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225 lines
7.6 KiB
Plaintext
225 lines
7.6 KiB
Plaintext
[section:nc_t_dist Noncentral T Distribution]
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``#include <boost/math/distributions/non_central_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 non_central_t_distribution;
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typedef non_central_t_distribution<> non_central_t;
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template <class RealType, class ``__Policy``>
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class non_central_t_distribution
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{
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public:
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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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non_central_t_distribution(RealType v, RealType delta);
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// Accessor to degrees_of_freedom parameter v:
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RealType degrees_of_freedom()const;
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// Accessor to non-centrality parameter delta:
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RealType non_centrality()const;
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};
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}} // namespaces
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The noncentral T distribution is a generalization of the __students_t_distrib.
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Let X have a normal distribution with mean [delta] and variance 1, and let
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[nu] S[super 2] have
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a chi-squared distribution with degrees of freedom [nu]. Assume that
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X and S[super 2] are independent. The
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distribution of t[sub [nu]]([delta])=X/S is called a
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noncentral t distribution with degrees of freedom [nu] and noncentrality
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parameter [delta].
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This gives the following PDF:
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[equation nc_t_ref1]
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where [sub 1]F[sub 1](a;b;x) is a confluent hypergeometric function.
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The following graph illustrates how the distribution changes
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for different values of [nu] and [delta]:
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[graph nc_t_pdf]
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[graph nc_t_cdf]
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[h4 Member Functions]
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non_central_t_distribution(RealType v, RealType delta);
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Constructs a non-central t distribution with degrees of freedom
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parameter /v/ and non-centrality parameter /delta/.
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Requires /v/ > 0 (including positive infinity) and finite /delta/, otherwise calls __domain_error.
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RealType degrees_of_freedom()const;
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Returns the parameter /v/ from which this object was constructed.
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RealType non_centrality()const;
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Returns the non-centrality parameter /delta/ from which this object was constructed.
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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 \[-[infin], +[infin]\].
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[h4 Accuracy]
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The following table shows the peak errors
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(in units of [@http://en.wikipedia.org/wiki/Machine_epsilon epsilon])
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found on various platforms with various floating-point types.
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Unless otherwise specified, any floating-point type that is narrower
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than the one shown will have __zero_error.
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[table_non_central_t_CDF]
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[table_non_central_t_CDF_complement]
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[caution The complexity of the current algorithm is dependent upon
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[delta][super 2]: consequently the time taken to evaluate the CDF
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increases rapidly for [delta] > 500, likewise the accuracy decreases
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rapidly for very large [delta].]
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Accuracy for the quantile and PDF functions should be broadly similar.
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The /mode/ is determined numerically and cannot
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in principal be more accurate than the square root of
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floating-point type FPT epsilon, accessed using `boost::math::tools::epsilon<FPT>()`.
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For 64-bit `double`, epsilon is about 1e-16, so the fractional accuracy is limited to 1e-8.
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[h4 Tests]
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There are two sets of tests of this distribution:
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Basic sanity checks compare this implementation to the test values given in
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"Computing discrete mixtures of continuous
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distributions: noncentral chisquare, noncentral t
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and the distribution of the square of the sample
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multiple correlation coefficient."
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Denise Benton, K. Krishnamoorthy,
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Computational Statistics & Data Analysis 43 (2003) 249-267.
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Accuracy checks use test data computed with this
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implementation and arbitary precision interval arithmetic:
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this test data is believed to be accurate to at least 50
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decimal places.
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The cases of large (or infinite) [nu] and/or large [delta] has received special
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treatment to avoid catastrophic loss of accuracy.
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New tests have been added to confirm the improvement achieved.
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From Boost 1.52, degrees of freedom [nu] can be +[infin]
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when the normal distribution located at [delta]
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(equivalent to the central Student's t distribution)
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is used in place for accuracy and speed.
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[h4 Implementation]
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The CDF is computed using a modification of the method
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described in
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"Computing discrete mixtures of continuous
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distributions: noncentral chisquare, noncentral t
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and the distribution of the square of the sample
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multiple correlation coefficient."
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Denise Benton, K. Krishnamoorthy,
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Computational Statistics & Data Analysis 43 (2003) 249-267.
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This uses the following formula for the CDF:
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[equation nc_t_ref2]
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Where I[sub x](a,b) is the incomplete beta function, and
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[Phi](x) is the normal CDF at x.
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Iteration starts at the largest of the Poisson weighting terms
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(at i = [delta][super 2] / 2) and then proceeds in both directions
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as per Benton and Krishnamoorthy's paper.
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Alternatively, by considering what happens when t = [infin], we have
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x = 1, and therefore I[sub x](a,b) = 1 and:
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[equation nc_t_ref3]
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From this we can easily show that:
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[equation nc_t_ref4]
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and therefore we have a means to compute either the probability or its
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complement directly without the risk of cancellation error. The
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crossover criterion for choosing whether to calculate the CDF or
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its complement is the same as for the
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__non_central_beta_distrib.
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The PDF can be computed by a very similar method using:
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[equation nc_t_ref5]
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Where I[sub x][super '](a,b) is the derivative of the incomplete beta function.
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For both the PDF and CDF we switch to approximating the distribution by a
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Student's t distribution centred on [delta] when [nu] is very large.
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The crossover location appears to be when [delta]/(4[nu]) < [epsilon],
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this location was estimated by inspection of equation 2.6 in
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"A Comparison of Approximations To Percentiles of the
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Noncentral t-Distribution". H. Sahai and M. M. Ojeda,
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Revista Investigacion Operacional Vol 21, No 2, 2000, page 123.
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Equation 2.6 is a Fisher-Cornish expansion by Eeden and Johnson.
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The second term includes the ratio [delta]/(4[nu]),
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so when this term become negligible, this and following terms can be ignored,
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leaving just Student's t distribution centred on [delta].
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This was also confirmed by experimental testing.
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See also
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* "Some Approximations to the Percentage Points of the Noncentral
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t-Distribution". C. van Eeden. International Statistical Review, 29, 4-31.
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* "Continuous Univariate Distributions". N.L. Johnson, S. Kotz and
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N. Balkrishnan. 1995. John Wiley and Sons New York.
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The quantile is calculated via the usual
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__root_finding_without_derivatives method
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with the initial guess taken as the quantile of a normal approximation
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to the noncentral T.
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There is no closed form for the mode, so this is computed via
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functional maximisation of the PDF.
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The remaining functions (mean, variance etc) are implemented
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using the formulas given in
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Weisstein, Eric W. "Noncentral Student's t-Distribution."
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From MathWorld--A Wolfram Web Resource.
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[@http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html
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http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html]
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and in the
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[@http://reference.wolfram.com/mathematica/ref/NoncentralStudentTDistribution.html
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Mathematica documentation].
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Some analytic properties of noncentral distributions
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(particularly unimodality, and monotonicity of their modes)
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are surveyed and summarized by:
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Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, 141 (2003) 3-12.
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[endsect] [/section:nc_t_dist]
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[/ nc_t.qbk
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Copyright 2008, 2012 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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