Hydrogen deuterium exchange mass spectrometry (HDX-MS) is a technique to explore differential protein structure by examining the rate of deuterium incorporation for specific peptides. This rate will be altered upon structural perturbation and detecting significant changes to this rate requires a statistical test. To determine rates of incorporation, HDX-MS measurements are frequently made over a time course. However, current statistical testing procedures ignore the correlations in the temporal dimension of the data. Using tools from functional data analysis, we develop a testing procedure that explicitly incorporates a model of hydrogen deuterium exchange. To further improve statistical power, we develop an empirical Bayes version of our method, allowing us to borrow information across peptides and stabilise variance estimates for low sample sizes. Our approach has increased power, reduces false positives and improves interpretation over linear model-based approaches. Due to the improved flexibility of our method, we can apply it to a multi-antibody epitope-mapping experiment where current approaches are inapplicable due insufficient flexibility. Hence, our approach allows HDX-MS to be applied in more experimental scenarios and reduces the burden on experimentalists to produce excessive replicates. Our approach is implemented in the R-package "hdxstats": https://github.com/ococrook/hdxstats .