Abstract
In density-based clustering methods, the clusters are defined as the connected components of the upper level sets of the underlying density $f$. In this setting, the practitioner fixes a probability $p$, and associates with it a threshold $t^{(p)}$ such that the level set $\{f\geq t^{(p)}\}$ has a probability $p$ with respect to the distribution induced by $f$. This paper is devoted to the estimation of the threshold $t^{(p)}$, of the level set $\{f\geq t^{(p)}\}$, as well as of the number $k(t^{(p)})$ of connected components of this level set. Given a nonparametric density estimate $\hat f_n$ of $f$ based on an i.i.d. $n$-sample drawn from $f$, we first propose a computationally simple estimate $t_n^{(p)}$ of $t^{(p)}$, and we establish a concentration inequality for this estimate. Next, we consider the plug-in level set estimate $\{\hat f_n\geq t_n^{(p)}\}$, and we establish the exact convergence rate of the Lebesgue measure of the symmetric difference between $\{f \geq t^{(p)}\}$ and $\{\hat f_n\geq t_n^{(p)}\}$. Finally, we propose a computationally simple graph-based estimate of $k(t^{(p)})$, which is shown to be consistent. Thus, the methodology yields a complete procedure for analyzing the grouping structure of the data, as $p$ varies over $(0;1)$.