论文标题
关于在Martingale差异错误下的非参数趋势估计中的带宽选择问题
On bandwidth selection problems in nonparametric trend estimation under martingale difference errors
论文作者
论文摘要
在本文中，我们对在因误差下的非参数曲线估计中平滑参数选择的问题感兴趣。我们专注于内核估计和误差形成符号差异随机变量的一般固定顺序，在这种变量中既不需要线性假设也不需要“所有矩”。作为绿色标准的广义交叉验证（GCV）扩展。我们证明，这三个最小化器和基于GCV家族的最小化器在概率上是一阶等效的。我们还给出了平均正方形误差的最小化器和绿色型标准的差距的正常渐近行为。这将扩展到GCV家族。在本文中，我们将理论结果应用于Martingale差异序列的特定情况，即进行ARCH（1）过程的蒙特卡罗模拟研究。
In this paper, we are interested in the problem of smoothing parameter selection in nonparametric curve estimation under dependent errors. We focus on kernel estimation and the case when the errors form a general stationary sequence of martingale difference random variables where neither linearity assumption nor "all moments are finite" are required.We compare the behaviors of the smoothing bandwidths obtained by minimizing either the unknown average squared error, the theoretical mean average squared error, a Mallows-type criterion adapted to the dependent case and the family of criteria known as generalized cross validation (GCV) extensions of the Mallows' criterion. We prove that these three minimizers and those based on the GCV family are first-order equivalent in probability. We give also a normal asymptotic behavior of the gap between the minimizer of the average square error and that of the Mallows-type criterion. This is extended to the GCV family.Finally, we apply our theoretical results to a specific case of martingale difference sequence, namely the Auto-Regressive Conditional Heteroscedastic (ARCH(1)) process.A Monte-carlo simulation study, for this regression model with ARCH(1) process, is conducted.
