Nonparametric estimation of the spectral density of amplitude-modulated time series with missing observations

Consider a real-valued and second-order stationary time series with mean zero. The aim is to estimate its spectral density. A minimax solution of this problem is known when either the time series is observed directly, or some observations are missed according to an independent Bernoulli process, or for some special cases when the time series is multiplied by an amplitude-modulating time series with known distribution. It is shown that if a time series of interest, a Bernoulli time series defining missing mechanism, and an amplitude-modulating time series are mutually independent, then the shape of spectral density of an underlying time series of interest can be estimated with the minimax rate known for the case of direct observations. Furthermore, in some special cases the spectral density can be estimated with the minimax rate known for directly observed time series of interest.