Abstract
In this article, we propose a new adaptive estimator for multivariate density functions defined on a bounded domain in the framework of multivariate mixing processes. Several procedures have been proposed in the literature to tackle the boundary bias issue encountered using classical kernel estimators. Most of them are designed to work in dimension $d=1$ or on the unit $d$-dimensional hypercube. We extend such results to more general bounded domains such as simple polygons or regular domains that satisfy a rolling condition. We introduce a specific family of kernel-type estimators devoid of boundary bias. We then propose a data-driven Goldenshluger and Lepski type procedure to jointly select a kernel and a bandwidth. We prove the optimality of our procedure in the adaptive framework, stating an oracle-type inequality. We illustrate the good behavior of our new class of estimators on simulated data. Finally, we apply our procedure to a real dataset.
Highlights
Studied the reflection of the data near the boundary
In the context of independent and identically distributed observations and twice differentiable density functions, we point out Bouezmarni and Rombouts (2010) who study the behavior of Beta kernels with a cross-validation selection procedure and Marshall and Hazelton (2010) who study the pointwise behavior of their estimators for a fixed bandwidth
To our best knowledge only Bertin et al (2018) proved adaptive results for integrated risks over D = [0, 1]d. They introduced a new family of kernel density estimators that do not suffer from the boundary bias problem and they proposed a data-driven procedure based on the Goldenshluger and Lepski approach that jointly selects a kernel and a bandwidth
Summary
Studied the reflection of the data near the boundary. Marron and Ruppert (1994) proposed a previous transformation of the data. To our best knowledge only Bertin et al (2018) proved adaptive results for integrated risks over D = [0, 1]d (in the sense that a single estimation procedure achieves the minimax rate of convergence over a large scale of regularity classes) They introduced a new family of kernel density estimators that do not suffer from the boundary bias problem and they proposed a data-driven procedure based on the Goldenshluger and Lepski (see Goldenshluger and Lepski, 2014) approach that jointly selects a kernel and a bandwidth. Other classes of methods that consider density estimation on bounded domains are that based on selection models of histograms or piecewise polynomials (Castellan, 2000, 2003; Birge and Rozenholc, 2006; Akakpo, 2012) using approach developed by Barron et al (1999) These papers studied above all estimation on the unit hypercube [0, 1]d ⊂ Rd in an independent framework. We aim at finding an accurate estimation procedure for f based on the observations X1, . . . , Xn, where n ∈ N
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