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137 lines
4.8 KiB
Plaintext
137 lines
4.8 KiB
Plaintext
[section:owens_t Owen's T function]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/owens_t.hpp>
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``
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namespace boost{ namespace math{
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template <class T>
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``__sf_result`` owens_t(T h, T a);
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template <class T, class ``__Policy``>
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``__sf_result`` owens_t(T h, T a, const ``__Policy``&);
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}} // namespaces
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[h4 Description]
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Returns the
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[@http://en.wikipedia.org/wiki/Owen%27s_T_function Owens_t function]
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of ['h] and ['a].
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[optional_policy]
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[sixemspace][sixemspace][equation owens_t]
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[$../graphs/plot_owens_t.png]
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The function `owens_t(h, a)` gives the probability
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of the event ['(X > h and 0 < Y < a * X)],
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where ['X] and ['Y] are independent standard normal random variables.
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For h and a > 0, T(h,a),
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gives the volume of an uncorrelated bivariate normal distribution
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with zero means and unit variances over the area between
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['y = ax] and ['y = 0] and to the right of ['x = h].
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That is the area shaded in the figure below (Owens 1956).
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[graph owens_integration_area]
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and is also illustrated by a 3D plot.
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[$../graphs/plot_owens_3d_xyp.png]
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This function is used in the computation of the __skew_normal_distrib.
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It is also used in the computation of bivariate and
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multivariate normal distribution probabilities.
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The return type of this function is computed using the __arg_promotion_rules:
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the result is of type `double` when T is an integer type, and type T otherwise.
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Owen's original paper (page 1077) provides some additional corner cases.
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[: ['T(h, 0) = 0]]
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[:['T(0, a) = [frac12][pi] arctan(a)]]
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[:['T(h, 1) = [frac12] G(h) \[1 - G(h)\]]]
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[:['T(h, [infin]) = G(|h|)]]
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where G(h) is the univariate normal with zero mean and unit variance integral from -[infin] to h.
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[h4 Accuracy]
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Over the built-in types and range tested,
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errors are less than 10 * std::numeric_limits<RealType>::epsilon().
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[table_owens_t]
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[h4 Testing]
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Test data was generated by Patefield and Tandy algorithms T1 and T4,
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and also the suggested reference routine T7.
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* T1 was rejected if the result was too small compared to `atan(a)` (ie cancellation),
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* T4 was rejected if there was no convergence,
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* Both were rejected if they didn't agree.
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Over the built-in types and range tested,
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errors are less than 10 std::numeric_limits<RealType>::epsilon().
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However, that there was a whole domain (large ['h], small ['a])
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where it was not possible to generate any reliable test values
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(all the methods got rejected for one reason or another).
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There are also two sets of sanity tests: spot values are computed using __Mathematica and __R.
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[h4 Implementation]
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The function was proposed and evaluated by
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[@http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177728074
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Donald. B. Owen, Tables for computing bivariate normal probabilities,
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Ann. Math. Statist., 27, 1075-1090 (1956)].
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The algorithms of Patefield, M. and Tandy, D.
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"Fast and accurate Calculation of Owen's T-Function", Journal of Statistical Software, 5 (5), 1 - 25 (2000)
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are adapted for C++ with arbitrary RealType.
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The Patefield-Tandy algorithm provides six methods of evalualution (T1 to T6);
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the best method is selected according to the values of ['a] and ['h].
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See the original paper and the source in
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[@../../../../boost/math/special_functions/owens_t.hpp owens_t.hpp] for details.
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The Patefield-Tandy algorithm is accurate to approximately 20 decimal places, so for
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types with greater precision we use:
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* A modified version of T1 which folds the calculation of ['atan(h)] into the T1 series
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(to avoid subtracting two values similar in magnitude), and then accelerates the
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resulting alternating series using method 1 from H. Cohen, F. Rodriguez Villegas, D. Zagier,
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"Convergence acceleration of alternating series", Bonn, (1991). The result is valid everywhere,
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but doesn't always converge, or may become too divergent in the first few terms to sum accurately.
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This is used for ['ah < 1].
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* A modified version of T2 which is accelerated in the same manner as T1. This is used for ['h > 1].
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* A version of T4 only when both T1 and T2 have failed to produce an accurate answer.
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* Fallback to the Patefiled Tandy algorithm when all the above methods fail: this happens not at all
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for our test data at 100 decimal digits precision. However, there is a difficult area when
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['a] is very close to 1 and the precision increases which may cause this to happen in very exceptional
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circumstances.
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Using the above algorithm and a 100-decimal digit type, results accurate to 80 decimal places were obtained
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in the difficult area where ['a] is close to 1, and greater than 95 decimal places elsewhere.
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[endsect] [/section:owens_t The owens_t Function]
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[/
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Copyright 2012 Bejamin Sobotta, 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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