Title: "Comparing many functional means", joint work with Colin Decker and Dehan Kong
Abstract: Many modern medical devices produce data with the structure of a multi-channel functional time series. Examples include medical imaging devices (fMRI, EEG, and ECG), high through-put time course gene sequencing devices, and high through-put devices that measure time-course microbiome composition. The typical number of channels can be of the same order or larger than the available sample size.
In this talk, we will present methodology to simultaneously test the equality of a growing number of functional means, in the example above, each mean corresponds to a channel. The number of channels can grow exponentially in the sample size. The proposed test is fully functional in the sense that we do not conduct any explicit dimension reduction or principal component analysis. The practical implementation is based on a Gaussian multiplier procedure and we provide explicit bounds on the speed of convergence of the rejection probability of our test to the nominal value under the null and power against local alternatives. Our theoretical analysis leverages recent advances in high-dimensional Gaussian approximation but requires several intricate modifications of those techniques.