# What is kernel in the kernel density estimate

## If the Epanechnikov kernel is theoretically optimal at kernel density estimation, why isn't it used more often?

The reason the Epanechnikov kernel is not widely used because of its theoretical optimality may well be that the Epanechnikov kernel is theoretically not optimal . Tsybakov explicitly criticizes the argument that the Epanechnikov kernel is "theoretically optimal" on pages 16-19 of the Introduction to nonparametric estimation (Section 1.2.4).

The attempt, under some assumptions about the kernel K and a fixed density p, takes the form of the mean integrated square error

1n h∫K2 (u) du + h44S2K∫ (p '' (x)) 2dx. (1)

The main criticism of Tsybakov seems to be the minimization versus non-negative kernels, as it is often possible to get more powerful estimators that are even non-negative without limiting yourself to non-negative kernels.

The first step of the argument for the Epanechnikov kernel begins with minimizing (1) over h and all non-negative kernels (and not all kernels of a broader class) to get an "optimal" bandwidth for K

hMichSE (K) = (∫K2n S2K∫ (p '') 2) 1/5

and the "optimal" kernel (Epanechnikov)

K ∗ (u) = 34 (1 - u2) +

whose mean integrated square error is:

hMichSE (K ∗) = (15n ∫ (p '') 2) 1/5.

However, these are not feasible decisions since they depend on the knowledge (about p ″) of the unknown density p - therefore they are "oracle" qualities.

A suggestion from Tsybakov implies that the asymptotic MISE for the Epanechnikov oracle is:

limn → ∞n4 / 5Ep∫ (pEn (x) - p (x)) 2dx = 34/551/54 (∫ (p '' (x)) 2dx) 1 / 5. (2)

SK = 0ε> 0

lim supn → ∞n4 / 5Ep∫ (p ^ n (x) −p (x)) 2dx≤ε.

Even if p n is not necessarily not negative, one still has the same result for the positive part of the estimator, p + p ^ np + n: = max (0,