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Detecting changes in the mean of functional observations


Address for correspondence: Piotr Kokoszka, Department of Mathematics and Statistics, Utah State University, 3900 Old Main Hill, Logan, UT 84322-3900, USA.


Summary.  Principal component analysis has become a fundamental tool of functional data analysis. It represents the functional data as Xi(t)=μ(t)+Σ1≤l<∞ηilvl(t), where μ is the common mean, vl are the eigenfunctions of the covariance operator and the ηil are the scores. Inferential procedures assume that the mean function μ(t) is the same for all values of i. If, in fact, the observations do not come from one population, but rather their mean changes at some point(s), the results of principal component analysis are confounded by the change(s). It is therefore important to develop a methodology to test the assumption of a common functional mean. We develop such a test using quantities which can be readily computed in the R package fda. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. The asymptotic test has excellent finite sample performance. Its application is illustrated on temperature data from England.