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<!DOCTYPE html>
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<meta name="description" content="StatsLib is a templated C++ library for fast computation of statistical distribution functions.">
<meta name="author" content="Keith O'Hara">
<meta name="keywords" content="Statistics, Probability, Distributions, C++, Cpp, C++11, constexpr, Econometrics" />
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<title>StatsLib: Student's t-Distribution</title>
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<a data-toggle="collapse" href="#collapse1"><h4><strong style="font-size: 120%;">StatsLib: Student's t-Distribution</strong></h4></a>
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<div id="collapse1" class="panel-collapse collapse">
<div class="panel-body">
<a href="#intro">Introduction</a> <br>
<a href="#density">Density</a> <br>
<a href="#probability">Probability</a> <br>
<a href="#quantile">Quantile</a> <br>
<a href="#random">Random Sampling</a>
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<h3 style="text-align: left;" id="intro"><strong style="font-size: 120%;">Introduction</strong></h3>
<p>The density function of Student's $t$ distribution:</p>
$$
f(x; \nu) = \dfrac{\Gamma \left( \frac{\nu + 1}{2} \right)}{ \sqrt{\nu \pi} \Gamma \left( \frac{\nu}{2} \right)} \left( 1 + \frac{t^2}{\nu} \right)^{- \frac{\nu+1}{2}}
$$
<p>Parameters:</p>
<ul>
<li><code>dof_par</code> is $\nu$. Note: this need not be an integer.</li>
</ul>
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<h3 style="text-align: left;" id="density"><strong style="font-size: 120%;">Density</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename Ta, typename Tb>
statslib_constexpr
return_t<Ta> dt(const Ta x, const Tb dof_par, const bool log_form = false);
</pre>
<p>Computes the density function.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double dof = 14.0;
// standard input
double dens_val = stats::dt(0.5,dof);
double log_dens_val = stats::dt(0.5,dof,true);
// Armadillo input
arma::mat X(10,1);
X.fill(0.5);
arma::mat dens_vals_mat = stats::dt(X,dof);
arma::mat log_dens_vals_mat = stats::dt(X,dof,true);
</pre>
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<h3 style="text-align: left;" id="probability"><strong style="font-size: 120%;">Probability</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename Ta, typename Tb>
statslib_constexpr
return_t<Ta> pt(const Ta x, const Tb dof_par, const bool log_form = false);
</pre>
<p>Computes the cumulative distribution function (CDF).</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double dof = 14.0;
// standard input
double prob_val = stats::pt(0.5,dof);
double log_prob_val = stats::pt(0.5,dof,true);
// Armadillo input
arma::mat X(10,1);
X.fill(0.5);
arma::mat prob_vals_mat = stats::pt(X,dof);
arma::mat log_prob_vals_mat = stats::pt(X,dof,true);
</pre>
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<h3 style="text-align: left;" id="quantile"><strong style="font-size: 120%;">Quantile</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename Ta, typename Tb>
statslib_constexpr
Ta qt(const Ta p, const Tb dof_par);
</pre>
<p>Computes the quantile function.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double dof = 14.0;
// standard input
double quant_val = stats::qt(0.7,dof);
// Armadillo input
arma::mat X(10,1);
X.fill(0.7);
arma::mat quant_vals_mat = stats::qt(X,dof);
</pre>
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<h3 style="text-align: left;" id="random"><strong style="font-size: 120%;">Random Sampling</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
// random engine seeding
template<typename T>
statslib_inline
return_t<T> rt(const T dof_par, rand_engine_t& engine);
// seeding values
template<typename T>
statslib_inline
return_t<T> rt(const T dof_par, uint_t seed_val = std::random_device{}());
// matrix output
template<typename mT, typename eT>
statslib_inline
mT rt(const uint_t n, const uint_t k, const eT dof_par);
</pre>
<p>Generates pseudo-random draws.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
double dof = 14.0;
// standard input
double rand_val = stats::rt(dof);
// Armadillo output: 10 x 1 matrix
arma::mat rand_mat = stats::rt<arma::mat>(10,1,dof);
</pre>
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