fnhat_SS2011() R function from [kader]

(Non-robust) Kernel Density Estimator of Srihera & Stute (2011)

Implementation of eq. (1.6) in Srihera & Stute (2011) for given and fixed scalars $\sigma$ and $\theta$ (and, of course, for fixed and given location(s) in $x$ , data $(X_1, \ldots, X_n)$ , a kernel function $K$ and a bandwidth $h$ ).


fnhat_SS2011(x, data, K, h, theta, 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 or infinite values are not allowed and entail an error.
K: A kernel function to be used for the estimator.
h: Numeric scalar for bandwidth $h$ .
theta: Numeric scalar for value of location parameter $\theta$ .
sigma: Numeric scalar for value of scale parameter $\sigma$ .

Returns

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. (1.6) in Srihera & Stute (2011).
`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:
`theta`	as in Arguments.
`sigma`	as in Arguments.

Details

The formula upon which the computational version implemented here is based is given in eq. (15.3) of Eichner (2017). This function does mainly only a simple preparatory computation and then calls compute_fnhat

which does the actual work.

Examples


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

 # Simulated N(0,1)-data and one sigma-value
set.seed(2017);     n <- 100;     d <- rnorm(n)
xgrid <- seq(-4, 4, by = 0.1)
(fit <- fnhat_SS2011(x = xgrid, data = d, K = dnorm, h = n^(-1/5),
  theta = mean(d), 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(tilde(f)[n], phi, "data")) 

 # The same data, but several sigma-values
sigmas <- seq(1, 4, length = 4)
(fit <- lapply(sigmas, function(sig)
  fnhat_SS2011(x = xgrid, data = d, K = dnorm, h = n^(-1/5),
    theta = mean(d), sigma = sig)))

ymat <- sapply(fit, "[[", "y")
matplot(x = xgrid, y = ymat, type = "l", lty = 1, col = 3:6,
  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", tilde(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_SS2011(x = xgrid, data = d, K = dnorm, h = n^(-1/5),
     theta = mean(d), 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 = 3:6)
legend("top", lty = 1, col = c("black", "red", NA), bty = "n",
  legend = expression("R's est.", "data", tilde(f)[n]~"in other colors"))

References

Srihera & Stute (2011) and Eichner (2017): see kader .

fnhat_SS2011 function