On the sample variogram and the sample autocovariance for non-stationary time series

We consider the estimation of the covariance structure in time series for which the classical conditions of both mean and variance stationary may not be satisfied. It is well known that the classical estimators of the autocovariance are biased even when the process is stationary; even for series of length 100–200 this bias can be surprisingly large. When the process is not mean stationary these estimators become hopelessly biased. When the process is not variance stationary the autocovariance is not even defined. By contrast the variogram is well defined for the much wider class of so-called intrinsic processes, its classical estimator is unbiased when the process is only mean stationary and an alternative but natural estimator has only a small bias even when the process is neither mean nor variance stationary. The basic theory is discussed and simulations presented. The procedures are illustrated in the context of a time series of the temperature of the Earth since the mid-19th century.