分数差分状态空间模型均方误差的相变与近似
Phase transitions and approximations of mean squared error for state-space models with fractional differencing
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中文总结 AI 辅助
本文研究分数差分状态空间模型的趋势估计，通过惩罚最小二乘估计量推导MSE近似，揭示$d=1/2$处的相变，并给出参数选择准则与数值验证。
中文摘要 AI 辅助
我们研究状态空间模型中的趋势估计，其中趋势具有阶数$d>0$的分数随机差分，观测误差构成短程依赖平稳过程。利用有限序列分数求和与差分算子，我们分析通过收缩趋势的分数差分得到的惩罚最小二乘估计量。我们推导了所有$d>0$情形的渐近均方误差（MSE）近似，并识别出在$d=1/2$处的尖锐相变。当$d>1/2$时，估计量是一致的，其最优平衡MSE的阶为$n^{-(2d-1)/(2d)}$。在边界$d=1/2$处，我们获得一个精细的有限样本近似，并证明MSE以较慢的阶$\log\log n/\log n$递减。当$0<d<1/2$时，MSE收敛到一个显式的正极限，因此在所考虑的尺度下无法一致恢复趋势。我们还描述了选择惩罚参数和差分阶数的实用准则，数值实验展示了MSE近似以及选择程序的行为。
英文摘要
We study trend estimation in state-space models in which the trend has a fractional stochastic difference of order $d>0$ and the observation errors form a short-range-dependent stationary process. Using finite-sequence fractional summation and differencing operators, we analyze the penalized least-squares estimator obtained by shrinking the fractional differences of the trend. We derive asymptotic mean squared error (MSE) approximations for all $d>0$ and identify a sharp phase transition at $d=1/2$. When $d>1/2$, the estimator is consistent and its optimally balanced MSE has order $n^{-(2d-1)/(2d)}$. At the boundary $d=1/2$, we obtain a refined finite-sample approximation and show that the MSE decreases at the slower order $\log\log n/\log n$. When $0<d<1/2$, the MSE converges to an explicit positive limit, so consistent recovery of the trend is impossible under the considered scaling. We also describe a practical criterion for choosing the penalty parameter and differencing order, and numerical experiments illustrate the MSE approximations and the behavior of the selection procedure.
发表机构
- University of California, Davis（加州大学戴维斯分校）
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