Kernel Adaptive Density Estimation and Regression

Implementation of various kernel adaptive methods in nonparametric curve estimation like density estimation as introduced in Stute and Srihera (2011) and Eichner and Stute (2013) for pointwise estimation, and like regression as described in Eichner and Stute (2012) .

kader

The goal of kader is to supply functions to compute nonparametric kernel estimators for

• density estimation using a data-adjusted kernel or an appropriate rank-transformation, and for
• regression using a data-adjusted kernel.

The functions are based on the theory introduced in

• Srihera, R., Stute, W. (2011): Kernel adjusted density estimation. Statistics and Probability Letters 81, 571 - 579, URL http://dx.doi.org/10.1016/j.spl.2011.01.013.
• Eichner, G., Stute, W. (2012): Kernel adjusted nonparametric regression. Journal of Statistical Planning and Inference 142, 2537 - 2544, URL http://dx.doi.org/10.1016/j.jspi.2012.03.011.
• Eichner, G., Stute, W. (2013): Rank Transformations in Kernel Density Estimation. Journal of Nonparametric Statistics 25(2), 427 - 445, URL http://dx.doi.org/10.1080/10485252.2012.760737.

A very brief summary of the theory and sort of a vignette is presented in Eichner, G. (2017): Kader - An R package for nonparametric kernel adjusted density estimation and regression. In: Ferger, D., et al. (eds.): From Statistics to Mathematical Finance, Festschrift in Honour of Winfried Stute. Springer International Publishing. To appear in Oct. 2017.

Installation

You can install kader from CRAN with:

install.packages("kader")

or from github with:

devtools::install_github("GerritEichner/kader")

Example

This example shows you how to estimate at x₀ = 2 the value of the density function of the probability distribution underlying Old-Faithful's eruptions data using the (nonrobust) method of Srihera & Stute (2011). The initial grid (given to Sigma) on which the minimization of the estimated MSE as a function of a (kernel-adjusting) scale parameter σ is started is rather coarse here to save computing time.

library(kader)
x0 <- 2
sigma <- seq(0.01, 10, length = 21)
fit <- kade(x = x0, data = faithful$eruptions, method = "nonrobust",
  Sigma = sigma, ticker = TRUE)
#> h set to n^(-1/5) with n = 272.
#> theta set to arithmetic mean of data in faithful$eruptions.
#> Using the adaptive method of Srihera & Stute (2011)
#> For each element in x: Computing estimated values of
#> bias and scaled variance on the sigma-grid.
#> Note: x has 1 element(s) and the sigma-grid 21.
#> 
#> As a little distraction, the 'ticker' documents the
#> computational progress (if you have set ticker = TRUE).
#> x[1]:sigma:1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21.
#> Minimizing MSEHat:
#> Step 1: Search smallest maximizer of VarHat.scaled on sigma-grid.
#> Step 2: Search smallest minimizer of MSEHat on sigma-grid to the
#>         LEFT of just found smallest maximizer of VarHat.scaled.
#> Step 3: Finer search for 'true' minimum of MSEHat using
#>         numerical minimization. (May take a while.)
#> sigma:1.sigma:1.sigma:1.sigma:1.sigma:1.sigma:1.sigma:1.sigma:1.sigma:1.sigma:1.sigma:1.sigma:1.sigma:1.
#> Step 4: Check if numerically determined minimum is smaller
#>         than discrete one.
#>         Yes, optimize() was 'better' than grid search.
#> 
#> .
print(fit)
#>   x         y sigma.adap  msehat.min discr.min.smaller sig.range.adj
#> 1 2 0.5478784   1.996629 0.003822793             FALSE             0