R package for univariate kernel density estimation with parametric starts and asymmetric kernels.
kdensity is an implementation of univariate kernel density estimation with support for parametric starts and asymmetric kernels. Its main function is
kdensity, which is has approximately the same syntax as
stats::density. Its new functionality is:
kdensityhas built-in support for many parametric starts, such as
gamma, but you can also supply your own.
gammakernels, but also the common symmetric ones. In addition, you can also supply your own kernels.
bw, again including an option to specify your own.
A reason to use
kdensity is to avoid boundary bias when estimating densities on the unit interval or the positive half-line. Asymmetric kernels such as
gcopula are designed for this purpose. The support for parametric starts allows you to easily use a method that is often superior to ordinary kernel density estimation.
First you need to install the package
CRAN. From inside
R, use the following command.
This installs the latest version of the package from GitHub. Call the
library function and use it just like
stats:density, but with optional additional arguments.
library("kdensity") plot(kdensity(mtcars$mpg, start = "normal"))
Kernel density estimation with a parametric start was introduced by Hjort and Glad in Nonparametric Density Estimation with a Parametric Start (1995). The idea is to start out with a parametric density before you do your kernel density estimation, so that your actual kernel density estimation will be a correction to the original parametric estimate. This is a good idea because the resulting estimator will be better than an ordinary kernel density estimator whenever the true density is close to your suggestion; and the estimator can be superior to the ordinary kernal density estimator even when the suggestion is pretty far off.
In addition to parametric starts, the package implements some asymmetric kernels. These kernels are useful when modelling data with sharp boundaries, such as data supported on the positive half-line or the unit interval. Currently we support the following asymmetric kernels:
Jones and Henderson’s Gaussian copula KDE, from Kernel-Type Density Estimation on the Unit Interval (2007). This is used for data on the unit interval. The bandwidth selection mechanism described in that paper is implemented as well. This kernel is called
Chen’s two beta kernels from Beta kernel estimators for density functions (1999). These are used for data supported on the on the unit interval, and are called
Chen’s two gamma kernels from Probability Density Function Estimation Using Gamma Kernels (2000). These are used for data supported on the positive half-line, and are called
These features can be combined to make asymmetric kernel densities estimators with parametric starts, see the example below. The package contains only one function,
kdensity, in addition to the generics
kdensity takes some
data, a kernel
kernel and a parametric start
start. You can optionally specify the
support parameter, which is used to find the normalizing constant.
The following example uses the data set plots both a gamma-kernel density estimate with a gamma start (black) and the the fully parametric gamma density. The underlying parameter estimates are always maximum likelood.
library("kdensity") kde = kdensity(airquality$Wind, start = "gamma", kernel = "gamma") plot(kde, main = "Wind speed (mph)") lines(kde, plot_start = TRUE, col = "red") rug(airquality$Wind)
Since the return value of
kdensity is a function, it is callable, as in:
kde(10) #>  0.09980471
You can access the parameter estimates by using
coef. You can also access the log likelihood (
logLik), AIC and BIC of the parametric start distribution.
coef(kde) #> shape rate #> 7.1872898 0.7217954 logLik(kde) #> 'log Lik.' 12.33787 (df=2) AIC(kde) #>  -20.67574
Hjort, Nils Lid, and Ingrid K. Glad. “Nonparametric density estimation with a parametric start.” The Annals of Statistics (1995): 882-904..
Jones, M. C., and D. A. Henderson. “Miscellanea kernel-type density estimation on the unit interval.” Biometrika 94.4 (2007): 977-984..
Chen, Song Xi. “Probability density function estimation using gamma kernels.” Annals of the Institute of Statistical Mathematics 52.3 (2000): 471-480..
Chen, Song Xi. “Beta kernel estimators for density functions.” Computational Statistics & Data Analysis 31.2 (1999): 131-145.