We propose and study a new global test, namely the Fmax-test, for the one-way ANOVA problem in functional data analysis. The test statistic is taken as the maximum value of the usual pointwise F -test statistics over the interval the functional responses are observed. A nonparametric bootstrap method is employed to approximate the null distribution of the test statistic and to obtain an estimated critical value for the test. The asymptotic random expression of the test statistic is derived and the asymptotic power is studied. In particular, under mild conditions, the Fmax-test asymptotically has the correct level and is root-n consistent in detecting local alternatives. Via some simulation studies, it is found that in terms of both level accuracy and power, the Fmax-test outperforms the Globalized Pointwise F (GPF) test of Zhang & Liang (2013) when the functional data are highly or moderately correlated, and its performance is comparable with the latter otherwise. An application to an ischemic heart real dataset suggests that, after proper manipulation, resting electrocardiogram (ECG) signals can be used as an effective tool in clinical ischemic heart screening, without the need of further stress tests as in the current standard procedure.keywords F -type test; L 2 -norm based test; local power; myocardial ischemia; nonparametric bootstrap, pointwise F -test.

In this paper, we propose two new tests for testing the equality of the covariance functions of several functional populations, namely a quasi GPF test and a quasi Fmax test. The asymptotic random expressions of the two tests under the null hypothesis are derived. We show that the asymptotic null distribution of the quasi GPF test is a chi-squared-type mixture whose distribution can be well approximated by a simple scaled chi-squared distribution. We also adopt a random permutation method for approximating the null distributions of the quasi GPF and Fmax tests. The random permutation method is applicable for both large and finite sample sizes. The asymptotic distributions of the two tests under a local alternative are investigated and they are shown to be root-n consistent.Simulation studies are presented to demonstrate the finite-sample performance of the new tests against three existing tests. They show that our new tests are more powerful than the three existing tests when the covariance functions at different time points have different scales. An illustrative example is also presented.which uses the ratio of the sum of squares between subjects (SSB) and the sum of squares due to errors (SSE) as its test statistic. That is F = SSB/(k−1) SSE/(n−k) where n and k are the sample size and the number of groups respectively, SSB and SSE measure the variations explained by the factors involved in the analysis and the variations due to measurement errors. Due to its robustness, the F -test is often recommended in practice. In the functional data analysis, we can define SSB and SSE for each time point and denote them as SSB(t) and SSE(t) respectively. The test statistic of the pointwise F -test described by Ramsay andSilverman (2005) can be defined as F (t) = SSB(t)/(k−1) SSE(t)/(n−k) which is a natural extension of the classical F -test to the field of functional data analysis; see more details in Section 2 below. However, this test is timeconsuming and cannot give a global conclusion. To overcome this difficulty, Cuevas et al. (2004) proposed an ANOVA test based on the L 2 -norm of SSB(t), i.e., the numerator of the pointwise F -test statistic but its asymptotic null distribution of the test statistic is not given. Zhang (2013) further investigated this test statistic which is called the L 2 -norm based test and showed that its null distribution is asymptotically a χ 2 -type mixture. Instead of only using the numerator of the pointwise F -test, Zhang and Liang (2013) studied a GPF test which is obtained via globalizing the pointwise F -test with integration. Alternatively, the pointwise F -test can be globalized via using its maximum value as a test statistic, resulting in the so-called F max -test as described by Cheng et al. (2012). It is shown that the F max test is powerful when the functional data are highly correlated and the GPF test is powerful when the functional data are less correlated. Besides its importance in functional ANOVA problems, the pointwise F -test can also be applied in functional linear models...

In the Introduction we note that the strong assumptions on the covariance matrix Σ imposed by Bai and Saranadasa (1996), Chen and Qin (2010) and Srivastava and Du (2008) may not hold. In that case, the null distributions of their test statistics may not be close to normality and may even depart from normality seriously. As an immediate consequence, those tests would suffer from severe inaccuracy in terms of size.To discuss this effect, we have the following observations from the histograms of simulated values of T BS plotted in Figure 1 with Σ taken as in Example 1: Σ = σ 2 [(1 − ρ)I p + ρJ p ], a compound symmetric matrix, where I p is the p × p identity matrix, J p is the p × p matrix of ones, 0 ≤ ρ ≤ 1 and σ 2 > 0.The shape of the histogram is less affected by the sample sizes n 1 , n 2 and dimension p, but is mainly determined by the correlation parameter ρ in Σ. The histogram is quite symmetric and bell-shaped when

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