weights_ES2012: Weights W_{ni} of Eichner & Stute (2012)
In kader: Kernel Adaptive Density Estimation and Regression

Description Usage Arguments Details Value References See Also Examples

Function, vectorized in its first argument sigma, to compute the “updated” weights W_{ni} in eq. (2.1) of Eichner & Stute (2012) for the kernel adjusted regression estimator.

1	weights_ES2012(sigma, xXh, thetaXh, K, h)

`sigma`	Numeric vector (σ_1, …, σ_s) with s ≥ 1 with values of the scale parameter σ.
`xXh`	Numeric vector expecting the pre-computed h-scaled differences (x - X_1)/h, ..., (x - X_n)/h where x is the single (!) location for which the weights are to be computed, the X_i's are the data values, and h is the numeric bandwidth scalar.
`thetaXh`	Numeric vector expecting the pre-computed h-scaled differences (θ - X_1)/h, ..., (θ - X_n)/h where θ is the numeric scalar location parameter, and the X_i's and h are as in `xXh`.
`K`	A kernel function (with vectorized in- & output) to be used for the estimator.
`h`	Numeric scalar for bandwidth h (as “contained” in `thetaXh` and `xXh`).

Note that it is not immediately obvious that W_{ni} in eq. (2.1) of Eichner & Stute (2012) is a function of σ. In fact, W_{ni} = W_{ni}(x; h, θ, σ) as can be seen on p. 2542 ibid. The computational version implemented here, however, is given in (15.19) of Eichner (2017). Pre-computed (x - X_i)/h and (θ - X_i)/h, i = 1, …, n are expected for efficiency reasons (and are currently prepared in function kare).

If length(sigma) > 1 a numeric matrix of the dimension length(sigma) by length(xXh) with elements (W_{ni}(x; h, θ, σ_r)) for r = 1, …, length(sigma) and i = 1, …, length(xXh); otherwise a numeric vector of the same length as xXh.

Eichner & Stute (2012) and Eichner (2017): see kader.

bias_ES2012 and var_ES2012 which both call this function, and kare which currently does the pre-computing.

require(stats)

 # Regression function:
m <- function(x, x1 = 0, x2 = 8, a = 0.01, b = 0) {
 a * (x - x1) * (x - x2)^3 + b
}
 # Note: For a few details on m() see examples in ?nadwat.

n <- 100       # Sample size.
set.seed(42)   # To guarantee reproducibility.
X <- runif(n, min = -3, max = 15)      # X_1, ..., X_n   # Design.
Y <- m(X) + rnorm(length(X), sd = 5)   # Y_1, ..., Y_n   # Response.

h <- n^(-1/5)
Sigma <- seq(0.01, 10, length = 51)   # sigma-grid for minimization.
x0 <- 5   # Location at which the estimator of m should be computed.

 # Weights (W_{ni}(x; \sigma_r))_{1<=r<=length(Sigma), 1<=i<=n} for
 # Var_n(sigma) and Bias_n(sigma) each at x0 on the sigma-grid:
weights_ES2012(sigma = Sigma, xXh = (x0 - X) / h,
  thetaXh = (mean(X) - X) / h, K = dnorm, h = h)