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StatsLib is a templated C++ library of statistical distribution functions, featuring unique compile-time computing capabilities and seamless integration with several popular linear algebra libraries.


  • A header-only library of probability density functions, cumulative distribution functions, quantile functions, and random sampling methods.
  • Functions are written in a specialized C++11 constexpr format, enabling the library to operate as both a compile-time and run-time computation engine.
  • Designed with a simple R-like syntax.
  • Optional vector-matrix functionality with wrappers to support:
  • Matrix-based operations are parallelizable with OpenMP.
  • Released under a permissive, non-GPL license.

Author: Keith O'Hara




Functions to compute the cdf, pdf, and quantile, as well as random sampling methods, are available for the following distributions:

In addition, pdf and randomization functions are available for several multivariate distributions:

Download and Installation

  • The source code is available on GitHub.Download the latest version with git:
  •     git clone -b master --single-branch ./stats

StatsLib is a header-only library. Simply add the header files to your project using

#include "stats.hpp"

The only dependencies are a C++11-compatible compiler and the GCE-Math library which comes pre-packaged with StatsLib.

To build the test files:

# clone stats
git clone -b master --single-branch ./stats
# compile tests
cd ./stats/tests
cd dens
./configure && make

Jupyter Notebook

You can test the library online using an interactive Jupyter notebook:



The following options should be declared before including the StatsLib header files.

  • For inline-only functionality (i.e., no constexpr specifiers):
  •     #define STATS_GO_INLINE
  • OpenMP functionality is enabled by default if the _OPENMP macro is detected (e.g., by invoking -fopenmp with a GCC or Clang compiler). To explicitly enable OpenMP features use:
  •     #define STATS_USE_OPENMP
  • To disable OpenMP functionality:
  •     #define STATS_DONT_USE_OPENMP
  • To use StatsLib with Armadillo, Blaze or Eigen:
  •     #define STATS_USE_ARMA
        #define STATS_USE_BLAZE
        #define STATS_USE_EIGEN

Syntax and Examples

Functions are called using an R-like syntax. Some general rules:

  • stats::d* density functions. For example, the Normal (Gaussian) density is called using
  •     stats::dnorm(<value>,<mean parameter>,<standard deviation>);
  • stats::p* cumulative distribution functions. For example, the Gamma CDF is called using
  •     stats::pgamma(<value>,<shape parameter>,<scale parameter>);
  • stats::q* quantile functions. For example, the Beta quantile is called using
  •     stats::qbeta(<value>,<a parameter>,<b parameter>);
  • stats::r* random sampling. For example, to generate a single draw from the Logistic distribution:
  •     stats::rlogis(<location parameter>,<scale parameter>,<seed value or random number engine>);


All of these functions have matrix-based equivalents using Armadillo, Blaze, and Eigen dense matrices.

  • The pdf, cdf, and quantile functions can take matrix-valued arguments. For example,
  •     // Using Armadillo:
        arma::mat norm_pdf_vals = stats::dnorm(arma::ones(10,20),1.0,2.0);
  • The randomization functions (r*) can output random matrices of arbitrary size. For example,
        // Armadillo:
        arma::mat gamma_rvs = stats::rgamma<arma::mat>(100,50,3.0,2.0);
        // Blaze:
        blaze::DynamicMatrix<double> gamma_rvs = stats::rgamma<blaze::DynamicMatrix<double>>(100,50,3.0,2.0);
        // Eigen:
        Eigen::MatrixXd gamma_rvs = stats::rgamma<Eigen::MatrixXd>(100,50,3.0,2.0);
    will generate a 100-by-50 matrix of iid draws from a Gamma(3,2) distribution.
  • All matrix-based operations are parallelizable with OpenMP. For GCC and Clang compilers, simply include the -fopenmp option during compilation.


Random number seeding is available in two formats: seed values and random number engines.

  • Seed values are passed as unsigned integers. For example, to generate a draw from a normal distribution N(1,2) with seed value 1776:
  •     stats::rnorm(1,2,1776);
  • Random engines in StatsLib use the 64-bit Mersenne-Twister generator (std::mt19937_64) and are passed by reference. Example:
  •     std::mt19937_64 engine(1776);


More examples with code:

    // evaluate the normal PDF at x = 1, mu = 0, sigma = 1
    double dval_1 = stats::dnorm(1.0,0.0,1.0)

    // evaluate the normal PDF at x = 1, mu = 0, sigma = 1, and return the log value
    double dval_2 = stats::dnorm(1.0,0.0,1.0,true)

    // evaluate the normal CDF at x = 1, mu = 0, sigma = 1
    double pval_1 = stats::pnorm(1.0,0.0,1.0)

    // evaluate the Laplacian quantile at p = 0.1, mu = 0, sigma = 1
    double qval_1 = stats::qlaplace(0.1,0.0,1.0)

    // draw from a t-distribution with dof = 30
    double rval = stats::rt(30);

    // matrix output
    arma::mat beta_rvs = stats::rbeta<arma::mat>(100,100,3.0,2.0);
    // matrix input
    arma::mat beta_cdf_vals = stats::pbeta(beta_rvs,3.0,2.0);

Compile-time Computation Capabilities

StatsLib is designed to operate equally well as a compile-time computation engine. Compile-time computation allows the compiler to replace function calls (e.g., dnorm(0,0,1)) with static values in the source code. That is, functions are evaluated during the compilation process, rather than at run-time. This capability is made possible due to the templated constexpr design of the library and can be verified by inspecting the assembly code generated by the compiler.

The compile-time features are enabled using the constexpr specifier. The example below computes the pdf, cdf, and quantile function of the Laplace distribution:

#include "stats.hpp"

int main()
    constexpr double dens_1  = stats::dlaplace(1.0,1.0,2.0); // answer = 0.25
    constexpr double prob_1  = stats::plaplace(1.0,1.0,2.0); // answer = 0.5
    constexpr double quant_1 = stats::qlaplace(0.1,1.0,2.0); // answer = -2.218875...

    return 0;

Assembly code generated by Clang without any optimization:

	.quad	-4611193153885729483    ## double -2.2188758248682015
	.quad	4602678819172646912     ## double 0.5
	.quad	4598175219545276417     ## double 0.25000000000000006
	.section	__TEXT,__text,regular,pure_instructions
	.globl	_main
	.p2align	4, 0x90
_main:                                  ## @main
	push	rbp
	mov	rbp, rsp
	xor	eax, eax
	movsd	xmm0, qword ptr [rip + LCPI0_0] ## xmm0 = mem[0],zero
	movsd	xmm1, qword ptr [rip + LCPI0_1] ## xmm1 = mem[0],zero
	movsd	xmm2, qword ptr [rip + LCPI0_2] ## xmm2 = mem[0],zero
	mov	dword ptr [rbp - 4], 0
	movsd	qword ptr [rbp - 16], xmm2
	movsd	qword ptr [rbp - 24], xmm1
	movsd	qword ptr [rbp - 32], xmm0
	pop	rbp