This article studies stationary functional time series with long-range dependence, and estimates the memory parameter involved. Semiparametric local Whittle estimation is used, where periodogram is constructed from the approximate first score, which is an inner product of the functional observation and estimated leading eigenfunction. The latter is obtained via classical functional principal component analysis. Under the restrictive condition of constancy of the memory parameter over the function support, and other conditions which include rather unprimitive ones on the first score, the estimate is shown to be consistent and asymptotically normal with asymptotic variance free of any unknown parameter, facilitating inference, as in the scalar time series case. Although the primary interest lies in long-range dependence, our methods and theory are relevant to short-range dependent or negative dependent functional time series. A Monte Carlo study of finite sample performance and an empirical example are included.

中文翻译：

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函数时间序列长期依赖的局部惠特尔估计

本文研究了具有长程相关性的平稳函数时间序列，并估计了所涉及的记忆参数。使用半参数局部惠特尔估计，其中周期图是从近似的第一个分数构建的，它是函数观察和估计的领先特征函数的内积。后者是通过经典的泛函主成分分析获得的。在记忆参数对函数支持的恒定性的限制条件下，以及在第一分数上包括相当非原始的其他条件下，估计被证明是一致且渐近正态的，渐近方差没有任何未知参数，便于推理，就像在标量时间序列的情况下一样。虽然主要兴趣在于长期依赖，我们的方法和理论与短期依赖或负依赖函数时间序列相关。包括有限样本性能的蒙特卡罗研究和经验示例。