CumulativeBehrensFisher Class Reference

Cumulative (generalized) BehrensFisher distribution. More...

#include <convolvedstudentt.hpp>

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Public Member Functions
	CumulativeBehrensFisher (const std::vector< Integer > &degreesFreedom=std::vector< Integer >(), const std::vector< Real > &factors=std::vector< Real >())
const std::vector< Integer > &	degreeFreedom () const
	Degrees of freedom of the Ts involved in the convolution. More...
const std::vector< Real > &	factors () const
	Factors in the linear combination. More...
Probability	operator() (Real x) const
	Returns the cumulative probability of the resulting distribution. More...
Probability	density (Real x) const
	Returns the probability density of the resulting distribution. More...

Private Member Functions
std::vector< Real >	polynCharactT (Natural n) const
	Student t characteristic polynomials. More...
std::vector< Real >	convolveVectorPolynomials (const std::vector< Real > &v1, const std::vector< Real > &v2) const

Private Attributes
std::vector< Integer >	degreesFreedom_
std::vector< Real >	factors_
std::vector< std::vector< Real > >	polynCharFnc_
std::vector< Real >	polyConvolved_
Real	a_ = 0.
Real	a2_

Detailed Description

Cumulative (generalized) BehrensFisher distribution.

Exact analitical computation of the cumulative probability distribution of the linear combination of an arbitrary number (not just two) of T random variables of odd integer order. Adapted from the algorithm in:

V. Witkovsky, Journal of Statistical Planning and Inference 94 (2001) 1-13

see also:

On the distribution of a linear combination of t-distributed variables; Glenn Alan Walker, Ph.D.thesis University of Florida 1977

'Convolutions of the T Distribution'; S. Nadarajah, D. K. Dey in Computers and Mathematics with Applications 49 (2005) 715-721

The last reference provides direct expressions for some of the densities when the linear combination of only two Ts is just an addition. It can be used for testing the results here.

Another available test on this algorithm stems from the realization that a linear convex ( \( \sum a_i=1\)) combination of Ts of order one is stable in the distribution sense (but this result is often of no practical use because of its non-finite variance).

This implementation is for two or more T variables in the linear combination albeit these must be of odd order. The case of exactly two T of odd order is known to be a finite mixture of Ts but that result is not used here. On this line see 'Linearization coefficients of Bessel polynomials' C.Berg, C.Vignat; February 2008; arXiv:math/0506458

\todo Implement the series expansion solution for the addition of
two Ts of even order described in: 'On the density of the sum of two
independent Student t-random vectors' C.Berg, C.Vignat; June 2009;
eprint arXiv:0906.3037

Definition at line 60 of file convolvedstudentt.hpp.

Constructor & Destructor Documentation

CumulativeBehrensFisher()

CumulativeBehrensFisher	(	const std::vector< Integer > &	degreesFreedom = std::vector<Integer>(),
		const std::vector< Real > &	factors = std::vector<Real>()
	)

Parameters

degreesFreedom	Degrees of freedom of the Ts convolved. The algorithm is limited to odd orders only.
factors	Factors in the linear combination of the Ts.

Definition at line 30 of file convolvedstudentt.cpp.

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Member Function Documentation

degreeFreedom()

const std::vector< Integer > & degreeFreedom

(

)

const

Degrees of freedom of the Ts involved in the convolution.

Definition at line 72 of file convolvedstudentt.hpp.

factors()

const std::vector< Real > & factors

(

)

const

Factors in the linear combination.

Definition at line 76 of file convolvedstudentt.hpp.

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polynCharactT()

std::vector< Real > polynCharactT ( Natural  n ) const

private

Student t characteristic polynomials.

Generates the polynomial coefficients defining the characteristic function of a T distribution \(T_\nu\) of odd order; \(\nu=2n+1\). In general the characteristic function is given by:

\[ \phi_{\nu}(t) = \varphi_{n}(t) \exp{-\nu^{1/2}|t|} ;\,where\,\nu = 2n+1 \]

where \( \varphi \) are polynomials that are computed recursively.

The convolved characteristic function is the product of the two previous characteristic functions and the problem is then the convolution (a product) of two polynomials.

@param n Natural number defining the order of the T for which
the characteristic function is to be computed. The order of the
 T is then \f$ \nu=2n+1 \f$

Definition at line 72 of file convolvedstudentt.cpp.

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convolveVectorPolynomials()

std::vector< Real > convolveVectorPolynomials ( const std::vector< Real > &  v1, const std::vector< Real > &  v2  ) const

private

Definition at line 94 of file convolvedstudentt.cpp.

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operator()()

Probability operator()

(

Real

x

)

const

Returns the cumulative probability of the resulting distribution.

To obtain the cumulative probability the Gil-Pelaez theorem is applied:

First compute the characteristic function of the linear combination variable by multiplying the individual characteristic functions. Then transform back integrating the characteristic function according to the GP theorem; this is done here analytically feeding in the expression of the total characteristic function this:

\[ \int_0^{\infty}x^n e^{-ax}sin(bx)dx = (-1)^n \Gamma(n+1) \frac{sin((n+1)arctg2(-b/a))} {(\sqrt{a^2+b^2})^{n+1}}; for\,a>0,\,b>0 \]

and for the first term I use:

\[ \int_0^{\infty} \frac{e^{-ax}sin(bx)}{x} dx = arctg2(b/a) \]

The GP complex integration is simplified thanks to the symetry of the distribution.

Definition at line 115 of file convolvedstudentt.cpp.

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density()

Probability density

(

Real

x

)

const

Returns the probability density of the resulting distribution.

Similarly to the cumulative probability, Gil-Pelaez theorem is applied, the integration is similar.

Definition at line 133 of file convolvedstudentt.cpp.

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Member Data Documentation

degreesFreedom_

std::vector<Integer> degreesFreedom_

mutableprivate

Definition at line 138 of file convolvedstudentt.hpp.

factors_

std::vector<Real> factors_

mutableprivate

Definition at line 139 of file convolvedstudentt.hpp.

polynCharFnc_

std::vector<std::vector<Real> > polynCharFnc_

mutableprivate

Definition at line 141 of file convolvedstudentt.hpp.

polyConvolved_

std::vector<Real> polyConvolved_

mutableprivate

Definition at line 142 of file convolvedstudentt.hpp.

a_

Real a_ = 0.

mutableprivate

Definition at line 145 of file convolvedstudentt.hpp.

a2_

Real a2_

private

Definition at line 145 of file convolvedstudentt.hpp.