On sequential spectral analysis of amplitude-modulated time series. Issue 2 (3rd April 2019) - Efromovich, Sam
Abstract: Consider a zero-mean and second-order stationary time series of interest { X t } that cannot be observed directly. Instead an amplitude-modulated time series { Y t } : = { A t U t X t } is observed where { A t } is a stationary Bernoulli time series and { U t } is a time series of independent variables satisfying P ( U t = 0 ) = 0 and μ : = E { U t } ≠ 0 . Time series { A t } 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 ( 2 π ) − 1 E { X t 2 } ) multiplied by factor μ 2 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.