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We consider an estimate of the mode $\theta$ of a multivariate probability density $f$ with support in ${ℝ}^d$ using a kernel estimate $f_n$ drawn from a sample $X_1,\cdots ,X_n$. The estimate ${\theta }_n$ is defined as any $x$ in $\{X_1,\cdots ,X_n\}$ such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that ${\theta }_n$ behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of $f_n$. More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of ${\theta }_n$ follows without further work.

Assessing the number of clusters of a statistical population is one of the essential issues of unsupervised learning. Given independent observations drawn from an unknown multivariate probability density , we propose a new approach to estimate the number of connected components, or clusters, of the -level set $ℒ\left(t\right)=\{x:f(x)\ge t\}$. The basic idea is to form a rough skeleton of the set $ℒ\left(t\right)$ using any preliminary estimator of , and to count the number of connected components of the resulting graph. Under mild analytic...

We consider an estimate of the mode of a multivariate probability density with support in ${ℝ}^d$ using a kernel estimate drawn from a sample . The estimate is defined as any in {} such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of . More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of follows without further work.