Data which are temporally or spatially close are often correlated. Common examples include data on a patient collected over time or environmental and demographic variables measured at locations in close spatial proximity. The correlation between neighboring sights makes analysis of summary statistics more difficult than for independent data, and for this reason subsampling procedures are attractive in this setting. The proposed research seeks to broaden the applicability of omnibus subsampling methods in the area of longitudinal and spatial statistics. Specifically, distribution theory (asymptotic normality) and variance estimation for general (potentially complicated) statistics computed from spatial data will be addressed, as will the problem of obtaining accurate confidence intervals for unknown parameters. The basic subsampling idea in the dependent data setting is to compute the statistics of interest on smaller subseries (longitudinal data) or subshapes (spatial data) as """"""""replicates"""""""" of the original statistics which retain the original dependence structure of the data. These replicates will be used to obtain (under appropriate conditions) asymptotic normality, variance estimation (convergence results and deriving effective choices of subseries, subshape size), and to obtain accurate confidence intervals by """"""""mixing"""""""" the empirical distribution of the replicates with that of the limiting normal distribution. The methodology is sufficiently general and will be extended to handle other data structures. For example, the methods (with modifications) will also be applied to statistics derived from estimation equations, which have direct applications to longitudinal studies, clinical trials, as well as studying relationships between spatially located variables; e.g., prevalence of disease.
