Time Series and Missing Observations

Chapter 3 provides results related to the asymptotic behavior of maxima in incomplete samples from time series. The motivation for this arises naturally given that real data are often incomplete, and therefore the theoretical results used in statistical inference should take this fact into account. The problem was first considered by Mittal 1978, who investigated the asymptotic behavior of maxima in complete and incomplete samples from Gaussian sequences. Note that only in very special cases is a sequence of observed random variables from a stationary sequence stationary as well. Hence, the results provided in this chapter are also related to non-stationary sequences. For weakly dependent stationary sequences it appears that the maxima of observed and non-observed random variables, among the first n terms, are asymptotically independent as n tends to infinity. Moreover, the limiting distribution of a random vector whose components are the maximum in a complete sample and the maximum of observed random variables is uniquely determined by the asymptotic relative frequency of observed random variables. This is not the case unless the conditions of weak dependence are satisfied. The asymptotic relative frequencies of different patterns of observed and non-observed random variables may play a significant role in this regard. Several results in Chapter 3 confirm this fact. Time series with heavy tailed marginal distributions are also considered. Original recent results are included. A non-trivial combination of combinatorics and point process theory was used to prove some of the results presented in this chapter.