Statistics and Computing manuscript No. (will be inserted by the editor) Fast parallel α-stable distribution function evaluation and parameter estimation using OpenCL in GPGPUs Guillermo Juli´ an-Moreno · Jorge E. L´ opez de Vergara · Iv´ an Gonz´ alez · Luis de Pedro · Javier Royuela-del-Val · Federico Simmross-Wattenberg Received: 19 th February 2016. Revised: 4 th August 2016. Accepted: 8 th August 2016. The final publication is available at Springer via http://dx.doi.org/10.1007/s11222-016-9691-9. Abstract α-stable distributions are a family of proba- bility distributions found to be suitable to model many complex processes and phenomena in several research fields, such as medicine, physics, finance and network- ing, among others. However, the lack of closed expres- sions makes their evaluation analytically intractable, and alternative approaches are computationally expen- sive. Existing numerical programs are not fast enough for certain applications and do not make use of the parallel power of general purpose graphic processing units (GPGPUs). In this paper, we develop novel par- allel algorithms for the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) – including a parallel Gauss-Kronrod quadrature–, quan- tile function, random number generator and maximum likelihood estimation of α-stable distributions using This work was partially supported by the Spanish Ministry of Economy and Competitiveness under the projects PackTrack (TEC2012-33754), TR ´ AFICA (MINECO/FEDER TEC2015- 69417-C2-1-R) and kt-WiSE-MR (TEC2014-57428R), and by the Universidad Aut´onoma de Madrid under the project “Implementaci´ on de Modelos Computacionales Masivamente Paralelos” (CEMU-2013-14). The authors also thank the Spanish Junta de Castilla y Le´ on for grant VA136U13 and the Universidad de Valladolid–Banco de Santander grant pro- gram 2012. Guillermo Juli´an-Moreno, Jorge E. L´ opez de Vergara, Iv´anGonz´alez,LuisdePedro Department of Electronics and Communication Technolo- gies, Escuela Polit´ ecnica Superior, Universidad Aut´onoma de Madrid, Spain E-mail: guillermo.julian@estudiante.uam.es, {jorge.lopez vergara, ivan.gonzalez, luis.depedro}@uam.es Javier Royuela-del-Val, Federico Simmross-Wattenberg Image Processing Lab, E.T.S.I. Telecomunicaci´on, Universi- dad de Valladolid, Spain E-mail: {jroyval, fedesim}@lpi.tel.uva.es OpenCL, achieving significant speedups and precision in all cases. Thanks to the use of OpenCL, we also eval- uate the results of our library with different GPU ar- chitectures. Keywords Gaussian quadrature · α-stable distribu- tion · Parallel algorithms · Numerical algorithms · OpenCL · GPGPU 1 Introduction The Central Limit Theorem is a well-known mathemat- ical result, which states that the standardized sum of a sufficiently large number of independent, identically distributed random variables with finite variance and mean will resemble a normal (Gaussian) distribution. This theorem can be generalized for random variables with infinite moments: the resulting distribution is then called an α-stable (or just stable) distribution (Gne- denko and Kolmogorov, 1968). Its name comes from an- other interesting property (Nolan, 2015): the fact that, given X 1 ,X 2 independent copies of a random variable X with stable distribution, then aX 1 + bX 2 dist. = cX + d (1) for some constants a, b, c > 0 and d ∈ R. These properties make stable distributions a suit- able model for many events in different fields that nat- urally exhibit such high variability rates that they can- not be adequately modeled using simple statistical dis- tributions. For example, in medicine they are used for segmentation of brain matter in Magnetic Resonance Imaging (MRI) (Salas-Gonz´alez et al., 2013) and as a model for ultrasound denoising (Achim et al., 2001); in physics they can be used to study and predict atomic