epanechnikov

The Epanechnikov kernel is a kernel function used in kernel density estimation and related nonparametric methods. It is defined by K(u) = (3/4)(1 − u^2) for |u| ≤ 1, and K(u) = 0 for |u| > 1. The kernel is symmetric, nonnegative, and has compact support on the interval [−1, 1]. Its integral over the real line equals 1, making it a valid probability density function on the line and a suitable kernel for smoothing.

In kernel density estimation, the density at a point x is estimated by f_hat(x) = (1/(n h)) ∑_{i=1}^n K((x − X_i)/h), where h is the bandwidth. The Epanechnikov kernel’s finite support leads to computational efficiency and can reduce boundary bias relative to kernels with broader support. It is also valued for its statistical efficiency properties.

A key theoretical result is that the Epanechnikov kernel minimizes the asymptotic mean integrated squared error (AMISE) among a broad class of kernels with finite support, making it asymptotically optimal for density estimation under standard smoothness assumptions. It is named after the Russian mathematician Vasiliy D. Epanechnikov, who introduced it in the context of nonparametric density estimation in 1969.

Compared with other kernels, the Epanechnikov kernel is not the only choice; Gaussian kernels have infinite support and different smoothing characteristics, while higher-order kernels such as the biweight (quartic) kernel offer alternative bias/variance trade-offs. The Epanechnikov kernel remains popular for its balance of simplicity, efficiency, and theoretical optimality.