fnhat_ES2013 {kader}

Robust Kernel Density Estimator of Eichner & Stute (2013)

Description

Implementation of eq. (4) in Eichner & Stute (2013) for a given and fixed scalar \sigma, for rank transformation function J (and, of course, for fixed and given location(s) in x, data (X_1, \ldots, X_n), a kernel function K, and a bandwidth h).

Usage

fnhat_ES2013(x, data, K, h, ranktrafo, sigma)

Arguments

x

Numeric vector with the location(s) at which the density estimate is to be computed.

data

Numeric vector (X_1, \ldots, X_n) of the data from which the estimate is to be computed. Missing values are not allowed and entail an error.

K

A kernel function to be used for the estimator.

h

Numeric scalar for bandwidth h.

ranktrafo

A function used for the rank transformation.

sigma

Numeric scalar for value of scale parameter \sigma.

Details

The formula upon which the computational version implemented here is based is given in eq. (15.9) of Eichner (2017). This function does mainly only a simple preparatory computation and then calls compute_fnhat which does the actual work.

Value

An object with class "density" whose underlying structure is a list containing the following components (as described in density), so that the print and plot methods for density-objects are immediately available):

x	the n coordinates of the points where the density is estimated.
y	the estimated density values from eq. (4) in Eichner & Stute (2013).
bw	the bandwidth used.
n	the sample size. (Recall: missing or infinite values are not allowed here.)
call	the call which produced the result.
data.name	the deparsed name of the x argument.
has.na	logical, for compatibility (always FALSE).
Additionally:
ranktrafo	as in Arguments.
sigma	as in Arguments.

References

Eichner & Stute (2013) and Eichner (2017): see kader.

See Also

fnhat_SS2011.

Examples

require(stats);   require(grDevices);   require(datasets)

 # Simulated N(0,1)-data and one sigma-value
set.seed(2016);     n <- 100;     d <- rnorm(n)
xgrid <- seq(-4, 4, by = 0.1)
(fit <- fnhat_ES2013(x = xgrid, data = d, K = dnorm, h = n^(-1/5),
  ranktrafo = J2, sigma = 1) )

plot(fit, ylim = range(0, dnorm(0), fit$y), col = "blue")
curve(dnorm, add = TRUE);   rug(d, col = "red")
legend("topleft", lty = 1, col = c("blue", "black", "red"),
  legend = expression(hat(f)[n], phi, "data"))


 # The same data, but several sigma-values
sigmas <- seq(1, 4, length = 4)
(fit <- lapply(sigmas, function(sig)
  fnhat_ES2013(x = xgrid, data = d, K = dnorm, h = n^(-1/5),
    ranktrafo = J2, sigma = sig)) )

ymat <- sapply(fit, "[[", "y")
matplot(x = xgrid, y = ymat, type = "l", lty = 1, col = 2 + seq(sigmas),
  ylim = range(0, dnorm(0), ymat), main = "", xlab = "", ylab = "Density")
curve(dnorm, add = TRUE);   rug(d, col = "red")
legend("topleft", lty = 1, col = c("black", "red", NA), bty = "n",
  legend = expression(phi, "data", hat(f)[n]~"in other colors"))


 # Old-Faithful-eruptions-data and several sigma-values
d <- faithful$eruptions;     n <- length(d);     er <- extendrange(d)
xgrid <- seq(er[1], er[2], by = 0.1);    sigmas <- seq(1, 4, length = 4)
(fit <- lapply(sigmas, function(sig)
   fnhat_ES2013(x = xgrid, data = d, K = dnorm, h = n^(-1/5),
     ranktrafo = J2, sigma = sig)) )

ymat <- sapply(fit, "[[", "y");     dfit <- density(d, bw = "sj")
plot(dfit, ylim = range(0, dfit$y, ymat), main = "", xlab = "")
rug(d, col = "red")
matlines(x = xgrid, y = ymat, lty = 1, col = 2 + seq(sigmas))
legend("top", lty = 1, col = c("black", "red", NA), bty = "n",
  legend = expression("R's est.", "data", hat(f)[n]~"in other colors"))