On sequential spectral analysis of amplitude-modulated time series

Abstract

Consider a zero-mean and second-order stationary time series of interest $\left\{X_t\right\}$ that cannot be observed directly. Instead an amplitude-modulated time series $\left\{Y_t\right\}:=\left\{A_t{U}_t{X}_t\right\}$ is observed where $\left\{A_t\right\}$ is a stationary Bernoulli time series and $\left\{U_t\right\}$ is a time series of independent variables satisfying $\mathbb{P}(U_t=0)=0$ and $\mu :=\mathbb{E}\left\{U_t\right\}\ne 0.$ Time series $\left\{A_t\right\}$ creates missing observations when A_t = 0, and U_t modulates not missed X_t. 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 ${(2\pi )}^{-1}\mathbb{E}\left\{X_t^2\right\}$) multiplied by factor ${\mu }^2$ 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.

Keywords:

• Adaptation
• ARMA
• nonparametric
• MISE
• minimal mean stopping time
• minimax
• shape

SUBJECT CLASSIFICATIONS:

• 62L10
• 62H10

Acknowledgments

Valuable comments of the Editor Nitis Mukhopadhyaya and the referees are gratefully appreciated.

Additional information

Funding

This research was supported in part by NSF Grant DMS-1513461.