High Dimensional Probability VIII pp 431-455 | Cite as

# Uniform in Bandwidth Estimation of the Gradient Lines of a Density

## Abstract

Let *X*_{1}, …, *X*_{n}, *n* ≥ 1, be independent identically distributed (i.i.d.) \(\mathbb {R}^{d}\) valued random variables with a smooth density function *f*. We discuss how to use these *X*^{′}*s* to estimate the gradient flow line of *f* connecting a point *x*_{0} to a local maxima point (mode) based on an empirical version of the gradient ascent algorithm using a kernel estimator based on a bandwidth *h* of the gradient ∇*f* of *f*. Such gradient flow lines have been proposed to cluster data. We shall establish a uniform in bandwidth *h* result for our estimator and describe its use in combination with plug in estimators for *h*.

## Keywords

Gradient lines Density estimation Nonparametric clustering Uniform in bandwidth## Notes

### Acknowledgements

David Mason thanks the editors for inviting him to contribute to this proceedings. He had intended give a talk based on the results in this paper at the 2017 High Dimensional Probability VIII conference. Unfortunately he had to cancel his participation due heath reasons. Part of this paper was written while he was a guest of Australian National University. All of the authors acknowledge the associate editor and referee for their useful comments and suggestions.

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