Pointwise adaptive estimation of a multivariate density under independence hypothesis

Abstract

In this paper, we study the problem of pointwise estimation of a multivariate density. We provide a data-driven selection rule from the family of kernel estimators and derive for it a pointwise oracle inequality. Using the latter bound, we show that the proposed estimator is minimax and minimax adaptive over the scale of anisotropic Nikolskii classes. It is important to emphasize that our estimation method adjusts automatically to eventual independence structure of the underlying density. This, in its turn, allows to reduce significantly the influence of the dimension on the accuracy of estimation (curse of dimensionality). The main technical tools used in our considerations are pointwise uniform bounds of empirical processes developed recently in Lepski [Math. Methods Statist. 22 (2013) 83-99].

Pointwise density estimation based on negatively associated data

“…It turns out that the convergence rate of our estimators coincides with the optimal convergence rate for pointwise estimation [2]. Furthermore, our theorem reduces to the corresponding results of Rebelles [14] when ω(y) ≡ 1 and the sample is independent.…”

Pointwise adaptive estimation of the marginal density of a weakly dependent process

“…This partially generalizes previous results obtained in i.i.d. case by Butucea (2000) and Rebelles (2015). To the best of our knowledge, this is the first adaptive result for pointwise density estimation in the context of dependent data.…”

Adaptive estimation of the hazard rate with multiplicative censoring

“…Indeed, the assumption that h belongs to H β ([0, a + ε], L) permits to state that f Y belongs to H β+1 ([a − ε ′ , a + ε ′ ], L ′ ) for an L ′ > 0 and a ε ′ ∈]0, a[ (see Proposition 6.1). This smoothness result ensures that the kernel estimate f Y has a pointwise risk upper-bounded by C(n/ ln(n)) −2(β+1)/(2β+3) (C a constant), for a "handchosen" bandwidth w ≍ n −1/(2β+1) , see Rebelles (2015) for example. Thus, as D m is set to 2β+3) .…”

“…We also choose w in the definition (10) with a Goldenshluger-Lepski type method over a collection of possible bandwidths larger than 1/ √ n, as described in Comte (2015) or Rebelles (2015). This permits to derive the following convergence rate.…”