WSJT-X/boost/libs/math/doc/distributions/students_t.qbk

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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;
// Construct:
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
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),
so he chose "Student". Given N independent measurements, let
[equation students_t_dist]
where /M/ is the population mean,[' ''' &#x3BC; '''] is the sample mean, and /s/ is the
sample variance.
Student's t-distribution is defined as the distribution of the random
variable t which is - very loosely - the "best" that we can do 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, 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: pdf = (v \/ (v + t[super 2]))[super (1+v)\/2 ] / (sqrt(v) * __beta(v\/2, 0.5)) ]]
[[cdf][Using the relations:
p = 1 - z /iff t > 0/
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: t = sign(p - 0.5) * sqrt(v * y \/ x)
where:
x = __ibeta_inv(v \/ 2, 0.5, 2 * min(p, q))
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();`
(For simplicity, we have not implemented the return of infinity in some cases
as suggested by [@http://en.wikipedia.org/wiki/Student%27s_t-distribution Wikipedia Student's t].
See also [@https://svn.boost.org/trac/boost/ticket/7177].)
[endsect] [/section:students_t_dist Students t]
[/ students_t.qbk
Copyright 2006, 2012 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).
]