Density Estimation

Synonyms

Kernel density estimation

Definition

Given a set of observations, x ₁, …, x _N , which is a random sample from a probability density function f _X x, density estimation attempts to approximate f _X x by $\widehat{{f}}_{X}\left ({x}_{0}\right )$.

A simple way of estimating a probability density function is to plot a histogram from a random sample drawn from the population. Usually, the range of data values is subdivided into equally sized intervals or bins. How well the histogram estimates the function depends on the bin width and the placement of the boundaries of the bins. The latter can be somewhat improved by modifying the histogram so that fixed boundaries are not used for the estimate. That is, the estimate of the probability density function at a point uses that point as the centre of a neighborhood. Following Hastie, Tibshirani and Friedman (2009), the estimate can be expressed as:

$$\widehat{{f}}_{X}\left ({x}_{0}\right ) = \frac{\#{x}_{i} \in N\left ({x}_{0}\right )}...