kernelweighted

Kernel weighting is a technique used to assign weights to observations based on a kernel function of the distance between their predictor values, enabling localized nonparametric estimation. It is central to kernel smoothing methods and widely used across statistics and machine learning.

In kernel density estimation, the estimated density at a point x is f_hat(x) = (1/(n h)) sum_i K((x - X_i)/h). In kernel regression, the estimated regression function at x is m_hat(x) = sum_i K((x - X_i)/h) Y_i / sum_j K((x - X_j)/h). The weights reflect proximity in the predictor space, with closer observations contributing more to the estimate.

Kernel functions are nonnegative, symmetric, and typically integrate to one. Common choices include Gaussian, Epanechnikov, and uniform kernels. The bandwidth parameter h controls the scale of weighting and thus the smoothness of the estimate: larger h yields smoother results, smaller h yields more detail but higher variance. In multivariate settings, kernels can be extended via product forms or multivariate kernels.

Kernel weighting is applied in a variety of tasks beyond density estimation and regression, including locally weighted scatterplot smoothing, time series smoothing, and spatial statistics. It can be used with different loss functions and in conjunction with local polynomial methods to reduce boundary bias.

Practical considerations include bandwidth selection (via cross-validation or plug-in methods), edge effects near the boundary of the data, and computational cost for large datasets. While the kernel choice matters, bandwidth selection typically has a greater impact on estimator performance.

See also: kernel smoothing, kernel density estimation, Nadaraya-Watson estimator, local polynomial regression.