Effect of dependence on stochastic measures of accuracy of density estimations
Effect of dependence on stochastic measures of accuracy of density estimations
复制标题
密度估计精度对随机测量的依赖性的影响
DOI：
10.1214/aos/1021379860
复制
发表时间：
2002
影响因子：
4.5
通讯作者：
P. Hall
中科院分区：
文献类型：
--
作者：
G. Claeskens;P. Hall
在核密度估计中，那些对估计器（在给定点计算）有非退化贡献的数据值往往间隔得很好。这一特性可以抑制长期依赖的许多传统后果，例如，收敛速度较慢，而传统的基于损失风险的性能评估可能会揭示这一点。从这个观点来看，在密度估计器经历任何一阶效应之前，依赖关系必须是非常远的。然而，根据特定实现的收敛率进行分析，而不是根据所有实现的平均速度进行分析，会显示出非常不同的情况。我们表明，从这个观点出发，在高斯过程函数的背景下，一旦跨越了短期和长期依赖之间的边界，对收敛速度的影响就会变得明显。例如，ISE-和mese -最佳带宽之间的距离对于依赖于远程的数据通常是更大的数量级。我们对交叉验证也有了新的认识。特别是，我们表明，对于远程依赖数据，交叉验证带宽的方差通常更大，并且该带宽的一阶属性并不取决于在构建交叉验证标准时遗漏了多少数据。此外，对于远程相关数据，交叉验证带宽通常在极限情况下与最优随机带宽完全负相关。
In kernel density estimation, those data values that make a nondegenerate contribution to the estimator (computed at a given point) tend to be spaced well apart. This property has the effect of suppressing many of the conventional consequences of long-range dependence, for example, slower rates of convergence, which might otherwise be revealed by a traditional lossor risk-based assessment of performance. From that viewpoint, dependence has to be very long-range indeed before a density estimator experiences any first-order effects. However, an analysis in terms of the convergence rate for a particular realization, rather than the rate averaged over all realizations, reveals a very different picture. We show that from that viewpoint, and in the context of functions of Gaussian processes, effects on rates of convergence can become apparent as soon as the boundary between short- and long-range dependence is crossed. For example, the distance between ISE- and MISE-optimal bandwidths is generally of larger order for long-range dependent data. We shed new light on cross-validation, too. In particular we show that the variance of the cross-validation bandwidth is generally larger for long-range dependent data, and that the first-order properties of this bandwidth do not depend on how many data are left out when constructing the cross-validation criterion. Moreover, for long-range dependent data the cross-validation bandwidth is usually perfectly negatively correlated, in the limit, with the optimal stochastic bandwidth.