Statistical theory and methodology for the combination of heterogeneous and distributed data

Data is now collected at unprecedented scales across many industries, meaning that there is huge potential for evidence-based advances in science, technology and public policy. However, to harness this potential we must navigate repositories that are often a far cry from the idealised datasets, carefully collected and curated under perfect conditions, that are usually imagined when new statistical methodology is introduced. Data are often gathered quickly and cheaply, patched together from multiple locations, with limited regard to enforcing experimental standards. We may have the large sample sizes we desire, but there will be missing values, misaligned datasets, contamination and, depending on the sector, there may be noise added purposefully to satisfy individuals' and regulatory bodies' privacy concerns.We propose to address such difficulties through the development of new statistical methodology and theoretical frameworks that explicitly incorporate various forms of data heterogeneity and measurement error. This will be divided into four main areas:1. Accounting for sampling bias when a complete dataset is complemented by additional incomplete datasets. This will be studied through the lens of semiparametric theory for functional estimation.2. Combining two or more datasets that record overlapping but distinct sets of variables, where few or no complete records of all variables are available. These file matching problems will be studied using new developments in statistical optimal transport.3. Examining the effect of the violation of missing data assumptions. Here we will introduce techniques from robust statistics to mitigate the error due to misspecifying assumptions about sampling bias.4. Securing individuals' private data through the intentional use of noisy measurement. Here we contribute to the growing field of differential privacy, specifically the user-level local variant, where distributed batches of observations are privatised simultaneously.