Minimax Estimation of a Discontinuity for the Density

• Published in: Journal of nonparametric statistics Vol. 14; no. 1-2; pp. 59 - 66
• Main Author: Gayraud, G.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01.04.2002
• Subjects: Histogram; Jump Point; Minimax Rate Of Convergence; Nonparametric Density Estimation
• ISSN: 1048-5252; 1029-0311
• DOI: 10.1080/10485250211390

We consider the problem of estimating the location, at which there is a discontinuity of the probability density f . We consider that the probability density is unknown but belongs to a large specified functional class {\cal F} . Due to the presence of the nuisance parameter f , the problem is nonparametric. Given a sample of size n , we propose a simple procedure to estimate which is based on differences of histograms. The rate of convergence n^{-1} of the proposed estimator \tilde{\theta}_{n} is obtained by analyzing the maximal risk of \tilde{\theta}_{n} over {\cal F} ; it turns out that n^{-1} is equal to the rate of convergence obtained in the parametric estimation of a location parameter. The choice of \tilde{\theta}_{n} is motivated by the minimax approach which gives a criterion of optimality for our estimate; indeed our last result shows that no estimate converge with a faster rate than n^{-1} .