# Kernel density estimators

March 5, 2016 — August 18, 2016

convolution
nonparametric
statistics

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.

• “Effective local sample size”

• understand the Frechet derivative + Wiener filtering construction used to derive the optimal kernel shape in .

### 0.1 Bandwidth/kernel selection in density estimation

Bernacchia and Pigolotti (2011) 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?

### 0.2 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. discuss some useful things common to various convex combination estimators.

### 0.3 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.

### 0.4 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.

### 0.5 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.

## 1 References

Aalen. 1978. The Annals of Statistics.
Adelfio, and Schoenberg. 2009. Annals of the Institute of Statistical Mathematics.
Baddeley, Adrian, and Turner. 2006. In Case Studies in Spatial Point Process Modeling. Lecture Notes in Statistics 185.
Baddeley, A., Turner, Møller, et al. 2005. Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Barnes, and Hut. 1986. Nature.
Bashtannyk, and Hyndman. 2001. Computational Statistics & Data Analysis.
Battey, and Liu. 2013. arXiv:1308.3968 [Stat].
Berman, and Diggle. 1989. Journal of the Royal Statistical Society. Series B (Methodological).
Bernacchia, and Pigolotti. 2011. Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Botev, Grotowski, and Kroese. 2010. The Annals of Statistics.
Cline. 1988. The Annals of Statistics.
Crisan, and Míguez. 2014. Bernoulli.
Cucala. 2008. Scandinavian Journal of Statistics.
Díaz-Avalos, Juan, and Mateu. 2012. Stochastic Environmental Research and Risk Assessment.
Diggle, Peter J. 1979. Biometrics.
Diggle, Peter. 1985. Journal of the Royal Statistical Society. Series C (Applied Statistics).
Doosti, and Hall. 2015. Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Ellis. 1991. Stochastic Processes and Their Applications.
Geenens. 2014. Journal of the American Statistical Association.
Greengard, and Strain. 1991. SIAM Journal on Scientific and Statistical Computing.
Hall. 1987. The Annals of Statistics.
Hall, and Park. 2002. The Annals of Statistics.
Helmers, Wayan Mangku, and Zitikis. 2003. Journal of Multivariate Analysis.
Ho, and Walker. 2020. arXiv:2012.14482 [Math, Stat].
Ibragimov. 2001. In Institute of Mathematical Statistics Lecture Notes - Monograph Series.
Jones, and Henderson. 2009. Computational Statistics & Data Analysis.
Koenker, and Mizera. 2006. Advances in Statistical Modeling and Inference.
Liu, Chang, and Ma. 2012. arXiv:1209.2082 [Cs].
Malec, and Schienle. 2014. Computational Statistics & Data Analysis.
Marshall, and Hazelton. 2010. Journal of Multivariate Analysis.
O’Brien, Kashinath, Cavanaugh, et al. 2016. Computational Statistics & Data Analysis.
Panaretos, and Konis. 2012. Journal of the American Statistical Association.
Rathbun. 1996. Biometrics.
Raykar, and Duraiswami. 2005.
Silverman. 1982. The Annals of Statistics.
Smith, and Lewicki. 2005. Neural Computation.
Stein. 2005. Journal of the American Statistical Association.
van de Geer. 1996. Journal of Nonparametric Statistics.
van Lieshout. 2011. Methodology and Computing in Applied Probability.
Yang, Duraiswami, and Davis. 2004. In Advances in Neural Information Processing Systems.
Yang, Duraiswami, Gumerov, et al. 2003. In Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2. ICCV ’03.
Zeevi, and Meir. 1997. Neural Networks: The Official Journal of the International Neural Network Society.
Zhang, and Karunamuni. 2010. Journal of Nonparametric Statistics.