Two-sample Hotelling's T² statistics based on the functional Mahalanobis semi-distance

The comparison of the means of two independent samples is one of the most popular problems in real-world data analysis. In the multivariate context, two-sample Hotelling's T² frequently used to test the equality of means of two independent Gaussian random samples assuming either the same or a different covariance matrix. In this paper, we derive two-sample Hotelling's T² from two functional distributions. The statistics that we propose are based on the functional Mahalanobis semi-distance and, under certain conditions, their asymptotic distributions are chisquared, regardless the distribution of the functional random samples. Additionally, we provide the link between the two-sample Hotelling's T² semi-distance and statistics based on the functional principal components semi-distance. A Monte Carlo study indicates that the twosample Hotelling's T² of power those based on the functional principal components semidistance. We analyze a data set of daily temperature records of 35 Canadian weather stations over a year with the goal of testing whether or not the mean temperature functions of the stations in the Eastern and Western Canada regions are equal. The results appear to indicate differences between both regions that are not found with statistics based on the functional principal components semi-distance.