### Une preuve «standard» du principe d'invariance de Stoll

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We consider an estimate of the mode $\theta $ of a multivariate probability density $f$ with support in ${\mathbb{R}}^{d}$ 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 ${r}_{n}\phantom{\rule{0.166667em}{0ex}}\parallel {\theta}_{n}-{\widehat{\theta}}_{n}\parallel \to 0$ 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 $\mathcal{L}\left(t\right)=\{x:f(x)\ge t\}$. The basic idea is to form a rough skeleton of the set $\mathcal{L}\left(t\right)$ 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 ${\mathbb{R}}^{d}$ 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 ${r}_{n}\phantom{\rule{0.166667em}{0ex}}\parallel {\theta}_{n}-{\widehat{\theta}}_{n}\parallel \to 0$ in probability. The asymptotic normality of follows without further work.

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