bias_AND_scaledvar function

Estimators of Bias and Scaled Variance

``Workhorse'' function for vectorized (in $\sigma$) computation of both the bias estimator and the scaled variance estimator of eq. (2.3) in Srihera & Stute (2011), and for the analogous computation of the bias and scaled variance estimator for the rank transformation method in the paragraph after eq. (6) in Eichner & Stute (2013).

bias_AND_scaledvar(sigma, Ai, Bj, h, K, fnx, ticker = FALSE)

Arguments

• sigma: Numeric vector $(\sigma_1, \ldots, \sigma_s)$ with $s \ge 1$.
• Ai: Numeric vector expecting $(x_0 - X_1, \ldots, x_0 - X_n) / h$, where (usually) $x_0$ is the point at which the density is to be estimated for the data $X_1, \ldots, X_n$ with $h = n^{-1/5}$.
• Bj: Numeric vector expecting $(-J(1/n), \ldots, -J(n/n))$ in case of the rank transformation method, but c("$(\\hat{\\theta} - X_1,\n$", "$\\ldots, \\hat{\\theta} - X_n)$") in case of the non-robust Srihera-Stute-method. (Note that this the same as argument Bj of adaptive_fnhat!)
• h: Numeric scalar, where (usually) $h = n^{-1/5}$.
• K: Kernel function with vectorized in- & output.
• fnx: $f_n(x_0) =$ mean(K(Ai))/h, where here typically $h = n^{-1/5}$.
• ticker: Logical; determines if a 'ticker' documents the iteration progress through sigma. Defaults to FALSE.

Returns

A list with components BiasHat and VarHat.scaled, both numeric vectors of same length as sigma.

Details

Pre-computed $f_n(x_0)$ is expected for efficiency reasons (and is currently prepared in function adaptive_fnhat).

Examples

require(stats)

set.seed(2017);     n <- 100;     Xdata <- sort(rnorm(n))
x0 <- 1;      Sigma <- seq(0.01, 10, length = 21)

h <- n^(-1/5)
Ai <- (x0 - Xdata)/h
fnx0 <- mean(dnorm(Ai)) / h   # Parzen-Rosenblatt estimator at x0.

 # non-robust method:
Bj <- mean(Xdata) - Xdata
# # rank transformation-based method (requires sorted data):
# Bj <- -J_admissible(1:n / n)   # rank trafo

kader:::bias_AND_scaledvar(sigma = Sigma, Ai = Ai, Bj = Bj, h = h,
  K = dnorm, fnx = fnx0, ticker = TRUE)

References

Srihera & Stute (2011) and Eichner & Stute (2013): see kader .