Definition:Kernel Density Estimation

Definition

Kernel density estimation is a nonparametric method for estimating a probability density function of a continuous random variable $X$ based on data forming a (usually large) sample.

Kernel

The kernel of a probability density function of a continuous random variable $X$ is a probability function $\map k u$ symmetric about $u = 0$.

Bandwidth

Let there be $n$ data points of a probability density function $f$:

$x_1, x_2, \ldots, x_n$

For each $x_i$ we can substitute:

$u = \dfrac {x - x_i} h$

to make a function of $x$.

In this context, $h$ is referred to as the bandwidth of $f$.

Hence the estimated probabiity density function is given by:

$\ds \map f x = \dfrac 1 {n h} \sum_{i \mathop = 1}^n \map k {\dfrac {x - x_i} h}$

Also see

• Definition:Lowess

• Results about kernel density estimation can be found here.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): kernel density estimation