In this paper, we derive the asymptotic properties of the density-weighted average derivative estimator when a regressor is contaminated with classical measurement error and the density of this error must be estimated. Average derivatives of conditional mean functions are used extensively in economics and statistics, most notably in semiparametric index models. As well as ordinary smooth measurement error, we provide results for supersmooth error distributions. This is a particularly important class of error distribution as it includes the Gaussian density. We show that under either type of measurement error, despite using nonparametric deconvolution techniques and an estimated error characteristic function, we are able to achieve a $\sqrt {n}$ -rate of convergence for the average derivative estimator. Interestingly, if the measurement error density is symmetric, the asymptotic variance of the average derivative estimator is the same irrespective of whether the error density is estimated or not. The promising finite sample performance of the estimator is shown through a Monte Carlo simulation.

中文翻译：

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测量误差下的平均导数估计

在本文中，我们推导了当回归量被经典测量误差污染并且必须估计该误差的密度时，密度加权平均导数估计量的渐近性质。条件平均函数的平均导数在经济学和统计学中被广泛使用，尤其是在半参数指数模型中。除了普通的平滑测量误差外，我们还提供了超平滑误差分布的结果。这是一类特别重要的误差分布，因为它包括高斯密度。我们表明，在任何一种测量误差下，尽管使用了非参数反卷积技术和估计的误差特征函数，我们都能够实现$\sqrt {n}$-平均导数估计器的收敛速度。有趣的是，如果测量误差密度是对称的，则无论是否估计误差密度，平均导数估计量的渐近方差都是相同的。通过蒙特卡罗模拟显示了估计器的有希望的有限样本性能。