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