[EN] Kernel density estimation is a non-parametric method to estimate the probability density function of a random quantity from a finite data sample. The estimator consists of a kernel function and a smoothing parameter called the bandwidth. Despite its undeniable usefulness, the convergence rate may be slow with the number of realizations and the discontinuity and peaked points of the target density may not be correctly captured. In this work, we analyze the applicability of a parametric method based on Monte Carlo simulation for the density estimation of certain random variable transformations. This approach has important applications in the setting of differential equations with input random parameters.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Cortés, J.; Jornet Sanz, M. (2020). Improving Kernel Methods for Density Estimation in Random Differential Equations Problems. Mathematical and Computational Applications (Online). 25(2):1-9. https://doi.org/10.3390/mca25020033S19252Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2020). Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Applied Numerical Mathematics, 151, 413-424. doi:10.1016/j.apnum.2020.01.012Calatayud, J., Cortés, J.-C., & Jornet, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A: Statistical Mechanics and its Applications, 512, 261-279. doi:10.1016/j.physa.2018.08.024Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2019). Density function of random differential equations via finite difference schemes: a theoretical analysis of a random diffusion-reaction Poisson-type problem. Stochastics, 92(4), 627-641. doi:10.1080/17442508.2019.1645849Calatayud, J., Cortés, J.-C., Dorini, F. A., & Jornet, M. (2020). Extending the study on the linear advection equation subject to stochastic velocity field and initial condition. Mathematics and Computers in Simulation, 172, 159-174. doi:10.1016/j.matcom.2019.12.014Jornet, M., Calatayud, J., Le Maître, O. P., & Cortés, J.-C. (2020). Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function. Journal of Computational and Applied Mathematics, 374, 112770. doi:10.1016/j.cam.2020.112770Tang, K., Wan, X., & Liao, Q. (2020). Deep density estimation via invertible block-triangular mapping. Theoretical and Applied Mechanics Letters, 10(3), 143-148. doi:10.1016/j.taml.2020.01.023Botev, Z., & Ridder, A. (2017). Variance Reduction. Wiley StatsRef: Statistics Reference Online, 1-6. doi:10.1002/9781118445112.stat0797
