  Most time-series models assume that the data come from observations that are
equally spaced in time. However, this assumption does not hold in many diverse
scientific fields, such as astronomy, finance, and climatology, among others.
There are some techniques that fit unequally spaced time series, such as the
continuous-time autoregressive moving average (CARMA) processes. These models
are defined as the solution of a stochastic differential equation. It is not
uncommon in astronomical time series, that the time gaps between observations
are large. Therefore, an alternative suitable approach to modeling astronomical
time series with large gaps between observations should be based on the
solution of a difference equation of a discrete process. In this work we
propose a novel model to fit irregular time series called the complex irregular
autoregressive (CIAR) model that is represented directly as a discrete-time
process. We show that the model is weakly stationary and that it can be
represented as a state-space system, allowing efficient maximum likelihood
estimation based on the Kalman recursions. Furthermore, we show via Monte Carlo
simulations that the finite sample performance of the parameter estimation is
accurate. The proposed methodology is applied to light curves from periodic
variable stars, illustrating how the model can be implemented to detect poor
adjustment of the harmonic model. This can occur when the period has not been
accurately estimated or when the variable stars are multiperiodic. Last, we
show how the CIAR model, through its state space representation, allows
unobserved measurements to be forecast.
