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

## Author

Listed:
• Joseph, Esdras
• Galeano San Miguel, Pedro
• Lillo Rodríguez, Rosa Elvira

## Abstract

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.

## Suggested Citation

• Joseph, Esdras & Galeano San Miguel, Pedro & Lillo Rodríguez, Rosa Elvira, 2015. "Two-sample Hotelling's T² statistics based on the functional Mahalanobis semi-distance," DES - Working Papers. Statistics and Econometrics. WS ws1503, Universidad Carlos III de Madrid. Departamento de Estadística.
• Handle: RePEc:cte:wsrepe:ws1503
as

File URL: https://e-archivo.uc3m.es/bitstream/handle/10016/20253/ws1503.pdf?sequence=1
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## References listed on IDEAS

as
1. Lajos Horváth & Piotr Kokoszka & Ron Reeder, 2013. "Estimation of the mean of functional time series and a two-sample problem," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(1), pages 103-122, January.
2. Joseph, Esdras & Galeano San Miguel, Pedro & Lillo Rodríguez, Rosa Elvira, 2013. "The Mahalanobis distance for functional data with applications to classification," DES - Working Papers. Statistics and Econometrics. WS ws131312, Universidad Carlos III de Madrid. Departamento de Estadística.
3. Mas, André, 2007. "Weak convergence in the functional autoregressive model," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1231-1261, July.
4. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2004. "An anova test for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 111-122, August.
5. Yehua Li & Naisyin Wang & Raymond J. Carroll, 2013. "Selecting the Number of Principal Components in Functional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(504), pages 1284-1294, December.
Full references (including those not matched with items on IDEAS)

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### Keywords

Functional data analysis;

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