Kernel Density Estimation and Convolution

Kernel Density Estimation (KDE) is a widely used technique for estimating the probability density function of a random variable. In this study, we revisit KDE through the lens of convolution and extend this perspective to special cases such as positive, bounded and heavy tailed random variables. Building on this foundation, we propose a novel simulation-based density estimation method that generates new data by adding noise to observed values and then smoothing the resulting histogram using splines. A minor adjustment to natural cubic splines is required to ensure nonnegative estimates. The noise is drawn from a class of bounded polynomial kernel densities obtained via convolution of uniform random variables, with the smoothing parameter naturally defined by the support bound. A practical choice for this parameter is determined by a percentile of the neighboring distances among sorted data. The proposed method offers enhanced flexibility for handling variables with specific support constraints (e.g., positive, bounded and heavy tailed) through simple transformations, and numerical studies demonstrate its competitive or superior performance compared to standard KDE across various scenarios.