nadwat function

The Classical Nadaraya-Watson Regression Estimator

In its arguments x and dataX vectorized function to compute the classical Nadaraya-Watson estimator (as it is $m_n$ in eq. (1.1) in Eichner & Stute (2012)).

nadwat(x, dataX, dataY, K, h)

Arguments

• x: Numeric vector with the location(s) at which the Nadaraya-Watson regression estimator is to be computed.
• dataX: Numeric vector $(X_1, \ldots, X_n)$ of the x-values from which (together with the pertaining y-values) the estimate is to be computed.
• dataY: Numeric vector $(Y_1, \ldots, Y_n)$ of the y-values from which (together with the pertaining x-values) the estimate is to be computed.
• K: A kernel function (with vectorized in- & output) to be used for the estimator.
• h: Numeric scalar for bandwidth $h$.

Returns

A numeric vector of the same length as x.

Details

Implementation of the classical Nadaraya-Watson estimator as in eq. (1.1) in Eichner & Stute (2012) at given location(s) in x for data c("$(X_1,\n$", "$Y_1), \\ldots, (X_n, Y_n)$"), a kernel function $K$ and a bandwidth $h$.

Examples

require(stats)

 # Regression function: a polynomial of degree 4 with one maximum (or
 # minimum), one point of inflection, and one saddle point.
 # Memo: for p(x) = a * (x - x1) * (x - x2)^3 + b the max. (or min.)
 # is at x = (3*x1 + x2)/4, the point of inflection is at x =
 # (x1 + x2)/2, and the saddle point at x = x2.
m <- function(x, x1 = 0, x2 = 8, a = 0.01, b = 0) {
 a * (x - x1) * (x - x2)^3 + b
 }
 # Note: for m()'s default values a minimum is at x = 2, a point
 # of inflection at x = 4, and a saddle point at x = 8.

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

x <- seq(-3, 15, length = 51)   # Where the Nadaraya-Watson estimator
                                # mn of m shall be computed.
mn <- nadwat(x = x, dataX = X, dataY = Y, K = dnorm, h = n^(-1/5))

plot(x = X, y = Y);   rug(X)
lines(x = x, y = mn, col = "blue")  # The estimator.
curve(m, add = TRUE, col = "red")   # The "truth".