bias_ES2012: Bias Estimator of Eichner & Stute (2012)

Description

Bias estimator $B i a s_{n} (σ)$ , vectorized in $σ$ , on p. 2540 of Eichner & Stute (2012).

Usage

bias_ES2012(sigma, h, xXh, thetaXh, K, mmDiff)

Arguments

sigma

Numeric vector $(σ_{1}, \dots, σ_{s})$ with $s \geq 1$ with values of the scale parameter $σ$ .

Numeric scalar for bandwidth $h$ (as ``contained'' in thetaXh and xXh).

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.

A kernel function (with vectorized in- & output) to be used for the estimator.

mmDiff

Numeric vector expecting the pre-computed differences $m_{n} (X_{1}) - m_{n} (x), \dots, m_{n} (X_{n}) - m_{n} (x)$ .

Value

A numeric vector of the length of sigma.

Details

The formula can also be found in eq. (15.21) of Eichner (2017). Pre-computed $(x - X_{i}) / h$ , $(θ - X_{i}) / h$ , and $m_{n} (X_{i}) - m_{n} (x)$ are expected for efficiency reasons (and are currently prepared in function kare).

References

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

Examples

Run this code

# NOT RUN {
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.

 # m_n(x_0) and m_n(X_i) for i = 1, ..., n:
mn <- nadwat(x = c(x0, X), dataX = X, dataY = Y, K = dnorm, h = h)

 # Estimator of Bias_x0(sigma) on the sigma-grid:
(Bn <- bias_ES2012(sigma = Sigma, h = h, xXh = (x0 - X) / h,
  thetaXh = (mean(X) - X) / h, K = dnorm, mmDiff = mn[-1] - mn[1]))

# }
# NOT RUN {
 # Visualizing the estimator of Bias_n(sigma) at x on the sigma-grid:
plot(Sigma, Bn, type = "o", xlab = expression(sigma), ylab = "",
  main = bquote(widehat("Bias")[n](sigma)~~"at"~~x==.(x0)))
# }
# NOT RUN {
# }

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