Abstract

This paper considers wavelet estimations of a multivariate density function based on stratified size-biased and strong mixing data. We provide upper bounds of the mean integrated squared error for linear and nonlinear wavelet estimators in Besov space $B_{p,q}^s({\mathbb{R}}^d).$ It is shown that the linear estimator achieves the optimal convergence rate in the case of $p\ge 2.$ Moreover, the convergence rate of nonlinear estimator coincides with the optimal convergence rate up to a $\text{ln}\hspace{0.17em}N$ factor for $p\in [1,+\infty ).$ In addition, the nonlinear wavelet estimator is adaptive.

中文翻译：

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强混合分层尺寸偏差样本的多元小波密度估计

摘要

本文考虑了基于分层尺寸偏差和强混合数据的多元密度函数的小波估计。我们为 Besov 空间中的线性和非线性小波估计量提供了平均积分平方误差的上限${乙}_{p,q}^{秒}({\mathbb{R}}^d).$结果表明，线性估计器在以下情况下达到最佳收敛速度$p\ge \mathrm{2个}.$此外，非线性估计器的收敛速度与最优收敛速度一致，直到$\text{在}\hspace{0.17em}否$因素$p\in [\mathrm{1个},+\infty ).$此外，非线性小波估计器是自适应的。