paper/paper.md

In JonasMoss/kdensity: Kernel Density Estimation with Parametric Starts and Asymmetric Kernels

--- title: 'kdensity: An R package for kernel density estimation with parametric starts and asymmetric kernels' tags: - R - statistics - kernel density estimation - non-parametric statistics - non-parametrics - non-parametric density estimation - boundary bias authors: - name: Jonas Moss orcid: 0000-0002-6876-6964 affiliation: 1 - name: Martin Tveten orcid: 0000-0002-4236-633X affiliation: 1 affiliations: - name: University of Oslo index: 1 date: 11 July 2019 bibliography: paper.bib

Summary

It is often necessary to estimate a probability density non-parametrically, that is, without making strong parametric assumptions such as normality. This R [@r] package provides a non-parametric density estimator that can take advantage of some of the knowledge the user has about the probability density.

Kernel density estimation [@silverman2018density] is a popular method for non-parametric density estimation based on placing kernels on each data point. @hjort_glad_1995 extended kernel density estimation with parametric starts. The parametric start is a parametric density that is multiplied with the kernel estimate. When the data-generating density is reasonably close to the parametric start density, kernel density estimation with that parametric start will outperform ordinary kernel density estimation.

Moreover, when estimating densities on the half-open interval $\left[0,\infty\right)$ and bounded intervals, such as $\left[0, 1\right]$, symmetric kernels are prone to serious boundary bias that should be corrected [@marron1994transformations]. Asymmetric kernels have been designed to avoid boundary bias and many of them are implemented in kdensity in addition to the classical symmetric kernels. For the unit interval, the Gaussian copula kernel of @jones2007miscellanea and the beta kernels of @chen1999beta are supported. The gamma kernel of @chen2000probability is available for the half-open interval.

The supported non-parametric starts include the normal, Laplace, Gumbel, exponential, gamma, log-normal, inverse Gaussian, Weibull, Beta, and Kumaraswamy densities. The parameters of all parametric starts are estimated using maximum likelihood. The implemented bandwidth selectors are the classical bandwidth selectors from stats, unbiased cross-validation, the Hermite polynomial method from @hjort_glad_1995, and the tailored bandwidth selector for the Gaussian copula method of @jones2007miscellanea. User-defined parametric starts, kernels and bandwidth selectors can also be set.

Several R packages deal with kernel estimation, see @deng2011 for an overview. While no other R package handles density estimation with parametric starts, several packages supports methods that handle boundary bias. @evmix provides a variety of boundary bias correction methods in the bckden functions. @kde1d corrects for boundary bias using probit or logarithmically transformed local polynomial kernel density estimation. @logKDE corrects for boundary bias on the half line using a logarithmic transform. @ks supports boundary correction through the kde.boundary function, while @Ake corrects for boundary bias using asymmetric kernels.

The following example uses the airquality data set from the built-in R package datasets. Since the data is positive we use Chen's gamma kernel. As the data is likely to be better approximated by a gamma distribution than a uniform distribution, we use the gamma parametric start. The plotted density is in figure 1, where the gamma distribution with parameters estimated by maximum likelihood is in red and the ordinary kernel density estimate in blue. Notice the boundary bias of the ordinary kernel density estimator.

# install.packages("kdensity")
library("kdensity")
kde = kdensity(airquality$Wind, start = "gamma", kernel = "gamma")
plot(kde, main = "Wind speed (mph)")
lines(kde, plot_start = TRUE, col = "red")
lines(density(airquality$Wind, adjust = 2), col = "blue")
rug(airquality$Wind)

References

JonasMoss/kdensity documentation built on Oct. 5, 2020, 2:30 p.m.