An extension of Chesneau’s theorem

Using a wavelet basis, we establish in this paper upper bounds of wavelet estimation on $$ L^{p}({\mathbb {R}}^{d}) $$
 risk of regression functions with strong mixing data for $$ 1\le p<\infty $$
 . In contrast to the independent case, these upper bounds have different analytic formulae for $$p\in [1, 2]$$
 and $$p\in (2, +\infty )$$
 . For $$p=2$$
 , it turns out that our result reduces to a theorem of Chaubey et al. (J Nonparametr Stat 25:53–71, 2013); and for $$d=1$$
 and $$p=2$$
 , it becomes the corresponding theorem of Chaubey and Shirazi (Commun Stat Theory Methods 44:885–899, 2015).

Wavelet regression estimations with strong mixing data

Using a wavelet basis, this article considers the mean integrated squared error (MISE) of linear and non linear wavelet estimators for regression derivatives r(d)(x) based on biased strong ...

MISE of wavelet estimators for regression derivatives with biased strong mixing data

Pointwise wavelet change-points estimation for dependent biased sample

Based on a data-driven selection of an estimator from a fixed family of kernel estimators, Goldenshluger and Lepski (Probab Theory Relat Fields 159:479–543, 2014) considered the problem of adaptive minimax un-compactly supported density estimation on $${\mathbb {R}}^{d}$$
 with $$L^{p}$$
 risk over Nikol’skii classes. This paper shows the same convergence rates by using a data-driven wavelet estimator over Besov spaces, because the wavelet estimations provide more local information and fast algorithm. Moreover, we explore better convergence rates under the independence hypothesis, which reduces the dimension disaster effectively.

Adaptive Wavelet Density Estimation Under Independence Hypothesis

Using a wavelet basis, Chesneau and Shirazi study the estimation of one dimensional regression functions in a biased nonparametric model over L2 risk (see Chesneau, C and Shirazi, E. Nonparametric wavelet regression based on biased data, Communication in Statistics — Theory and Methods, 43: 2642-2658, 2014). This paper considers d dimensional regression function estimation over Lp (1 ≤ p < ∞) risk. It turns out that our results reduce to the corresponding theorems of Chesneau and Shirazi's theorems, when d = 1 and p = 2.

Non parametric regression estimations over Lp risk based on biased data

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Non parametric regression estimations over Lp risk based on biased data