The present paper studies density deconvolution in the presence of small Berkson errors, in particular, when the variances of the errors tend to zero as the sample size grows. It is known that when the Berkson errors are present, in some cases, the unknown density estimator can be obtained by simple averaging without using kernels. However, this may not be the case when Berkson errors are asymptotically small. By treating the former case as a kernel estimator with the zero bandwidth, we obtain the optimal expressions for the bandwidth. We show that the density of Berkson errors acts as a regularizer, so that the kernel estimator is unnecessary when the variance of Berkson errors lies above some threshold that depends on the shapes of the densities in the model and the number of observations.

中文翻译：

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具有较小Berkson误差的密度反​​卷积

本文研究存在小Berkson误差时的密度反卷积，特别是当误差的方差随着样本大小的增加而趋于零时。众所周知，当存在伯克森误差时，在某些情况下，可以通过不使用核的简单平均来获得未知密度估计量。但是，当Berkson误差渐近较小时，情况可能并非如此。通过将前一种情况视为带宽为零的核估计器，我们获得了带宽的最佳表达式。我们证明了Berkson误差的密度起着正则化的作用，因此当Berkson误差的方差高于某个阈值时，就不需要核估计器，该阈值取决于模型中密度的形状和观测值的数量。