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

Abstract

a b s t r a c tThis paper studies the estimation of a density in the convolution density model from strong mixing observations. The ordinary smooth case is considered. Adopting the minimax approach under the mean integrated square error over Besov balls, we explore the performances of two wavelet estimators: a linear one based on projections and a non-linear one based on a hard thresholding rule. The feature of the non-linear one is to be adaptive, i.e., it does not require any prior knowledge of the smoothness class of the unknown density in its construction. We prove that it attains a fast rate of convergence which corresponds to the optimal one obtained in the standard i.i.d. case up to a logarithmic term.

Wavelet deconvolution in a periodic setting with long-range dependent errors

“…Some analysis has been done for the direct model (ν = 0) with LRD errors in works such as Wang (1996); Kulik and Raimondo (2009b). The topic of density deconvolution with LRD has been studied by Kulik (2008); Chesneau (2012).…”

A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator underα-Mixing Dependence

“…Theorem 4 improves ( [30], Proposition 5.1) in terms of rate of convergence; we gain a logarithmic term.…”

“…setting, hard thresholding wavelet estimators and important results can be found in, for example, Donoho and Johnstone [14,15], Donoho et al [16,17], Delyon and Juditsky [35], Kerkyacharian and Picard [36], and Fan and Koo [42]. In the -mixing context,̂defined by (7) is a general and improved version of the estimator considered in Chesneau [30,31]. The main differences are the presence of the tuning parameter and the global definition of the function offering numerous possibilities of applications.…”

“…In the context of -mixing dependence, various wavelet methods have been elaborated for a wide variety of nonparametric problems. Recent developments can be found in, for example, Leblanc [20], Tribouley and Viennet [21], Masry [22], Patil and Truong [23], Doosti et al [24], Doosti and Niroumand [25], Doosti et al [26], Cai and Liang [27], Niu and Liang [28], Benatia and Yahia [29], Chesneau [30][31][32], Chaubey and Shirazi [33], and Abbaszadeh and Emadi [34].…”