UniformKern

UniformKern is a simple kernel function used for smoothing in nonparametric statistics and signal processing. It is defined as a uniform (rectangular) kernel with bandwidth h > 0: K_h(x) = 1/(2h) for |x| ≤ h, and 0 otherwise. In multiple dimensions, the uniform kernel on R^d corresponds to a rectangular hypercube support, often defined as K_h(x) = ∏_{i=1}^d 1/(2h_i) for |x_i| ≤ h_i.

Properties of UniformKern include compact support, constant weight across the window, and computational simplicity. Because the kernel is flat within the window and zero outside, smoothing effects are easy to interpret: the estimate at a point is the average of observed values within a window of width 2h. However, its discontinuities at the window boundary can induce less smooth estimates compared with kernels such as the Gaussian or Epanechnikov, and edge effects may be more pronounced near the boundaries of the domain.

Applications of UniformKern appear in kernel density estimation, where the density at a point is estimated by counting data within a window and normalizing by n h. It is also used in kernel regression (local averaging) and in image or signal smoothing through discrete convolution with a rectangular kernel. In higher dimensions, the separable form allows efficient computation via product windows, and cumulative-sum techniques can accelerate counting within moving hypercubes.

Compared with other kernels, UniformKern is often used as a baseline due to its simplicity. Choice of bandwidth h remains critical, balancing bias and variance; cross-validation or plug-in methods are commonly employed to select it.