NW_boundary: Boundary-adjusted Nadaraya-Watson estimator

In svjaco/lpreba: Local Polynomial Regression with Option for Explicit Boundary Adjustment

Description

Explicit boundary adjustment via boundary kernels for the Nadaraya-Watson estimator (local constant regression).

Usage

NW_boundary(x, X, Y, bw,
            kernel_interior = epanechnikov, kernel_left = epanechnikov_left,
            boundary_left, boundary_right)

Arguments

x	Evaluation points (vector).
X	Data for the regressor (vector).
Y	Data for the regressand (vector).
bw	Bandwidth (scalar).
kernel_interior	Kernel used in the interior (function). Default is epanechnikov.
kernel_left	Left boundary kernels (function). Default is epanechnikov_left.
boundary_left	Lower boundary of the support of X (scalar).
boundary_right	Upper boundary of the support of X (scalar).

Details

When applying the Nadaraya-Watson estimator with a compactly supported (on the interval [-1, 1]) kernel to data of a regressor that is compactly supported, boundary effects arise over the regions [boundary_left, boundary_left + bw) and (boundary_right - bw, boundary_right]. That is, the bias is of larger order compared to the interior. The original order can be preserved by using special so-called boundary kernels when applying Nadaraya-Watson at the boundaries (e.g. Gasser and Müller, 1979).

“Smooth optimum boundary kernels” (Müller, 1991, Table 1) are available: uniform_left, epanechnikov_left, biweight_left, triweight_left. Moreover, an alternative version of these kernels with greater asymptotic efficiency at the cost of discontinuity at their endpoints (Müller and Wang, 1994, Table 1) is available: uniform_alt_left, epanechnikov_alt_left, biweight_alt_left, triweight_alt_left.

The effective kernel at an evaluation point is the set of effectively assigned weights in the smoothing process (see Hastie and Loader, 1993).

Value

List containing:

estimates	Estimates for the regression function at the evaluation points (vector).
effective_kernels	Effective kernels at the evaluation points (matrix).

References

Gasser, T. and H.-G. Müller (1979). “Kernel estimation of regression functions”. In: Smoothing Techniques for Curve Estimation. Ed. by T. Gasser and M. Rosenblatt. Berlin: Springer, pp. 23–68.

Müller, H.-G. (1991). “Smooth optimum kernel estimators near endpoints”. Biometrika 78 (3), pp. 521–530.

Müller, H.-G. and J.-L. Wang (1994). “Hazard rate estimation under random censoring with varying kernels and bandwidths”. Biometrics 50 (1), pp. 61–76.

Hastie, T. and C. Loader (1993). “Local regression: Automatic kernel carpentry”. Statistical Science 8 (2), pp. 120–129.

Examples

m_fun <- function(x) {sin(2*pi*x)} # True regression function
n <- 100 # Sample size
X <- seq(0, 1, length.out = n) # Data for the regressor
m_X <- m_fun(X) # True values of regression function
epsilon <- rnorm(n, sd = 0.25) # Error term
Y <- m_X + epsilon # Data for the regressand
bw <- 0.2 # Bandwidth
x <- seq(0, 1, length.out = n/2) # Evaluation points

output_NW_boundary <- NW_boundary(x = x, X = X, Y = Y, bw = bw,
                                  kernel_interior = epanechnikov, kernel_left = epanechnikov_left,
                                  boundary_left = 0, boundary_right = 1)
estimates_NW_boundary <- output_NW_boundary$estimates
effective_kernels_NW_boundary <- output_NW_boundary$effective_kernels

svjaco/lpreba documentation built on March 4, 2022, 12:42 a.m.