Estimation on unevenly spaced time series

In many different fields realizations of stationary time series might be recorded at irregular points in time, resulting in observed unevenly spaced samples. These missing observations can happen for several reasons, depending on the mechanisms that record the data or external conditions that force the missing observations. In this article, we first focus on the question if we can estimate the mean of a stationary time series when data are not equally spaced. We show that any unevenly spaced sample can be used to estimate the mean of an underlying stationary linear time series. Specifically, we do not impose any restrictions on sampling structure and times, as long as they are independent of the underlying time series. We provide an expression for the sample mean estimator and we establish its asymptotic properties and the central limit theorem. Subsequently we studentize estimation which allows to build confidence intervals for the mean. Finite sample properties of the estimator for the mean are investigated in a Monte Carlo study which confirms good performance of such estimation procedure.