Adaptive Semiparametric Estimation of the Memory Parameter
Adaptive Semiparametric Estimation of the Memory Parameter
复制标题
记忆参数的自适应半参数估计
DOI：
10.1006/jmva.1999.1865
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
A. Samarov
中科院分区：
文献类型：
--
作者：
L. Giraitis;P. Robinson;A. Samarov
在Giraitis，罗宾逊，and Samarov（1997）中，我们证明了具有局部光滑度s的半参数长记忆模型中记忆参数估计的最优速率是n？r（s），r（s）=s/（2s+1），最大频数m=m（s）？n2 r（s）是速率最优的。我们在本文中解决的问题是什么是最好的可获得率时，s是未知的，所以估计不能依赖于s。我们得到一个下界的渐近二次风险的任何这样的自适应估计，这原来是大于最佳非自适应率n？r（s）的对数因子。然后，我们考虑一个修改的对数周期图回归估计的基础上，锥形数据和数据依赖的最大频率m=m（s），这取决于一个自适应选择的估计s的s，并显示，使用Lepskii（1990）在另一个上下文中提出的方法，该估计达到了一个对数因子的下限。一方面，这意味着该估计量在所有自适应（不含s）估计量中具有接近最优的速率，另一方面，它显示了我们对修改后的对数周期图回归估计量的最大频率速率的数据依赖性选择的接近最优性。证明包含的结果，也是独立的利益：一个结果表明，数据锥形给一个显着的改善的渐近性质的离散傅立叶变换的长记忆时间序列的协方差，而另一个给出了一个指数不等式的修改后的对数周期图回归估计。
In Giraitis, Robinson, and Samarov (1997), we have shown that the optimal rate for memory parameter estimators in semiparametric long memory models with degree of “local smoothness” s is n?r(s), r(s)=s/(2s+1), and that a log-periodogram regression estimator (a modified Geweke and Porter-Hudak (1983) estimator) with maximum frequency m=m(s)?n2r(s) is rate optimal. The question which we address in this paper is what is the best obtainable rate when s is unknown, so that estimators cannot depend on s. We obtain a lower bound for the asymptotic quadratic risk of any such adaptive estimator, which turns out to be larger than the optimal nonadaptive rate n?r(s) by a logarithmic factor. We then consider a modified log-periodogram regression estimator based on tapered data and with a data-dependent maximum frequency m=m(s), which depends on an adaptively chosen estimator s of s, and show, using methods proposed by Lepskii (1990) in another context, that this estimator attains the lower bound up to a logarithmic factor. On one hand, this means that this estimator has nearly optimal rate among all adaptive (free from s) estimators, and, on the other hand, it shows near optimality of our data-dependent choice of the rate of the maximum frequency for the modified log-periodogram regression estimator. The proofs contain results which are also of independent interest: one result shows that data tapering gives a significant improvement in asymptotic properties of covariances of discrete Fourier transforms of long memory time series, while another gives an exponential inequality for the modified log-periodogram regression estimator.