Create a smooth density estimate for some data:

Compute probabilities from the distribution:

Increase the bandwidth for smoother estimates:

Allow the bandwidth to vary adaptively with local density:

Identify features in data to aid in parametric model fitting:

The estimate suggests both the form and starting values for maximum likelihood estimation:

Use kernel density estimation in higher dimensions:

A four-dimensional kernel density estimate:

Sample from the distribution:

Explore properties of kernel density estimators using custom kernel functions:

Specify radial or product type kernels for multivariate estimates:

Estimate distribution functions:

The first few terms of the PDF and CDF:

Compute moments of the distribution:

Special moments:

General moments:

Moments can often be computed in closed form:

Compute a closed form expression for the variance with a symbolic adaptive bandwidth:

Quantile function:

Special quantile values:

Generate random numbers:

Compute probabilities and expectations:

Generating functions:

Estimate bivariate distribution functions:

Compute moments of a bivariate distribution:

Special moments:

General moments:

Generate random numbers:

Automatically select the bandwidth to use:

More data yields better approximations to the underlying distribution:

Explicitly specify the bandwidth to use:

Use bandwidths of

and

:

Larger bandwidths yield smoother estimates:

The bandwidth need not be numeric:

The PDF and CDF of the estimate:

Specify bandwidths in units of standard deviation:

Use bandwidths of

and

of the standard deviation:

Allow the bandwidth to vary adaptively with local density:

Vary the local sensitivity from

(none) to

(full):

Setting the sensitivity to

Automatic uses

where

is the dimension of the data:

The PDFs are equivalent:

Vary the initial bandwidth for an adaptive estimate:

Specify an initial bandwidth of

and

, respectively:

Use any of several automatic bandwidth selection methods:

Silverman's method is used by default:

The PDFs are equivalent:

In the multivariate case, the bandwidth is a symmetric positive definite

×

matrix:

Giving a scalar

h effectively uses

h IdentityMatrix[p]:

Specifying diagonal elements

effectively uses

DiagonalMatrix[d]:

Any

×

matrix that could be symmetric positive definite can be given:

By default, Silverman's method is used to independently select bandwidths in each dimension:

Any automated method can be used to independently select diagonal bandwidth elements:

Methods used to estimate the diagonal need not be the same:

Use adaptive, oversmoothed, and constant bandwidths in the respective dimensions:

Plot the univariate marginal PDFs:

Give a scalar value to use the same bandwidth in all dimensions:

To use nonzero off-diagonal elements, give a fully specified bandwidth matrix:

The bandwidth matrix controls the variance and orientation of individual kernels:

Scalar bandwidths:

Dimension-wise bandwidths:

Fully specified bandwidth matrices:

Some named bandwidth methods follow a rule-of-thumb approach:

Formulas for some named bandwidth methods:

The estimates are equivalent:

The method of least squares cross-validation:

The expectation of the PDF using a Gaussian kernel and bandwidth

:

The expectation of the PDF of the leave-one-out density estimator:

The bandwidth is found by minimizing the least squares cross-validation function over

:

The method of Sheather and Jones uses a plug-in estimator to solve for the bandwidth:

The Sheather and Jones estimator:

The estimates are equivalent:

Specify any one of several kernel functions:

Define the kernel function as a pure function:

By default, the Gaussian kernel is used:

This is equivalent to using the PDF of a

NormalDistribution:

Shapes of some univariate kernel functions:

Specify any one of several kernel functions for multivariate data:

Shapes of some bivariate product kernels:

Choose between product and radial-type kernel functions for multivariate data:

Computation of a single biweight kernel in two dimensions:

The radial version:

Bandwidths have similar effect for both radial and product type kernels:

Scalar bandwidths stretch the kernel equally in each dimension:

Diagonal elements stretch the kernel independently along each axis:

Nonzero off-diagonal elements change the orientation:

The PDFs of the various kernel functions:

The efficiency of kernels under the assumption of normally distributed data:

The built-in kernel functions all have relatively high statistical efficiency: