1()const;
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RealType degrees_of_freedom2()const;
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};
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}} //namespaces
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The F distribution is a continuous distribution that arises when testing
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whether two samples have the same variance. If [chi][super 2][sub m][space] and
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[chi][super 2][sub n][space] are independent variates each distributed as
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Chi-Squared with /m/ and /n/ degrees of freedom, then the test statistic:
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F[sub n,m][space] = ([chi][super 2][sub n][space] / n) / ([chi][super 2][sub m][space] / m)
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Is distributed over the range \[0, [infin]\] with an F distribution, and
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has the PDF:
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[equation fisher_pdf]
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The following graph illustrates how the PDF varies depending on the
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two degrees of freedom parameters.
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[graph fisher_f_pdf]
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[h4 Member Functions]
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fisher_f_distribution(const RealType& df1, const RealType& df2);
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Constructs an F-distribution with numerator degrees of freedom /df1/
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and denominator degrees of freedom /df2/.
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Requires that /df1/ and /df2/ are both greater than zero, otherwise __domain_error
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is called.
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RealType degrees_of_freedom1()const;
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Returns the numerator degrees of freedom parameter of the distribution.
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RealType degrees_of_freedom2()const;
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Returns the denominator degrees of freedom parameter of the distribution.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
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that are generic to all distributions are supported: __usual_accessors.
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The domain of the random variable is \[0, +[infin]\].
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[h4 Examples]
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Various [link math_toolkit.stat_tut.weg.f_eg worked examples] are
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available illustrating the use of the F Distribution.
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[h4 Accuracy]
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The normal distribution is implemented in terms of the
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[link math_toolkit.sf_beta.ibeta_function incomplete beta function]
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and its [link math_toolkit.sf_beta.ibeta_inv_function inverses],
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refer to those functions for accuracy data.
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[h4 Implementation]
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In the following table /v1/ and /v2/ are the first and second
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degrees of freedom parameters of the distribution,
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/x/ is the random variate, /p/ is the probability, and /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][The usual form of the PDF is given by:
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[equation fisher_pdf]
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However, that form is hard to evaluate directly without incurring problems with
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either accuracy or numeric overflow.
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Direct differentiation of the CDF expressed in terms of the incomplete beta function
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led to the following two formulas:
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f[sub v1,v2](x) = y * __ibeta_derivative(v2 \/ 2, v1 \/ 2, v2 \/ (v2 + v1 * x))
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with y = (v2 * v1) \/ ((v2 + v1 * x) * (v2 + v1 * x))
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and
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f[sub v1,v2](x) = y * __ibeta_derivative(v1 \/ 2, v2 \/ 2, v1 * x \/ (v2 + v1 * x))
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with y = (z * v1 - x * v1 * v1) \/ z[super 2]
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and z = v2 + v1 * x
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The first of these is used for v1 * x > v2, otherwise the second is used.
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The aim is to keep the /x/ argument to __ibeta_derivative away from 1 to avoid
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rounding error. ]]
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[[cdf][Using the relations:
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p = __ibeta(v1 \/ 2, v2 \/ 2, v1 * x \/ (v2 + v1 * x))
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and
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p = __ibetac(v2 \/ 2, v1 \/ 2, v2 \/ (v2 + v1 * x))
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The first is used for v1 * x > v2, otherwise the second is used.
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The aim is to keep the /x/ argument to __ibeta well away from 1 to
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avoid rounding error. ]]
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[[cdf complement][Using the relations:
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p = __ibetac(v1 \/ 2, v2 \/ 2, v1 * x \/ (v2 + v1 * x))
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and
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p = __ibeta(v2 \/ 2, v1 \/ 2, v2 \/ (v2 + v1 * x))
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The first is used for v1 * x < v2, otherwise the second is used.
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The aim is to keep the /x/ argument to __ibeta well away from 1 to
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avoid rounding error. ]]
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[[quantile][Using the relation:
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x = v2 * a \/ (v1 * b)
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where:
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a = __ibeta_inv(v1 \/ 2, v2 \/ 2, p)
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and
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b = 1 - a
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Quantities /a/ and /b/ are both computed by __ibeta_inv without the
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subtraction implied above.]]
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[[quantile
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from the complement][Using the relation:
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x = v2 * a \/ (v1 * b)
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where
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a = __ibetac_inv(v1 \/ 2, v2 \/ 2, p)
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and
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b = 1 - a
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Quantities /a/ and /b/ are both computed by __ibetac_inv without the
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subtraction implied above.]]
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[[mean][v2 \/ (v2 - 2)]]
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[[variance][2 * v2[super 2 ] * (v1 + v2 - 2) \/ (v1 * (v2 - 2) * (v2 - 2) * (v2 - 4))]]
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[[mode][v2 * (v1 - 2) \/ (v1 * (v2 + 2))]]
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[[skewness][2 * (v2 + 2 * v1 - 2) * sqrt((2 * v2 - 8) \/ (v1 * (v2 + v1 - 2))) \/ (v2 - 6)]]
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[[kurtosis and kurtosis excess]
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[Refer to, [@http://mathworld.wolfram.com/F-Distribution.html
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Weisstein, Eric W. "F-Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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]
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[endsect][/section:f_dist F distribution]
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[/ fisher.qbk
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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