1 results
On the asymptotic properties of a simple estimate of the Mode
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- Journal:
- ESAIM: Probability and Statistics / Volume 8 / August 2004
- Published online by Cambridge University Press:
- 15 September 2004, pp. 1-11
- Print publication:
- August 2004
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- Article
- Export citation
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We consider an estimate of the mode θ of a multivariate probability density f with support in $\mathbb R^d$
using a kernel estimate f n drawn from a sample X1,...,Xn . The estimate θn is defined as any x in {X1,...,Xn } such that $f_n(x)=\max_{i=1, \hdots,n} f_n(X_i)$ 
. It is shown that θn behaves asymptotically as any maximizer ${\hat{\theta}}_n$ 
of f n . More precisely, we prove that for any sequence $(r_n)_{n\geq 1}$ 
of positive real numbers such that $r_n\to\infty$ 
and $r_n^d\log n/n\to 0$ 
, one has $r_n\,\|\theta_n-{\hat{\theta}}_n\| \to 0$ 
in probability. The asymptotic normality of θn follows without further work.
