minimize_MSEHat: Minimization of Estimated MSE

In kader: Kernel Adaptive Density Estimation and Regression

Description

Minimization of the estimated MSE as function of σ in four steps.

Usage

minimize_MSEHat(VarHat.scaled, BiasHat.squared, sigma, Ai, Bj, h, K, fnx,
  ticker = FALSE, plot = FALSE, ...)

Arguments

VarHat.scaled	Vector of estimates of the scaled variance (for values of σ in sigma).
BiasHat.squared	Vector of estimates of the squared bias (for values of σ in sigma).
sigma	Numeric vector (σ_1, …, σ_s) with s ≥ 1.
Ai	Numeric vector expecting (x_0 - X_1, …, 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, …, X_n with h = n^{-1/5}.
Bj	Numeric vector expecting (-J(1/n), …, -J(n/n)) in case of the rank transformation method, but (\hat{θ} - X_1, …, \hat{θ} - 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.
plot	Should graphical output be produced? Defaults to FALSE.
...	Currently ignored.

Details

Step 1: determine first (= smallest) maximizer of VarHat.scaled (!) on the grid in sigma. Step 2: determine first (= smallest) minimizer of estimated MSE on the σ-grid LEFT OF the first maximizer of VarHat.scaled. Step 3: determine a range around the yet-found (discrete) minimizer of estimated MSE within which a finer search for the “true” minimum is continued using numerical minimization. Step 4: check if the numerically determined minimum is indeed better, i.e., smaller than the discrete one; if not keep the first.

Value

A list with components sigma.adap, msehat.min and discr.min.smaller whose meanings are as follows:

sigma.adap	Found minimizer of MSE estimator.
msehat.min	Found minimum of MSE estimator.
discr.min.smaller	TRUE iff the numerically found minimum was smaller than the discrete one.

Examples

require(stats)

set.seed(2017);     n <- 100;     Xdata <- sort(rnorm(n))
x0 <- 1;      Sigma <- seq(0.01, 10, length = 11)

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

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

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

kader:::minimize_MSEHat(VarHat.scaled = BV$VarHat.scaled,
  BiasHat.squared = (BV$BiasHat)^2, sigma = Sigma, Ai = Ai, Bj = Bj,
  h = h, K = dnorm, fnx = fnx0, ticker = TRUE, plot = FALSE)

kader documentation built on May 1, 2019, 10:13 p.m.