Adaptive density estimation under weak dependence

Abstract

Abstract. Assume that (Xt) t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue's measure. [Donoho et al. Ann. Stat. 24 (1996) 508-539] propose nearminimax estimators fn based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators fn depend on weak dependence properties of the sequence (Xt) t∈Z through the constant. If these properties are unknown, we propose cross-validation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds.Mathematics Subject Classification. 62G07, 60G10, 60G99, 62G20.

On the adaptive wavelet deconvolution of a density for strong mixing sequences

“…Neumann and von Sachs (1995) for the spectral density estimation under various noni.i.d. situations, Kulik and Raimondo (2009) and Li, Liu, and Xiao (2009) for the nonparametric regression with long range dependence and Gannaz and Wintenberger (2010) for the standard density model under λ-dependence.…”

Adaptive nonparametric estimation in the presence of dependence

“…Therefore we make use of the framework introduced by Gannaz and Wintenberger [2010] which has also been studied, for example, by Bertin and Klutchnikoff [2014]. In the simulations we generate observations Z 1 , .…”

“…where G is the marginal distribution of Y i (see for details Gannaz and Wintenberger [2010]) and the Y i , i ∈ Z is given by…”

“…Considering again the non-parametric estimation of a density as well as a regression function we derive mixing conditions such that the fully data-driven estimator based on dependent observations can attain the minimax-rates for independent data. Finally, considering the framework used by Gannaz and Wintenberger [2010] and Bertin and Klutchnikoff [2014] results of a simulation study are reported in Section 5 which allow to compare the finite sample performance of different data-driven estimators of a density as well as a regression function given independent or dependent observations.…”

“…A comparison, however, allows us to state conditions on the mixing coefficients which ensure that the fully data-driven estimator still attains the minimax-optimal rates for a wide variety of classes of F, and hence, is adaptive. The adaptive non-parametric estimation based on weakly dependent observations of either a density or a regression function has been consider by Tribouley and Viennet [1998], Comte and Merlevede [2002], Comte and Rozenholc [2002], Gannaz and Wintenberger [2010], Comte et al [2008] or Bertin and Klutchnikoff [2014], to name but a view. However, our conditions to derive rates of convergence of the fully-data driven estimator can be verified for both, non-parametric density estimation and non-parametric regression with random design.…”