This paper considers wavelet estimations for a multivariate density function based on strongly mixing data. We first construct a linear wavelet estimator and provide a convergence rate over \(L^{p} (1\le p<\infty )\) risk in Besov space \(B^{s}_{r,q}(\mathbb {R}^{d})\). However, this estimator depends on the smoothness of density function, which means that the estimator is not adaptive. A nonlinear adaptive wavelet estimator is proposed by thresholding method. Moreover, the convergence rate of nonlinear estimator is better than the linear one in the case of \(r\le p\).

中文翻译：

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强混合样本的小波密度估计器的收敛速度

本文考虑基于强混合数据的多元密度函数的小波估计。我们首先构造一个线性小波估计器，并在Besov空间\（B ^ {s} _ {r，q}（\\}中提供\（L ^ {p}（1 \ le p <\ infty）\）风险的收敛速度。 mathbb {R} ^ {d}）\）。然而，该估计器取决于密度函数的平滑度，这意味着该估计器不是自适应的。提出了一种基于阈值的非线性自适应小波估计器。此外，在\（r \ le p \）的情况下，非线性估计的收敛速度要好于线性估计。