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Kernel density estimation of 100 normally distributed random numbers using different smoothing bandwidths.

In statistics, kernel density estimation (or Parzen window method, named after Emanuel Parzen) is a way of estimating the probability density function of a random variable. As an illustration, given some data about a sample of a population, kernel density estimation makes it possible to extrapolate the data to the entire population.

A histogram can be thought of as a collection of point samples from a kernel density estimate for which the kernel is a uniform box the width of the histogram bin.

If x₁, x₂, ..., x_N ~ ƒ is an IID sample of a random variable, then the kernel density approximation of its probability density function is

where K is some kernel and h is the bandwidth (smoothing parameter). Quite often K is taken to be a standard Gaussian function with mean zero and variance 1:

Although less smooth density estimators such as the histogram density estimator can be made to be asymptotically consistent, others are often either discontinuous or converge at slower rates than the kernel density estimator. Rather than grouping observations together in bins, the kernel density estimator can be thought to place small "bumps" at each observation, determined by the kernel function. The estimator consists of a "sum of bumps" and is clearly smoother as a result (see below image).

Six Gaussians (red) and their sum (blue). The Parzen window density estimate f(x) is obtained by dividing this sum by 6, the number of Gaussians. The variance of the Gaussians was set to 0.5. Note that where the points are denser the density estimate will have higher values.

Let be the L² risk function for ƒ. Under weak assumptions on ƒ and K,

By minimizing the theoretical risk function, it can be shown that the optimal bandwidth is

where

When the optimal choice of bandwidth is chosen, the risk function is for some constant c₄ > 0. It can be shown that, under weak assumptions, there cannot exist a non-parametric estimator that converges at a faster rate than the kernel estimator. Note that the n^−4/5 rate is slower than the typical n⁻¹ convergence rate of parametric methods.

• In Stata, it is implemented through kdensity; for example histogram x, kdensity.
• In R, it is implemented through the density function.

• kernel (statistics)
• kernel (mathematics)
• density estimation
• smoothing

• Parzen E. (1962). On estimation of a probability density function and mode, Ann. Math. Stat. 33, pp. 1065-1076.
• Duda, R. and Hart, P. (1973). Pattern Classification and Scene Analysis. John Wiley & Sons. ISBN 0-471-22361-1.
• Wasserman, L. (2005). All of Statistics: A Concise Course in Statistical Inference, Springer Texts in Statistics.

• Introduction to kernel density estimation
• Free Online Software (Calculator) computes the Kernel Density Estimation for any data series according to the following Kernels: Gaussian, Epanechnikov, Rectangular, Triangular, Biweight, Cosine, and Optcosine.