1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
|
[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:
BOOST_MATH_GPU_ENABLED students_t_distribution(const RealType& v);
// Accessor:
BOOST_MATH_GPU_ENABLED RealType degrees_of_freedom()const;
// degrees of freedom estimation:
BOOST_MATH_GPU_ENABLED 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]
BOOST_MATH_GPU_ENABLED 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.
BOOST_MATH_GPU_ENABLED RealType degrees_of_freedom()const;
returns the number of degrees of freedom of this distribution.
BOOST_MATH_GPU_ENABLED 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.
For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
be run on both host and device.
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).
]
|
