On the asymptotic properties of a simple estimate of the Mode

Christophe Abraham; Gérard Biau; Benoît Cadre

ESAIM: Probability and Statistics (2010)

  • Volume: 8, page 1-11
  • ISSN: 1292-8100

Abstract

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We consider an estimate of the mode θ of a multivariate probability density f with support in ℝ d using a kernel estimate fn 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 , ⋯ , n f n ( X i ) . It is shown that θn behaves asymptotically as any maximizer θ ^ n of fn. More precisely, we prove that for any sequence ( r n ) n ≥ 1 of positive real numbers such that r n → ∞ and r n d log n / n → 0 , one has r n ∥ θ n - θ ^ n ∥ → 0 in probability. The asymptotic normality of θn follows without further work.

How to cite

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Abraham, Christophe, Biau, Gérard, and Cadre, Benoît. "On the asymptotic properties of a simple estimate of the Mode." ESAIM: Probability and Statistics 8 (2010): 1-11. <http://eudml.org/doc/104318>.

@article{Abraham2010,
abstract = { We consider an estimate of the mode θ of a multivariate probability density f with support in $\mathbb R^d$ using a kernel estimate fn 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 fn. 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. },
author = {Abraham, Christophe, Biau, Gérard, Cadre, Benoît},
journal = {ESAIM: Probability and Statistics},
keywords = {Multivariate probability density; mode; kernel estimate; central limit theorem.; multivariate probability density; central limit theorem},
language = {eng},
month = {3},
pages = {1-11},
publisher = {EDP Sciences},
title = {On the asymptotic properties of a simple estimate of the Mode},
url = {http://eudml.org/doc/104318},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Abraham, Christophe
AU - Biau, Gérard
AU - Cadre, Benoît
TI - On the asymptotic properties of a simple estimate of the Mode
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 1
EP - 11
AB - We consider an estimate of the mode θ of a multivariate probability density f with support in $\mathbb R^d$ using a kernel estimate fn 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 fn. 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.
LA - eng
KW - Multivariate probability density; mode; kernel estimate; central limit theorem.; multivariate probability density; central limit theorem
UR - http://eudml.org/doc/104318
ER -

References

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  10. J.P. Romano, On weak convergence and optimality of kernel density estimates of the mode. Ann. Statist.16 (1988) 629-647.  
  11. M. Rosenblatt, Remarks on some nonparametric estimates of a density function. Ann. Math. Statist.27 (1956) 832-837.  
  12. T.W. Sager, Estimating modes and isopleths. Comm. Statist. – Theory Methods12 (1983) 529-557.  
  13. M. Samanta, Nonparametric estimation of the mode of a multivariate density. South African Statist. J.7 (1973) 109-117.  
  14. B. Silverman, Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Statist.6 (1978) 177-184.  
  15. P. Vieu, A note on density mode estimation. Statist. Probab. Lett.26 (1996) 297-307.  

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