Kernel Estimator and Bandwidth Selection for Density and its Derivatives

In statistics, the univariate kernel density estimation (KDE) is a non-parametric way to estimate the probability density function f(x) of a random variable X, is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. This techniques are widely used in various inference procedures such as signal processing, data mining and econometrics, see e.g., Silverman [1986], Wand and Jones [1995], Jeffrey [1996], Wolfgang et all [2004], Alexandre [2009]. The kernel estimator are standard in many books with applications and computer vision, see Wolfgang [1991], Scott [1992], Bowman and Azzalini [1997], Venables and Ripley [2002], for computational complexity and with implementation in S, for an overview. Estimation of the density derivatives also comes up in various other applications like estimation of modes and inflexion points of densities, a good list of applications which require the estimation of density derivatives can be found in Singh [1977]. There already exist a number of packages that can perform kernel density estimation in R (density in R base); see for example KernSmooth [Wand and Ripley, 2013], sm [Bowman and Azzalini, 2013], np [Tristen and Jeffrey, 2008] and feature [Duong and Matt, 2013], they exist also of functions for kernel density derivative estimation (KDDE), e.g., kdde in ks package [Duong, 2007]. We introduce in this vignette a new R package kedd [Guidoum, 2015] for use with the statistical programming environment R Development Core Team [2015], which implements smoothing techniques and computing bandwidth selectors of the rth derivative of a probability density f(x) for univariate data, using several kernels functions.