标题： 带有异方差误差的密度估计 显示英文标题

标题： Density estimation with heteroscedastic error

摘要： 在反卷积问题中，通常假设测量误差是同分布的。然而，在许多实际应用中，这一条件并不满足，因此为同方差误差开发的反卷积估计量变得不一致。在本文中，我们引入了在异方差污染情况下密度的核估计量。我们建立了该估计量的一致性，并证明在相当一般的条件下，它能够达到最优收敛速度。我们研究了该方法在某些极端情况下的应用极限，在这些情况下，我们证明当误差的缩放参数无界时，我们的估计量仍然是一致的。我们提出了一个修改后的估计量，用于误差分布未知但有重复观测的情况。最后，提出了一种选择平滑参数的自适应过程，并在模拟例子中研究了其有限样本性质。 显示英文摘要

摘要： It is common, in deconvolution problems, to assume that the measurement errors are identically distributed. In many real-life applications, however, this condition is not satisfied and the deconvolution estimators developed for homoscedastic errors become inconsistent. In this paper, we introduce a kernel estimator of a density in the case of heteroscedastic contamination. We establish consistency of the estimator and show that it achieves optimal rates of convergence under quite general conditions. We study the limits of application of the procedure in some extreme situations, where we show that, in some cases, our estimator is consistent, even when the scaling parameter of the error is unbounded. We suggest a modified estimator for the problem where the distribution of the errors is unknown, but replicated observations are available. Finally, an adaptive procedure for selecting the smoothing parameter is proposed and its finite-sample properties are investigated on simulated examples.