Statistics and Its Interface

Volume 12 (2019)

Number 1

An $L^2$-norm based ANOVA test for the equality of weakly dependent functional time series

Pages: 167 – 180



Jia Guo (College of Economics and Management, Zhejiang University of Technology, Hangzhou, Zhejiang, China)

Ying Chen (Department of Statistics and Applied Probability, National University of Singapore)


We propose an $L^2$-norm based test for testing the equality of the mean functions of $k$ groups of weakly dependent stationary functional time series. The proposed testing procedure is flexible and can be applied to both homoscedastic and heteroscedastic cases. Under the null hypothesis, the asymptotic random expression of the test statistic is a $\chi^2$-type mixture, which is approximated by a two-cumulant and a three-cumulant matched $\chi^2$ approximation methods, respectively. Under a local alternative hypothesis, the asymptotic random expression is also derived and the test is shown to be root-$n$ consistent. Simulation studies are performed to compare the finite sample performance of the proposed test under various scenarios with alternatives, e.g., an existing FPCA based test and some respective ANOVA tests. It is shown that the proposed test generally outperforms the alternative tests in terms of empirical sizes and powers. Two real data examples help to illustrate the implementation of our test based on the US yield curves and Google flu trends, respectively.


$\chi^2$-type mixture, equality of the mean functions, root-$n$ consistency

2010 Mathematics Subject Classification

Primary 62M10. Secondary 62G10.

This work is supported by Singapore Ministry of Education Academic Research Fund Tier 1 and Institute of Data Science at National University of Singapore. The first author also would like to thank Professor Wen-Lung Shiau, advanced data analysis center (PLS-SEM of Zhejiang University of Technology) for his help on this research.

Received 24 February 2018

Published 26 October 2018