Abstract
Let X1, …, Xn, 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 x0 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.
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