Adaptive Warped Kernel Estimators

Abstract

In this work, we develop a method of adaptive non‐parametric estimation, based on ‘warped’ kernels. The aim is to estimate a real‐valued function s from a sample of random couples (X,Y). We deal with transformed data (Φ(X),Y), with Φ a one‐to‐one function, to build a collection of kernel estimators. The data‐driven bandwidth selection is performed with a method inspired by Goldenshluger and Lepski (Ann. Statist., 39, 2011, 1608). The method permits to handle various problems such as additive and multiplicative regression, conditional density estimation, hazard rate estimation based on randomly right‐censored data, and cumulative distribution function estimation from current‐status data. The interest is threefold. First, the squared‐bias/variance trade‐off is automatically realized. Next, non‐asymptotic risk bounds are derived. Lastly, the estimator is easily computed, thanks to its simple expression: a short simulation study is presented.

Adaptive kernel estimation of the baseline function in the Cox model with high-dimensional covariates

“…As for Bouaziz et al [3], our kernel estimator depends on another estimator, so we need Assumption 3.8(ii) in order to control the difference between the kernel estimator (8) and the pseudo-estimator (11). If our kernel estimator did not involve another estimator, we would have considered condition  1/h ≤ k 0 n a 0 , as in Chagny [8], instead of Assumption 3.8(ii). The term in ln(p)/n in the remaining term comes from the control of |β − β 0 | 1 given by Proposition 3.4.…”

“…In this paper, we do not assume that the kernel K has a compact support, unlike Bouaziz et al [3]. The Breslow estimator (8) and the pseudo-estimator (11) are then well-defined for all t ∈ [0, τ ].…”

“…Chagny [8], in the context of an additive regression model, established an oracle inequality for the kernel estimator of the real-value regression function with a term remaining of order 1/n. In the context of estimation of the intensity of a recurrent event process observed under a standard censoring scheme but without covariates, Bouaziz et al [3] have a logarithm term which appears in their oracle inequality, with a term remaining of order ln 1+a (n)/n instead of the expected 1/n.…”

“…For this, we use a relatively recent method introduced by Goldenshluger and Lepski [14] for the problem of density estimation. The ''Goldenshluger and Lepski method'' has only been considered in two different settings: Bouaziz et al [3] has applied it to provide an adaptive kernel function estimator of the intensity function of a recurrent event process, and Chagny [8] has used it to estimate a real-valued function from a sample of random pairs. Recently, Chagny [7] has also proposed a ''mixed strategy'', which consists in applying the ''Goldenshluger and Lepski method'' to select the relevant model in model selection methods for real-valued function in regression models.…”

“…In the latter, the problem of bandwidth selection in kernel density estimation is addressed and an adaptive estimator is derived, which satisfies non-asymptotic minimax bounds. This method was then considered by Doumic et al [11] for estimating the division rate of a size-structured population in a non-parametric setting, by Bouaziz et al [3] to estimate the intensity function of a recurrent event process, and by Chagny [8] for the estimation of a real function via a warped kernel strategy. In the present paper, we consider this method in order to obtain an adaptive kernel estimator of the baseline function with a data-driven bandwidth.…”