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

## TL;DR

This paper develops new statistical tests for comparing mean surfaces of two-sample bivariate functional data, addressing limitations of previous methods by focusing on profile and global differences with interpretability and effective power.

## Contribution

It introduces novel profile and globe tests for mean surfaces in bivariate functional data using pooled projection and profile functional PCA, with proven asymptotic properties.

## Key findings

- Tests effectively control type I error.
- High power in detecting mean surface differences.
- Successful application to real-world data sets.

## Abstract

Multivariate functional data has received considerable attention but testing for equality of mean surfaces and its profile has limited progress. The existing literature has tested equality of either mean curves of univariate functional samples directly, or mean surfaces of bivariate functional data samples but turn into functional curves comparison again. In this paper, we aim to develop both the profile and globe tests of mean surfaces for two-sample bivariate functional data. We present valid approaches of tests by employing the idea of pooled projection and by developing a novel profile functional principal component analysis tool. The proposed methodology enjoys the merit of readily interpretability and implementation. Under mild conditions, we derive the asymptotic behaviors of test statistics under null and alternative hypotheses. Simulations show that the proposed tests have a good control of the type I error by the size and can detect difference in mean surfaces and its profile effectively in terms of power in finite samples. Finally, we apply the testing procedures to two real data sets associated with the precipitation change affected jointly by time and locations in the Midwest of USA, and the trends in human mortality from European period life tables.