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Introduction

The density function of the Logistic distribution:

$$ f(x; \mu, \sigma) = \dfrac{\exp\left( - \frac{x-\mu}{\sigma} \right)}{\sigma \left( 1 + \exp\left( - \frac{x-\mu}{\sigma} \right) \right)^2 } $$

Parameters:

• mu_par is $\mu$
• sigma_par is $\sigma$

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Density

Definition:

 
template<typename Ta, typename Tb>
statslib_constexpr
return_t<Ta> dlogis(const Ta x, const Tb mu_par, const Tb sigma_par, const bool log_form = false);

Computes the density function.

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Examples:

// parameters
double mu = 0.0;
double sigma = 1.0;

// standard input
double dens_val = stats::dlogis(0.5,mu,sigma);
double log_dens_val = stats::dlogis(0.5,mu,sigma,true);

// Armadillo input
arma::mat X(10,1);
X.fill(0.5);

arma::mat dens_vals_mat = stats::dlogis(X,mu,sigma);
arma::mat log_dens_vals_mat = stats::dlogis(X,mu,sigma,true);

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Probability

Definition:

 
template<typename Ta, typename Tb>
statslib_constexpr
return_t<Ta> plogis(const Ta x, const Tb mu_par, const Tb sigma_par, const bool log_form = false);

Computes the cumulative distribution function (CDF).

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Examples:

// parameters
double mu = 0.0;
double sigma = 1.0;

// standard input
double prob_val = stats::plogis(0.5,mu,sigma);
double log_prob_val = stats::plogis(0.5,mu,sigma,true);

// Armadillo input
arma::mat X(10,1);
X.fill(0.5);

arma::mat prob_vals_mat = stats::plogis(X,mu,sigma);
arma::mat log_prob_vals_mat = stats::plogis(X,mu,sigma,true);

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Quantile

Definition:

 
template<typename Ta, typename Tb>
statslib_constexpr
Ta qlogis(const Ta p, const Tb mu_par, const Tb sigma_par);

Computes the quantile function.

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Examples:

// parameters
double mu = 0.0;
double sigma = 1.0;

// standard input
double quant_val = stats::qlogis(0.7,mu,sigma);

// Armadillo input
arma::mat X(10,1);
X.fill(0.7);

arma::mat quant_vals_mat = stats::qlogis(X,mu,sigma);

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Random Sampling

Definition:

 
// random engine seeding
template<typename T>
statslib_inline
return_t<T> rlogis(const T mu_par, const T sigma_par, rand_engine_t& engine);

// seeding values
template<typename T>
statslib_inline
return_t<T> rlogis(const T mu_par, const T sigma_par, uint_t seed_val = std::random_device{}());

// matrix output
template<typename mT, typename eT>
statslib_inline
mT rlogis(const uint_t n, const uint_t k, const eT mu_par, const eT sigma_par);

Generates pseudo-random draws.

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Examples:

// parameters
double mu = 0.0;
double sigma = 1.0;

// standard input
double rand_val = stats::rlogis(mu,sigma);

// Armadillo output: 10 x 1 matrix
arma::mat rand_mat = stats::rlogis<arma::mat>(10,1,mu,sigma);

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