On the Covariance Between Related Horvitz-Thompson Estimators

National Statistical Institutes often use repeating surveys to estimate measures of change in the levels of the statistics estimated by the surveys. In order to assess the significance of any measured change it is necessary to have an estimate of the variance of this change. An important component of this variance is the covariance between successive estimates of level. This covariance depends very much on the rotation scheme used to determine the overlap between successive samples. This article presents a general solution for this covariance in situations where measures of change are based on Horvitz-Thompson estimators. This solution applies exactly or approximately for a very large class of rotation schemes, including the most commonly applied ones. The article also presents an unbiased estimator for the covariance and compares the general solution with specific solutions previously published.