Abstract Consider a zero-mean and second-order stationary time series of interest that cannot be observed directly. Instead an amplitude-modulated time series is observed where is a stationary Bernoulli time series and is a time series of independent variables satisfying and Time series creates missing observations when At = 0, and Ut modulates not missed Xt. There is bad and good news about spectral analysis of amplitude-modulated time series. The bad news is that in general consistent estimation of the spectral density is impossible. The good news is that the spectral shape (which is the spectral density minus ) multiplied by factor may be consistently estimated. This article, for the first time in the literature, explores a classical problem of sequential nonparametric estimation of the scaled shape with assigned mean integrated square error. It proposes an adaptive sequential estimator that solves the problem and whose mean stopping time matches the performance of a minimax oracle that knows an underlying spectral density and the amplitude-modulating mechanism. The asymptotic theory is complemented by numerical examples.

中文翻译：

---

调幅时间序列的序贯谱分析

摘要 考虑一个不能直接观察的感兴趣的零均值和二阶平稳时间序列。相反，观察幅度调制的时间序列，其中 是平稳的伯努利时间序列，并且是满足自变量的时间序列，当 At = 0 时，时间序列会创建缺失的观测值，并且 Ut 调制未缺失的 Xt。调幅时间序列的频谱分析有好有坏。坏消息是，通常不可能对谱密度进行一致的估计。好消息是频谱形状（即频谱密度减去 ）乘以因子可以一致地估计。本文首次在文献中探讨了具有指定均值积分平方误差的缩放形状的顺序非参数估计的经典问题。它提出了一种自适应顺序估计器来解决该问题，其平均停止时间与知道潜在频谱密度和幅度调制机制的极小极大预言机的性能相匹配。渐近理论由数值例子补充。