# (Convolution) kernel function estimation

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. But you can estimate anything.

• “Effective local sample size”
• understand the Frechet derivative + Wiener filtering construction used to derive the optimal kernel shape in BePi11 and OKCC16.

## Estimating Densities

What you are usually doing with this tool.

### 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 things 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 typical Bayesian one.

### Does this work with asymmetric kernels?

Almost all the kernel estimates I’ve seen require KDEs to be symmetric, because of Cline’s argument that asymmetric kernels are inadmissible in the class of all (possibly multivariate) densities. Presumably this implies $\mathcal(C)_1$ distributions, i.e. once-differentiable ones. In particular admissible kernels are those which have “nonnegative Fourier transforms bounded by 1”, which implies symmetry about the axis. If we have an a priori constrained class of densities, this might not apply.

## Estimating general functions

i.e. not densities. Keyword: Nadarya-Watson estimator.

For regression this is generally not incredibly popular, but sometimes it is what you need - e.g. conditional intensity point processes, some DSP resampling problems. You can try more sophisticated things like loess or smoothing splines. I won’t stop you. Try using the phrase Sobolev space while you do so, you’ll feel ever so clever. Regardles, at this point it becomes a spline basis method, so read on over there for some extra details.

### Connection to rate estimation

The Ogata special case of the Lamperti transform gives a suggestive connection to rate estimation as a kind of inverse-method kernel estimation. See change-of-time

### Fast Gauss Transform and Fast multipole methods

How to make these methods computationally feasible at scale. See Fast Gauss Transform and other related fast multipole methods.

## Refs

BaHy01
Bashtannyk, D. M., & Hyndman, R. J.(2001) Bandwidth selection for kernel conditional density estimation. Computational Statistics & Data Analysis, 36(3), 279–298. DOI.
BaLi13
Battey, H., & Liu, H. (2013) Smooth projected density estimation. arXiv:1308.3968 [Stat].
BePi11
Bernacchia, A., & Pigolotti, S. (2011) Self-consistent method for density estimation. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(3), 407–422.
BoGK10
Botev, Z. I., Grotowski, J. F., & Kroese, D. P.(2010) Kernel density estimation via diffusion. The Annals of Statistics, 38(5), 2916–2957. DOI.
Clin88
Cline, D. B. H.(1988) Admissible Kernel Estimators of a Multivariate Density. The Annals of Statistics, 16(4), 1421–1427. DOI.
CrMí14
Crisan, D., & Míguez, J. (2014) Particle-kernel estimation of the filter density in state-space models. Bernoulli, 20(4), 1879–1929. DOI.
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Doosti, H., & Hall, P. (2015) Making a non-parametric density estimator more attractive, and more accurate, by data perturbation. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(2), 445–462. DOI.
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Ellis, S. P.(1991) Density estimation for point processes. Stochastic Processes and Their Applications, 39(2), 345–358. DOI.
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Geenens, G. (2014) Probit Transformation for Kernel Density Estimation on the Unit Interval. Journal of the American Statistical Association, 109(505), 346–358. DOI.
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Hall, P., & Park, B. U.(2002) New Methods for Bias Correction at Endpoints and Boundaries. The Annals of Statistics, 30(5), 1460–1479. DOI.
Ibra01
Ibragimov, I. (2001) Estimation of analytic functions. In Institute of Mathematical Statistics Lecture Notes - Monograph Series (pp. 359–383). Beachwood, OH: Institute of Mathematical Statistics
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Koenker, R., & Mizera, I. (2006) Density estimation by total variation regularization. Advances in Statistical Modeling and Inference, 613–634.
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OKCC16
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Park, B. U., Jeong, S.-O., Jones, M. C., & Kang, K.-H. (2003) Adaptive variable location kernel density estimators with good performance at boundaries. Journal of Nonparametric Statistics, 15(1), 61–75. DOI.
Silv82
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SmLe05
Smith, E., & Lewicki, M. S.(2005) Efficient Coding of Time-Relative Structure Using Spikes. Neural Computation, 17(1), 19–45. DOI.
Geer96
van de Geer, S. (1996) Rates of convergence for the maximum likelihood estimator in mixture models. Journal of Nonparametric Statistics, 6(4), 293–310. DOI.
WaWa07
Wang, B., & Wang, X. (2007) Bandwidth selection for weighted kernel density estimation. arXiv Preprint arXiv:0709.1616.
WeWu15
Wen, K., & Wu, X. (2015) An Improved Transformation-Based Kernel Estimator of Densities on the Unit Interval. Journal of the American Statistical Association, 110(510), 773–783. DOI.
ZeMe97
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ZhKa10
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