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Multivariate functional data has received considerable attention but testingfor equality of mean surfaces and its profile has limited progress. Theexisting literature has tested equality of either mean curves of univariatefunctional samples directly, or mean surfaces of bivariate functional datasamples but turn into functional curves comparison again. In this paper, we aimto develop both the profile and globe tests of mean surfaces for two-samplebivariate functional data. We present valid approaches of tests by employingthe idea of pooled projection and by developing a novel profile functionalprincipal component analysis tool. The proposed methodology enjoys the merit ofreadily interpretability and implementation. Under mild conditions, we derivethe asymptotic behaviors of test statistics under null and alternativehypotheses. Simulations show that the proposed tests have a good control of thetype I error by the size and can detect difference in mean surfaces and itsprofile effectively in terms of power in finite samples. Finally, we apply thetesting procedures to two real data sets associated with the precipitationchange affected jointly by time and locations in the Midwest of USA, and thetrends in human mortality from European period life tables.

Profile and Globe Tests of Mean Surfaces for Two-Sample Bivariate Functional Data