Abstract: The paper considers simultaneous statistical inference for mean curves of functional and longitudinal data in a unified framework. We establish the asymptotic distribution for the normalized maximum deviations of the local linear estimators from the true mean functions. The asymptotic distribution leads to simultaneous confidence bands with asymptotically correct coverage probabilities. A Gaussian multiplier bootstrap procedure is proposed to obtain the cutoff values and our fully data-driven approach has a good finite sample performance. All the results obtained in the present paper are unified with respect to the sampling schemes.

Key words and phrases: Functional data analysis, Gaussian approximation, Gaussian multiplier bootstrap, local linear smoothing, longitudinal data analysis, simultaneous confidence bands.