Description Usage Arguments Details Value References See Also Examples
Bias estimator Bias_n(σ), vectorized in σ, on p. 2540 of Eichner & Stute (2012).
1  bias_ES2012(sigma, h, xXh, thetaXh, K, mmDiff)

sigma 
Numeric vector (σ_1, …, σ_s) with s ≥ 1 with values of the scale parameter σ. 
h 
Numeric scalar for bandwidth h (as “contained” in

xXh 
Numeric vector expecting the precomputed hscaled 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 precomputed hscaled 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 
K 
A kernel function (with vectorized in & output) to be used for the estimator. 
mmDiff 
Numeric vector expecting the precomputed differences m_n(X_1)  m_n(x), …, m_n(X_n)  m_n(x). 
The formula can also be found in eq. (15.21) of Eichner (2017).
Precomputed (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
).
A numeric vector of the length of sigma
.
Eichner & Stute (2012) and Eichner (2017): see kader
.
kare
which currently does the precomputing.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  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) # sigmagrid 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 sigmagrid:
(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 sigmagrid:
plot(Sigma, Bn, type = "o", xlab = expression(sigma), ylab = "",
main = bquote(widehat("Bias")[n](sigma)~~"at"~~x==.(x0)))
## End(Not run)

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