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190 lines
6.6 KiB
Plaintext
190 lines
6.6 KiB
Plaintext
[section:students_t_dist Students t Distribution]
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``#include <boost/math/distributions/students_t.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 students_t_distribution;
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typedef students_t_distribution<> students_t;
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template <class RealType, class ``__Policy``>
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class students_t_distribution
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{
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typedef RealType value_type;
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typedef Policy policy_type;
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// Constructor:
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BOOST_MATH_GPU_ENABLED students_t_distribution(const RealType& v);
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// Accessor:
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BOOST_MATH_GPU_ENABLED RealType degrees_of_freedom()const;
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// degrees of freedom estimation:
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BOOST_MATH_GPU_ENABLED static RealType find_degrees_of_freedom(
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RealType difference_from_mean,
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RealType alpha,
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RealType beta,
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RealType sd,
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RealType hint = 100);
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};
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}} // namespaces
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Student's t-distribution is a statistical distribution published by William Gosset in 1908.
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His employer, Guinness Breweries, required him to publish under a
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pseudonym (possibly to hide that they were using statistics to improve beer quality),
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so he chose "Student".
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Given N independent measurements, let
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[equation students_t_dist]
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where /M/ is the population mean, [mu] is the sample mean, and /s/ is the sample variance.
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[@https://en.wikipedia.org/wiki/Student%27s_t-distribution Student's t-distribution]
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is defined as the distribution of the random
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variable t which is - very loosely - the "best" that we can do while not
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knowing the true standard deviation of the sample. It has the PDF:
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[equation students_t_ref1]
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The Student's t-distribution takes a single parameter: the number of
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degrees of freedom of the sample. When the degrees of freedom is
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/one/ then this distribution is the same as the Cauchy-distribution.
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As the number of degrees of freedom tends towards infinity, then this
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distribution approaches the normal-distribution. The following graph
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illustrates how the PDF varies with the degrees of freedom [nu]:
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[graph students_t_pdf]
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[h4 Member Functions]
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BOOST_MATH_GPU_ENABLED students_t_distribution(const RealType& v);
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Constructs a Student's t-distribution with /v/ degrees of freedom.
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Requires /v/ > 0, including infinity (if RealType permits),
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otherwise calls __domain_error. Note that
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non-integral degrees of freedom are supported,
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and are meaningful under certain circumstances.
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BOOST_MATH_GPU_ENABLED RealType degrees_of_freedom()const;
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returns the number of degrees of freedom of this distribution.
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BOOST_MATH_GPU_ENABLED static RealType find_degrees_of_freedom(
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RealType difference_from_mean,
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RealType alpha,
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RealType beta,
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RealType sd,
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RealType hint = 100);
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returns the number of degrees of freedom required to observe a significant
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result in the Student's t test when the mean differs from the "true"
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mean by /difference_from_mean/.
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[variablelist
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[[difference_from_mean][The difference between the true mean and the sample mean
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that we wish to show is significant.]]
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[[alpha][The maximum acceptable probability of rejecting the null hypothesis
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when it is in fact true.]]
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[[beta][The maximum acceptable probability of failing to reject the null hypothesis
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when it is in fact false.]]
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[[sd][The sample standard deviation.]]
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[[hint][A hint for the location to start looking for the result, a good choice for this
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would be the sample size of a previous borderline Student's t test.]]
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]
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[note
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Remember that for a two-sided test, you must divide alpha by two
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before calling this function.]
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For more information on this function see the
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[@http://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm
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NIST Engineering Statistics Handbook].
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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] that are generic to all
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distributions are supported: __usual_accessors.
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For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
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be run on both host and device.
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The domain of the random variable is \[-[infin], +[infin]\].
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[h4 Examples]
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Various [link math_toolkit.stat_tut.weg.st_eg worked examples] are available illustrating the use of the Student's t
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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 [link math_toolkit.sf_beta.ibeta_inv_function its inverses],
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refer to accuracy data on those functions for more information.
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[h4 Implementation]
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In the following table /v/ is the degrees of freedom of the distribution,
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/t/ 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][Using the relation: [role serif_italic pdf = (v \/ (v + t[super 2]))[super (1+v)\/2 ] / (sqrt(v) * __beta(v\/2, 0.5))] ]]
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[[cdf][Using the relations:
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[role serif_italic p = 1 - z /iff t > 0/]
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[role serif_italic p = z /otherwise/]
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where z is given by:
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__ibeta(v \/ 2, 0.5, v \/ (v + t[super 2])) \/ 2 ['iff v < 2t[super 2]]
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__ibetac(0.5, v \/ 2, t[super 2 ] / (v + t[super 2]) \/ 2 /otherwise/]]
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[[cdf complement][Using the relation: q = cdf(-t) ]]
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[[quantile][Using the relation: [role serif_italic t = sign(p - 0.5) * sqrt(v * y \/ x)]
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where:
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[role serif_italic x = __ibeta_inv(v \/ 2, 0.5, 2 * min(p, q)) ]
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[role serif_italic y = 1 - x]
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The quantities /x/ and /y/ are both returned by __ibeta_inv
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without the subtraction implied above.]]
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[[quantile from the complement][Using the relation: t = -quantile(q)]]
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[[mode][0]]
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[[mean][0]]
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[[variance][if (v > 2) v \/ (v - 2) else NaN]]
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[[skewness][if (v > 3) 0 else NaN ]]
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[[kurtosis][if (v > 4) 3 * (v - 2) \/ (v - 4) else NaN]]
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[[kurtosis excess][if (v > 4) 6 \/ (df - 4) else NaN]]
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]
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If the moment index /k/ is less than /v/, then the moment is undefined.
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Evaluating the moment will throw a __domain_error unless ignored by a policy,
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when it will return `std::numeric_limits<>::quiet_NaN();`
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[h5:implementation Implementation]
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(By popular demand, we now support infinite argument and random deviate.
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But we have not implemented the return of infinity
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as suggested by [@http://en.wikipedia.org/wiki/Student%27s_t-distribution Wikipedia Student's t],
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instead throwing a domain error or return NaN.
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See also [@https://svn.boost.org/trac/boost/ticket/7177].)
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[endsect] [/section:students_t_dist Students t]
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[/ students_t.qbk
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Copyright 2006, 2012, 2017 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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