Student’s t-Distribution¶

Table of contents

Density Function ¶

The density function of the Student’s t distribution:

\[f(x; \nu) = \dfrac{\Gamma \left( \frac{\nu + 1}{2} \right)}{ \sqrt{\nu \pi} \Gamma \left( \frac{\nu}{2} \right)} \left( 1 + \frac{x^2}{\nu} \right)^{- \frac{\nu+1}{2}}\]

where \(\Gamma(\cdot)\) denotes the gamma function.

Methods for scalar input, as well as for vector/matrix input, are listed below.

Scalar Input ¶

template<typename T1, typename T2> constexpr common_return_t<T1, T2> dt(const T1 x, const T2 dof_par, const bool log_form = false) noexcept¶

Density function of the t-distribution.

Example:

stats::dt(0.37,11,false); 

Parameters

x – a real-valued input.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

the density function evaluated at x.

Vector/Matrix Input ¶

STL Containers ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>> inline std::vector<rT> dt(const std::vector<eT> &x, const T1 dof_par, const bool log_form = false)¶

Density function of the t-distribution.

Example:

std::vector<double> x = {1.8, 0.7, 4.2};
stats::dt(x,4,false);

Parameters

x – a standard vector.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

a vector of density function values corresponding to the elements of x.

Armadillo ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>> inline ArmaMat<rT> dt(const ArmaMat<eT> &X, const T1 dof_par, const bool log_form = false)¶

Density function of the t-distribution.

Example:

arma::mat X = { {1.8, 0.7, 4.2},
                {0.3, 5.3, 3.7} };
stats::dt(X,4,false);

Parameters

X – a matrix of input values.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

a matrix of density function values corresponding to the elements of X.

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>, bool To = blaze::columnMajor> inline BlazeMat<rT, To> dt(const BlazeMat<eT, To> &X, const T1 dof_par, const bool log_form = false)¶

Density function of the t-distribution.

Example:

stats::dt(X,4,false);

Parameters

X – a matrix of input values.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

a matrix of density function values corresponding to the elements of X.

Eigen ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic> inline EigenMat<rT, iTr, iTc> dt(const EigenMat<eT, iTr, iTc> &X, const T1 dof_par, const bool log_form = false)¶

Density function of the t-distribution.

Example:

stats::dt(X,4,false);

Parameters

X – a matrix of input values.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

a matrix of density function values corresponding to the elements of X.

Cumulative Distribution Function ¶

The cumulative distribution function of the Student’s t distribution:

\[F(x; \nu) = \int_{-\infty}^x f(z; \nu) dz = \frac{1}{2} + x \Gamma \left( \frac{\nu + 1}{2} \right) + \dfrac{ {}_2 F_1 \left( \frac{1}{2}, \frac{\nu+1}{2}; \frac{3}{\nu}; - \frac{x^2}{\nu} \right) }{ \sqrt{\nu \pi} \Gamma \left( \frac{\nu}{2} \right)}\]

where \(\Gamma(\cdot)\) denotes the gamma function and \({}_2 F_1\) denotes the hypergeometric function.

Methods for scalar input, as well as for vector/matrix input, are listed below.

Scalar Input ¶

template<typename T1, typename T2> constexpr common_return_t<T1, T2> pt(const T1 x, const T2 dof_par, const bool log_form = false) noexcept¶

Distribution function of the t-distribution.

Example:

stats::pt(0.37,11,false); 

Parameters

x – a real-valued input.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

the cumulative distribution function evaluated at x.

Vector/Matrix Input ¶

STL Containers ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>> inline std::vector<rT> pt(const std::vector<eT> &x, const T1 dof_par, const bool log_form = false)¶

Distribution function of the t-distribution.

Example:

std::vector<double> x = {0.0, 1.0, 2.0};
stats::pt(x,4,false);

Parameters

x – a standard vector.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

a vector of CDF values corresponding to the elements of x.

Armadillo ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>> inline ArmaMat<rT> pt(const ArmaMat<eT> &X, const T1 dof_par, const bool log_form = false)¶

Distribution function of the t-distribution.

Example:

arma::mat X = { {0.2, -1.7,  0.1},
                {0.9,  4.0, -0.3} };
stats::pt(X,4,false);

Parameters

X – a matrix of input values.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

a matrix of CDF values corresponding to the elements of X.

Blaze ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>, bool To = blaze::columnMajor> inline BlazeMat<rT, To> pt(const BlazeMat<eT, To> &X, const T1 dof_par, const bool log_form = false)¶

Distribution function of the t-distribution.

Example:

stats::pt(X,4,false);

Parameters

X – a matrix of input values.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

a matrix of CDF values corresponding to the elements of X.

Eigen ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic> inline EigenMat<rT, iTr, iTc> pt(const EigenMat<eT, iTr, iTc> &X, const T1 dof_par, const bool log_form = false)¶

Distribution function of the t-distribution.

Example:

stats::pt(X,4,false);

Parameters

X – a matrix of input values.
dof_par – the degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

a matrix of CDF values corresponding to the elements of X.

Quantile Function ¶

The quantile function of the Student’s t distribution:

\[q(p; \nu) = \inf \left\{ x : p \leq F(x; \nu) \right\}\]

Methods for scalar input, as well as for vector/matrix input, are listed below.

Scalar Input ¶

template<typename T1, typename T2> constexpr common_return_t<T1, T2> qt(const T1 p, const T2 dof_par) noexcept¶

Quantile function of the t-distribution.

Example:

stats::qt(0.5,11); 

Parameters

p – a real-valued input.
dof_par – the degrees of freedom parameter, a real-valued input.

Returns

the quantile function evaluated at p.

Vector/Matrix Input ¶

STL Containers ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>> inline std::vector<rT> qt(const std::vector<eT> &x, const T1 dof_par)¶

Quantile function of the t-distribution.

Example:

std::vector<double> x = {0.3, 0.5, 0.8};
stats::qt(x,4);

Parameters

x – a standard vector.
dof_par – the degrees of freedom parameter, a real-valued input.

Returns

a vector of quantile values corresponding to the elements of x.

Armadillo ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>> inline ArmaMat<rT> qt(const ArmaMat<eT> &X, const T1 dof_par)¶

Quantile function of the t-distribution.

Example:

arma::mat X = { {0.2, 0.7, 0.9},
                {0.1, 0.8, 0.3} };
stats::qt(X,4);

Parameters

X – a matrix of input values.
dof_par – the degrees of freedom parameter, a real-valued input.

Returns

a matrix of quantile values corresponding to the elements of X.

Blaze ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>, bool To = blaze::columnMajor> inline BlazeMat<rT, To> qt(const BlazeMat<eT, To> &X, const T1 dof_par)¶

Quantile function of the t-distribution.

Example:

stats::qt(X,4);

Parameters

X – a matrix of input values.
dof_par – the degrees of freedom parameter, a real-valued input.

Returns

a matrix of quantile values corresponding to the elements of X.

Eigen ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic> inline EigenMat<rT, iTr, iTc> qt(const EigenMat<eT, iTr, iTc> &X, const T1 dof_par)¶

Quantile function of the t-distribution.

Example:

stats::qt(X,4);

Parameters

X – a matrix of input values.
dof_par – the degrees of freedom parameter, a real-valued input.

Returns

a matrix of quantile values corresponding to the elements of X.

Random Sampling ¶

Scalar Output ¶

Random number engines

template<typename T> inline return_t<T> rt(const T dof_par, rand_engine_t &engine)¶

Random sampling function for Student’s t-distribution.

Example:

stats::rand_engine_t engine(1776);
stats::rt(4,engine);

Parameters

dof_par – the degrees of freedom parameter, a real-valued input.
engine – a random engine, passed by reference.

Returns

a pseudo-random draw from Student’s t-distribution.

Seed values

template<typename T> inline return_t<T> rt(const T dof_par, const ullint_t seed_val = std::random_device{}())¶

Random sampling function for Student’s t-distribution.

Example:

stats::rt(4,1776);

Parameters

dof_par – the degrees of freedom parameter, a real-valued input.
seed_val – initialize the random engine with a non-negative integral-valued seed.

Returns

a pseudo-random draw from Student’s t-distribution.

Vector/Matrix Output ¶

Random number engines

template<typename mT, typename T1> inline mT rt(const ullint_t n, const ullint_t k, const T1 dof_par, rand_engine_t &engine)¶

Random matrix sampling function for Student’s t-distribution.

Example:

stats::rand_engine_t engine(1776);
// std::vector
stats::rt<std::vector<double>>(5,4,12,engine);
// Armadillo matrix
stats::rt<arma::mat>(5,4,12,engine);
// Blaze dynamic matrix
stats::rt<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,12,engine);
// Eigen dynamic matrix
stats::rt<Eigen::MatrixXd>(5,4,12,engine);

Note

This function requires template instantiation; acceptable output types include: std::vector, with element type float, double, etc., as well as Armadillo, Blaze, and Eigen dense matrices.

Parameters

n – the number of output rows
k – the number of output columns
dof_par – the degrees of freedom parameter, a real-valued input.
engine – a random engine, passed by reference.

Returns

a matrix of pseudo-random draws from the t-distribution.

Seed values

template<typename mT, typename T1> inline mT rt(const ullint_t n, const ullint_t k, const T1 dof_par, const ullint_t seed_val = std::random_device{}())¶

Random matrix sampling function for Student’s t-distribution.

Example:

// std::vector
stats::rt<std::vector<double>>(5,4,12);
// Armadillo matrix
stats::rt<arma::mat>(5,4,12);
// Blaze dynamic matrix
stats::rt<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,12);
// Eigen dynamic matrix
stats::rt<Eigen::MatrixXd>(5,4,12);

Note

This function requires template instantiation; acceptable output types include: std::vector, with element type float, double, etc., as well as Armadillo, Blaze, and Eigen dense matrices.

Parameters

n – the number of output rows
k – the number of output columns
dof_par – the degrees of freedom parameter, a real-valued input.
seed_val – initialize the random engine with a non-negative integral-valued seed.

Returns

a matrix of pseudo-random draws from Student’s t-distribution.