## (Convolution) kernel density estimation

**dan mackinlay**

A nonparametric method of approximating something from data

by assuming that it’s close to the data distribution

convolved with some kernel.

This is especially popular the target is a probability density function;

Then you are working with a kernel density estimator.

## Bandwidth/kernel selection in density estimation

Bernacchia (BePi11) has a neat hack:

“self consistency” for simultaneous

kernel and distribution inference,

i.e. simultaneous deconvolution and

bandwidth selection.

The idea is removing bias by using simple spectral methods,

thereby estimating a kernel which in a certain sense would generate the data that you just observed.

The results look similar to finite-sample corrections for Gaussian scale parameter estimates, but are not quite Gaussian.

Question: could it work with mixture models too?

## Mixture models

Where the number of kernels does *not* grow as fast as the number of data points,

this becomes a mixture model; Or, if you’d like, kernel density estimates are a limiting case of mixture model estimates,

They are so clearly similar that I think it best we not make them both

feel awkward by dithering about where the free parameters are.

Anyway, they are filed separately.

BaLi13, ZeMe97 and Geer96 discuss some useful common to various convex combination estimators.

## Does this work with uncertain point locations?

The fact we can write the kernel density estimate as an integral with

a convolution of Dirac deltas immediately suggests

that we could write it as a convolution of something else, such as Gaussians.

Can we recover well-behaved estimates in that case?

This would be a kind of hierarchical model, possibly a very normal Bayesian one.

## Does this work with asymmetric kernels?

Almost all the kernel estimates I’ve seen require KDEs to be symmetric, because of

Daren Cline’s argument that asymmetric kernels are inadmissible

in a decision-theoretic context in the class of all

(possibly multivariate) densities.

Presumably this implies \(\mathcal(C)_1\) distributions,

i.e. once-differentiable ones, without atoms.

In particular admissible kernels are those

which have “nonnegative Fourier transforms bounded by 1”,

which implies symmetry about the axis.

If we have a constrained class of densities, this might not apply.

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