Random variate generation by numerical inversion when only the density is known

Abstract

We present a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and nearly the same for all distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. This makes our algorithm especially attractive for the simulation of copulas and for quasi-Monte Carlo applications.

Generating generalized inverse Gaussian random variates by fast inversion

“…In [5] we suggest a new algorithm that combines Newton's interpolation formula with adaptive Gauss-Lobatto integration to evaluate the differences of the CDF. Thus only the probability density function (PDF) of the target distribution is required.…”

“…In particular it can be computed during the setup and it can be interpreted with respect to the resolution of the underlying uniform pseudo-random number generator or low discrepancy set (see [5] for details). In fact goodness-of-fit tests like the Kolmogorov-Smirnov test or the χ 2 test look exactly at that deviation.…”

“…We used the names HINV for the Hermite interpolation algorithm of [4] and PINV for the Newton interpolation with Gauss-Lobatto integration algorithm of [5]. The UNU.RAN library is also accessible from the R statistical programming language [9] by using our package Runuran [10].…”

“…However, due to its slow convergence the bisection method is usually considered as a last resort. For the case of fixed parameters we propose two much faster methods which we presented in two papers [4,5]. These are based on polynomial approximations of the inversion CDF of the (generalized) inversion Gaussian distribution In the remaining part of this note we shortly describe these algorithms and report our computational experience.…”

Efficient Numerical Inversion for Financial Simulations

“…The slow setup is due to the fact that this algorithm requires the CDF which is extremely expensive for the generalized hyperbolic distribution. This talk motivated us to demonstrate the practical application of our recently proposed algorithm [5] to such distributions. It only requires the probability density function (PDF) of the target distribution and computes the CDF during the setup.…”