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Smooth Location-Dependent Bandwidth Selection for Local Polynomial Regression

机译:平滑的局部多项式回归与位置相关的带宽选择

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摘要

A bandwidth function for local polynomial models is commonly obtained by optimizing a pointwise penalty criterion, such as an estimated mean squared error (MSE), over a grid of predictor locations. A resultant regression estimate may suffer from irregularities, such as discontinuities, and contextual information over nearby predictor locations is not used. To mediate these difficulties, ad hoc postprocessing is sometimes carried out in the form of smoothing of the penalty criterion and/or the bandwidth estimates. In this work a technique is developed for choosing a smooth bandwidth function that uses a smoothing spline selected based on new "fit"and "roughness" penalties. The fit penalty pushes the bandwidth estimate to adhere to the chosen pointwise criterion, whereas the roughness penalty is imposed on the fitted regression estimate as opposed to the bandwidth estimate, which usually is not of direct interest. The technique can be used in conjunction with various adaptive bandwidth selection methods and provides a systematic way of incorporating contextual information into bandwidth estimation. To justify a spline bandwidth function, we show that under mild regularity conditions, there exists a smooth, asymptotically optimal bandwidth function. We also demonstrate empirically that the technique outperforms the empirical-bias bandwidth selector (EBBS) of Ruppert when using an EBBS MSE pointwise penalty estimate.
机译:通常通过在预测器位置的网格上优化逐点惩罚标准(例如估计的均方误差(MSE))来获得局部多项式模型的带宽函数。结果回归估计可能会遇到不连续性(例如不连续性),并且不使用附近预测变量位置上的上下文信息。为了调解这些困难,有时以平滑惩罚标准和/或带宽估计的形式进行临时后处理。在这项工作中,开发了一种用于选择平滑带宽函数的技术,该函数使用基于新的“拟合”和“粗糙度”惩罚选择的平滑样条。拟合罚分使带宽估计符合所选的逐点标准,而粗糙度罚分则强加于拟合的回归估计上,而不是带宽估计,而带宽估算通常不具有直接意义。该技术可以与各种自适应带宽选择方法结合使用,并提供一种将上下文信息合并到带宽估计中的系统方法。为了证明样条带宽函数的合理性,我们表明在中等规律性条件下,存在一个平滑的渐近最优带宽函数。我们还从经验上证明,当使用EBBS MSE点惩罚评估时,该技术优于Ruppert的经验偏置带宽选择器(EBBS)。

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