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Coding Standards

#include <boost/math/distributions/non_central_t.hpp>

namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > class non_central_t_distribution; typedef non_central_t_distribution<> non_central_t; template <class RealType, class Policy> class non_central_t_distribution { public: typedef RealType value_type; typedef Policy policy_type; // Constructor: non_central_t_distribution(RealType v, RealType delta); // Accessor to degrees_of_freedom parameter v: RealType degrees_of_freedom()const; // Accessor to non-centrality parameter lambda: RealType non_centrality()const; }; }} // namespaces

The noncentral T distribution is a generalization of the Students
t Distribution. Let X have a normal distribution with mean δ and
variance 1, and let ν S^{2} have a chi-squared distribution with degrees of
freedom ν. Assume that X and S^{2} are independent. The distribution of t_{ν}(δ)=X/S
is called a noncentral t distribution with degrees of freedom ν and noncentrality
parameter δ.

This gives the following PDF:

where _{1}F_{1}(a;b;x) is a confluent hypergeometric function.

The following graph illustrates how the distribution changes for different values of δ:

non_central_t_distribution(RealType v, RealType lambda);

Constructs a non-central t distribution with degrees of freedom parameter
*v* and non-centrality parameter *delta*.

Requires v > 0 and finite delta, otherwise calls domain_error.

RealType degrees_of_freedom()const;

Returns the parameter *v* from which this object was
constructed.

RealType non_centrality()const;

Returns the non-centrality parameter *delta* from
which this object was constructed.

All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.

The domain of the random variable is [-∞, +∞].

The following table shows the peak errors (in units of epsilon) found on various platforms with various floating-point types. Unless otherwise specified, any floating-point type that is narrower than the one shown will have effectively zero error.

**Table 18. Errors In CDF of the Noncentral T Distribution**

Significand Size |
Platform and Compiler |
ν,δ < 600 |
---|---|---|

53 |
Win32, Visual C++ 8 |
Peak=120 Mean=26 |

64 |
RedHat Linux IA32, gcc-4.1.1 |
Peak=121 Mean=26 |

64 |
Redhat Linux IA64, gcc-3.4.4 |
Peak=122 Mean=25 |

113 |
HPUX IA64, aCC A.06.06 |
Peak=115 Mean=24 |

Caution | |
---|---|

The complexity of the current algorithm is dependent upon δ |

Accuracy for the quantile and PDF functions should be broadly similar,
note however that the *mode* is determined numerically
and can not in principal be more accurate than the square root of machine
epsilon.

There are two sets of tests of this distribution: basic sanity checks compare this implementation to the test values given in "Computing discrete mixtures of continuous distributions: noncentral chisquare, noncentral t and the distribution of the square of the sample multiple correlation coefficient." Denise Benton, K. Krishnamoorthy, Computational Statistics & Data Analysis 43 (2003) 249-267. While accuracy checks use test data computed with this implementation and arbitary precision interval arithmetic: this test data is believed to be accurate to at least 50 decimal places.

The CDF is computed using a modification of the method described in "Computing discrete mixtures of continuous distributions: noncentral chisquare, noncentral t and the distribution of the square of the sample multiple correlation coefficient." Denise Benton, K. Krishnamoorthy, Computational Statistics & Data Analysis 43 (2003) 249-267.

This uses the following formula for the CDF:

Where I_{x}(a,b) is the incomplete beta function, and Φ(x) is the normal
CDF at x.

Iteration starts at the largest of the Poisson weighting terms (at i
= δ^{2} / 2) and then proceeds in both directions as per Benton and Krishnamoorthy's
paper.

Alternatively, by considering what happens when t = ∞, we have x = 1,
and therefore I_{x}(a,b) = 1 and:

From this we can easily show that:

and therefore we have a means to compute either the probability or its complement directly without the risk of cancellation error. The crossover criterion for choosing whether to calculate the CDF or its complement is the same as for the Noncentral Beta Distribution.

The PDF can be computed by a very similar method using:

Where I_{x}^{'}(a,b) is the derivative of the incomplete beta function.

The quantile is calculated via the usual derivative-free root-finding techniques, with the initial guess taken as the quantile of a normal approximation to the noncentral T.

There is no closed form for the mode, so this is computed via functional maximisation of the PDF.

The remaining functions (mean, variance etc) are implemented using the formulas given in Weisstein, Eric W. "Noncentral Student's t-Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html and in the Mathematica documentation.

Some analytic properties of noncentral distributions (particularly unimodality, and monotonicity of their modes) are surveyed and summarized by:

Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, 141 (2003) 3-12.