ABSTRACT
 Top of page
 ABSTRACT
 Introduction
 Main results
 Simulation studies
 An illustrative example
 Concluding remarks
 Acknowledgements
 References
 Appendix: proofs of the main results
In this paper, we propose and study a new global test, namely, GPF test, for the oneway anova problem for functional data, obtained via globalizing the usual pointwise Ftest. The asymptotic random expressions of the test statistic are derived, and its asymptotic power is investigated. The GPF test is shown to be rootn consistent. It is much less computationally intensive than a parametric bootstrap test proposed in the literature for the oneway anova for functional data. Via some simulation studies, it is found that in terms of sizecontrolling and power, the GPF test is comparable with two existing tests adopted for the oneway anova problem for functional data. A real data example illustrates the GPF test.
Introduction
 Top of page
 ABSTRACT
 Introduction
 Main results
 Simulation studies
 An illustrative example
 Concluding remarks
 Acknowledgements
 References
 Appendix: proofs of the main results
Let denote k groups of random functions defined over a given finite interval . Let SP(μ,γ) denote a stochastic process with mean function and covariance function . Assuming that
 (1)
it is often interesting to test the equality of the k mean functions
 (2)
against the usual alternative that at least two mean functions are not equal. The aforementioned problem is known as the ksample testing problem or the oneway anova problem for functional data. For the aforementioned oneway anova problem, the k mean functions are often decomposed as μ_{i}(t) = μ_{0}(t) + α_{i}(t),i = 1,2, ⋯ ,k, where μ_{0}(t) is the grand mean function, and α_{i}(t),i = 1,2, ⋯ ,k are the k maineffect functions so that (2) is often equivalently written as the problem for testing if the maineffect functions are all 0, that is, .
For testing (2), some interesting work has been performed in the literature. The L^{2}normbased test and the Ftype test proposed and studied by Faraway (1997), Shen & Faraway (2004), Zhang & Chen (2007), and Zhang (2011) may be adopted; see Section 2.2 for some details. The most related literature is because of Cuevas et al. (2004) who proposed and studied a L^{2}normbased test directly for (2), derived the random expression of their test statistic, and approximated the null distribution by a parametric bootstrap (PB) method via resampling the Gaussian processes involved in the limit random expression of the test statistic under the null hypothesis. Although their method worked reasonably well in their numerical implementation, Cuevas et al. (2004) noted that their PB test is timeconsuming. In this paper, we aim to develop a new global test, namely, GPF test, obtained via globalizing the pointwise Ftest. The GPF test is much less computationally intensive than the aforementioned PB test because no bootstrapping is needed in conducting the GPF test.
The pointwise Ftest was proposed by Ramsay & Silverman (2005, p.227), naturally extending the classical Ftest to the context of functional data analysis. The test statistic of the pointwise Ftest for (2) is defined as
where and throughout, denotes the total sample size, and denote the pointwise betweensubject and withinsubject variations, respectively, and are the sample grand mean function and the sample group mean functions, respectively. There are a few advantages for using the pointwise Ftest. When the functional data (1) are realizations of Gaussian processes, for any given , F_{n}(t) ∼ F_{k − 1,n − k} for all , where F_{k − 1,n − k} denotes the Fdistribution with k − 1 and n − k DOF. Therefore, we can test (2) at all points of using the same critical value F_{k − 1,n − k}(α) for any predetermined significant level α, where F_{k − 1,n − k}(α) denotes the upper 100 α percentile of F_{k − 1,n − k}. However, the aforementioned pointwise Ftest has some limitations too. For example, it is timeconsuming to conduct the pointwise Ftest at all and it is not guaranteed that the oneway anova problem (2) is overall significant for a given significance level even when the pointwise Ftest is significant for all at the same significance level. To overcome this difficulty, in this paper, we propose the aforementioned GPF test via globalizing the pointwise Ftest using its integral over
 (3)
The main contributions of the paper are as follows. First of all, we derived several asymptotic random expressions of T_{n} under the null hypothesis (2) and very general conditions. This allows describing several methods for approximating the null distribution of T_{n}. We recommend the socalled Welch–Satterthwaite χ^{2}approximation method. This method allows to conduct the GPF test quickly without involving bootstrapping. Second, we derived the asymptotic power of the GPF test under some local alternatives and show that the GPF test is rootn consistent. On the contrary, Cuevas et al. (2004) did not study the asymptotic power of their testing procedure. Third, via some simulation studies, we found that in terms of sizecontrolling and power, the GPF test is generally comparable with the aforementioned L^{2}normbased and Ftype tests adopted for (2). The computational effort needed for the three testing procedures are also comparable although the GPF test is constructed via directly summarizing the pointwise Ftest as in (3), whereas the L^{2}normbased and Ftype tests are defined in different ways. Finally, we described how to implement the GPF test when the k functional samples (1) are sampled densely and noisily, irrespective of whether the sampling design is regular or irregular across subjects.
Alternatively, we may globalize the pointwise Ftest using its maximum value over
 (4)
For any given significance level α, we can simulate the critical value of the resulting F_{max}test via bootstrapping. This F_{max}test is somewhat similar to the one suggested in Ramsay & Silverman (2005, p.234), where they used the square root of F_{n}(t) as their test statistic and used a permutationbased critical value. A further study on the F_{max}test is important but is skipped here for space saving because the F_{max}test is not the focus of this paper although we shall illustrate its use in our real data example presented in Section 4.
The rest of the paper is organized as follows. The main results are presented in Section 2. Simulation studies and a real data example are given in Sections 3 and 4, respectively. Some concluding remarks are given in Section .
Simulation studies
 Top of page
 ABSTRACT
 Introduction
 Main results
 Simulation studies
 An illustrative example
 Concluding remarks
 Acknowledgements
 References
 Appendix: proofs of the main results
In this section, we present some simulation studies, aiming to check if the GPF test is comparable, in terms of size controlling and powers, with the L^{2}normbased test (Faraway, 1997; Zhang & Chen, 2007) and the Ftype test (Shen & Faraway, 2004; Zhang, 2011) adopted for the oneway anova problem (2) as briefly described in Section 2.
We use the following model to generate k discrete functional samples:
 (32)
where the k coefficient vectors c_{i} = [c_{i1},c_{i2},c_{i3},c_{i4}]^{T} are used to flexibly specify the k group mean functions μ_{i}(t) = c_{i1} + c_{i2}t + c_{i3}t^{2} + c_{i4}t^{3}; the i.i.d. random variables z_{ijr} (having mean 0 and variance 1), the orthonormal basis vector Ψ(t) = [ψ_{1}(t), ⋯ ,ψ_{q}(t)]^{T}, and the q decreasingordered variance components λ_{r} are used to flexibly specify the common covariance function so that the subject–effect functions ; and the measurement errors ε_{ijl} are i.i.d normal with mean 0 and variance σ^{2}. The unequal design time points t_{ijl} are obtained via randomly removing some design time points at a rate r_{miss} from a common grid of time points
 (33)
where M is some positive integer and r_{miss} is some small number, measuring the missing rate of the design time points.
The tuning parameters are specified as follows. For specifying the k group mean functions μ_{i}(t) = c_{i1} + c_{i2}t + c_{i3}t^{2} + c_{i4}t^{3}, we set c_{1} = [1,2.3,3.4,1.5]^{T} and c_{i} = c_{1} + (i − 1)δu,i = 2, ⋯ ,k where δ controls the differences μ_{i}(t) − μ_{1}(t),i = 2, ⋯ ,k, and u controls the direction of these differences. We set δ properly as listed in Table 1 so that the null hypothesis (when δ = 0) and the three alternatives (when δ > 0) are considered. In addition, we set . For specifying the common covariance function , we set λ_{r} = aρ^{r},r = 1,2, ⋯ ,q with a = 1.5, ρ = 0.10,0.50 or 0.90, and q = 7 and set the basis functions as r = 1, ⋯ ,(q − 1) ∕ 2. Notice that the values 0.10, 0.50, and 0.90 of ρ represent the three cases when the simulated functional data have high, moderate, and low correlations. We set r_{miss} = 10% and set the noise variance σ^{2} = aρ ∕ 16,aρ ∕ 4,aρ ∕ 2, representing the three cases of small, moderate, and large measurement errors. We set k = 3 and specify three cases of n = [n_{1},n_{2},n_{3}] as n_{1} = [20,30,30],n_{2} = [40,60,60], and n_{3} = [60,90,90], representing the small, moderate, and large sample size cases. We specify two cases of the distribution of the i.i.d. random variables z_{ijr}'s as follows: and , allowing to generate Gaussian and nonGaussian functional data, respectively. Notice that the distribution is chosen because it has nearly the heaviest tails among the tdistributions with finite first two moments. Finally, we specify two cases of the number of distinct design time points, M = 80 and M = 150.
Table 1. Empirical sizes and powers (in percentages) of the L^{2}normbased, Ftype, and GPF tests for the oneway anova problem (2) when and M = 80. The associated standard deviations (in percentages) for the GPF test are given in parenthesesρ  n  σ^{2}  L^{2}  F  GPF  L^{2}  F  GPF  L^{2}  F  GPF  L^{2}  F  GPF 


   δ = 0  δ = 0.07  δ = 0.15  δ = 0.22 
0.10  n_{1}  0.15/16  5.42  4.94  5.30(0.31)  15.35  14.28  15.52(0.51)  59.74  57.70  60.02(0.69)  92.16  91.48  92.22(0.37) 
  0.15/4  5.30  4.92  5.26(0.31)  15.28  14.04  15.12(0.50)  61.26  59.81  61.50(0.68)  92.88  92.08  92.80(0.36) 
  0.15/2  5.40  4.92  5.40(0.32)  15.04  14.09  15.20(0.51)  61.36  60.08  61.16(0.69)  92.76  92.02  92.60(0.37) 

 n_{2}  0.15/16  5.52  5.30  5.40(0.31)  26.14  25.18  26.14(0.62)  90.96  90.58  91.18(0.40)  99.66  99.66  99.66(0.08) 
  0.15/4  5.44  5.22  5.42(0.32)  27.72  27.02  27.74(0.63)  91.02  90.62  90.92(0.40)  99.64  99.64  99.66(0.08) 
  0.15/2  5.52  5.34  5.60(0.33)  28.04  27.13  27.90(0.63)  91.02  90.64  90.80(0.41)  99.66  99.64  99.64(0.08) 

 n_{3}  0.15/16  5.46  5.32  5.46(0.32)  38.32  37.76  38.40(0.68)  98.70  98.64  98.78(0.15)  100.0  100.0  100.0(0.00) 
  0.15/4  5.22  5.00  5.28(0.31)  38.74  38.20  38.82(0.68)  98.84  98.74  98.82(0.15)  99.96  99.96  99.96(0.02) 
  0.15/2  5.30  5.18  5.30(0.32)  38.82  38.27  38.47(0.69)  98.82  98.72  98.82(0.15)  99.96  99.96  99.96(0.02) 

   δ = 0  δ = 0.10  δ = 0.20  δ = 0.30 
0.50  n_{1}  0.75/16  5.08  4.68  5.42(0.32)  8.88  8.38  9.50(0.41)  20.10  19.18  21.20(0.57)  44.76  43.22  46.52(0.70) 
  0.75/4  5.00  4.76  5.28(0.31)  8.54  7.96  8.80(0.40)  21.02  20.04  21.86(0.58)  43.06  41.66  44.72(0.70) 
  0.75/2  5.12  4.80  5.16(0.31)  8.46  7.96  8.88(0.40)  21.02  19.94  22.08(0.59)  42.78  41.34  44.40(0.70) 

 n_{2}  0.75/16  4.92  4.80  5.04(0.30)  11.04  10.78  11.56(0.45)  39.16  38.64  40.26(0.69)  76.34  75.78  77.96(0.58) 
  0.75/4  4.54  4.29  4.86(0.30)  12.38  11.92  12.30(0.46)  39.66  39.04  40.48(0.69)  78.42  77.78  79.42(0.57) 
  0.75/2  4.42  4.20  4.84(0.30)  12.44  12.02  12.26(0.46)  39.53  39.00  40.34(0.69)  78.28  77.54  79.00(0.57) 

 n_{3}  0.75/16  5.20  5.09  5.30(0.31)  16.61  16.20  16.94(0.53)  57.44  56.92  59.04(0.69)  92.54  92.40  93.10(0.35) 
  0.75/4  5.16  5.08  5.40(0.31)  16.60  16.34  17.14(0.53)  58.06  57.46  59.38(0.69)  91.98  91.86  92.50(0.37) 
  0.75/2  5.32  5.20  5.44(0.32)  16.52  16.26  17.08(0.53)  57.54  56.94  58.64(0.70)  91.80  91.66  92.16(0.38) 

   δ = 0  δ = 0.23  δ = 0.46  δ = 0.69 
0.90  n_{1}  1.35/16  4.04  3.70  4.56(0.29)  9.00  8.26  9.88(0.42)  34.36  33.26  35.84(0.67)  72.11  71.18  73.74(0.62) 
  1.35/4  3.80  3.54  4.28(0.28)  9.44  8.84  10.19(0.42)  32.54  31.42  34.54(0.67)  72.30  71.31  73.34(0.62) 
  1.35/2  4.50  3.94  4.94(0.31)  9.26  8.86  10.08(0.43)  34.02  32.84  35.24(0.68)  71.68  70.60  72.72(0.63) 
 n_{2}  1.35/16  4.66  4.48  4.82(0.30)  16.88  16.34  18.10(0.54)  67.16  66.50  67.64(0.66)  97.40  97.24  97.70(0.21) 
  1.35/4  4.88  4.78  5.24(0.31)  17.74  17.42  18.50(0.54)  67.62  67.06  68.58(0.65)  97.30  97.22  97.64(0.21) 
  1.35/2  5.16  5.06  5.44(0.32)  17.30  16.86  17.90(0.54)  66.40  66.00  67.70(0.66)  96.80  96.76  97.00(0.24) 
 n_{3}  1.35/16  4.97  4.82  5.28(0.31)  24.88  24.50  25.46(0.61)  88.04  87.82  88.66(0.44)  99.82  99.82  99.84(0.05) 
  1.35/4  5.12  4.96  5.20(0.31)  25.50  25.24  26.12(0.62)  86.78  86.52  87.02(0.47)  99.84  99.78  99.84(0.05) 
  1.35/2  4.68  4.58  4.90(0.31)  26.68  26.46  27.20(0.63)  87.24  87.06  87.38(0.47)  99.86  99.86  99.86(0.05) 
For a given set of tuning parameters, the k discrete functional samples (32) are generated. Because the design time points t_{ijl} for different functions are different, as suggested in Section 2.4, some smoothing technique should be used to reconstruct the individual functions before the L^{2}normbased, Ftype, and GPF tests can be applied. For simplicity, the regression spline reconstruction method (Zhang, 2013, ch.3) is applied to the simulated data of each individual function with a common set of equally spaced interior knots. The number of interior knots is selected by the GCV (generalized crossvalidation) rule. As described in Section 2.4, the three tests are then applied to the k reconstructed functional samples evaluated at the common grid of the distinct design time points (33). Their pvalues are then recorded. When the pvalues are smaller than the nominal significance level α (5% here), the null hypothesis (2) is rejected. The aforementioned process is repeated N = 5000 times. For each case, the empirical size or power of a test is then computed as the proportion of the number of rejections (out of N = 5000 replications) based on the calculated pvalue.
Table 1 presents the empirical sizes and powers (in percentages) of the three tests under various tuning parameters and when and M = 80. The associated standard deviations (in percentages) of the GPFtest's empirical sizes or powers are given in the parentheses. Under the same tuning parameters, the associated standard deviations of the other two tests’ empirical sizes or powers are approximately the same as that of the GPFtest's empirical size or power and are hence omitted for space saving. In fact, they can be easily calculated on the basis of the associated empirical sizes or powers. From the columns associated with δ = 0, we see that in terms of size controlling, when we take the associated standard deviations of the empirical sizes into account, the three tests are overall comparable. This is because when the functional data are highly and moderately correlated (when ρ = 0.10,0.50), the L^{2}normbased and GPF tests are slightly more liberal than the Ftype test, but when the functional data are less correlated (when ρ = 0.90), they are also less conservative than the Ftype test. In addition, the differences between the empirical sizes of the three tests are generally not significant at 5% level and with the sample sizes increasing, these differences generally become smaller and smaller. From the columns associated with δ > 0, we also see that in terms of power, when we take the associated standard deviations of the powers into account, the three tests are also overall comparable. This is because the differences between the powers of the three tests are generally not significant even at 5% level although we do observe, in numerical values, that the GPF test generally have the highest powers, followed by the L^{2}normbased test and then by the Ftype test. In addition, we also see that (i) with increasing the sample sizes, the empirical sizes of the three tests generally become better in size controlling and the empirical powers of the three tests generally become higher; and (ii) for different noise variance σ^{2}, the empirical sizes and powers of the three tests are generally comparable. Observation (i) is expected, whereas observation (ii) is probably because the effect of the measurement errors are largely removed by regression spline smoothing.
In the aforementioned, we only present the simulation results when and M = 80 because the main conclusions based on the simulation results in other cases are similar. From all these simulation results, we conclude that in terms of size controlling and powers, the GPF test is generally comparable with the L^{2}normbased and Ftype tests adopted for the oneway anova problem (2).
An illustrative example
 Top of page
 ABSTRACT
 Introduction
 Main results
 Simulation studies
 An illustrative example
 Concluding remarks
 Acknowledgements
 References
 Appendix: proofs of the main results
An orthotic is an orthopedic device applied externally to limb or body to provide support, stability, and prevention of deformity from getting worse or replacement of lost function. Depending on the diagnosis and physical needs of an individual, a large variety of orthosis is available. According to Abramovich et al. (2004), the orthosis data were acquired and computed in an experiment by Dr. Amarantini David & Dr. Martin Luc (Laboratoire Sport et Performance Motrice, EA 597, UFRAPS, Grenoble University, France). The aim of the experiment was to analyze how muscle copes with an external perturbation. The experiment recruited seven young male volunteers who wore a springloaded orthosis of adjustable stiffness under the following four experimental conditions: a control condition (without orthosis), an orthosis condition (with orthosis), and two spring conditions (with spring 1 or with spring 2) in which steppinginplace was perturbed by fitting a springloaded orthosis onto the right knee joint. All the seven volunteers tried all four conditions 10 times for 20 seconds each, whereas only the central 10 seconds were used in the study in order to avoid possible perturbations in the initial and final parts of the experiment. The resultant moment of force at the knee was derived by means of body segment kinematics recorded with a sampling frequency of 200 Hz. For each steppinginplace replication, the resultant moment was computed at 256 time points equally spaced and scaled to [0,1] so that a time interval corresponds to an individual gait cycle.
In this section, for illustrative purpose, we use the first volunteer's orthosis data under the four experimental conditions only. Panels (A)–(D) of Figure 1 present the 40 raw orthosis curves of the first volunteer under the four experimental conditions with each 10 panel curves. Of interest is to test if the mean orthosis curves of the first volunteer are different under the four experimental conditions. This is equivalent to testing the null hypothesis H_{0} : μ_{1}(t) ≡ μ_{2}(t) ≡ μ_{3}(t) ≡ μ_{4}(t), t ∈ [a,b], where μ_{i}(t),i = 1,2,3,4 denote the underlying mean curves of the first volunteer's orthosis curves under the four experimental conditions, and [a,b] is a finite interval of interest.
Because the 256 design time points are common for all the orthosis curves, we can apply the pointwise Ftest, the L^{2}normbased, Ftype, GPF, and F_{max} tests directly to the raw orthosis data of the first volunteer. The resulting Fvalue line (solid) of the pointwise Ftest over the whole range [0,1], together with the associated 95% pointwise Ftest critical line (dashed) and the 95% F_{max}test critical line (dotdashed), is displayed in Figure 1(E). Both the pointwise Ftest and F_{max}test suggest that the mean orthosis curves of the first volunteer under the four experimental conditions are unlikely the same, but they may be the same at the last stage of the experiment over the interval [0.8,1]. Table 2 displays the test results of the L^{2}normbased, Ftype, GPF, and F_{max} tests. All the four global tests suggest that the mean orthosis curves of the first volunteer under the four experimental conditions are unlikely the same over [0,1] but they are very likely to be the same at the last stage of the experiment over [0.8,1]. This is in agreement with what we observed from Figure 1(E). Notice that throughout this section, the critical values and pvalues of the F_{max}test were obtained on the basis of 10,000 bootstrap replicates and we use 7.6e5 to denote 7.6 × 10^{5} for simplicity.
Table 2. Oneway anova of the first volunteer's raw orthosis data by the L^{2}normbased, Ftype, GPF, and F_{max} tests[a,b]  L^{2}normbased test  Ftype test  GPF test  F_{max} test 

S_{n}  ()  pvalue  F_{n}  ()  pvalue  T_{n}  ()  pvalue  F_{max}  pvalue 

[0,1]  7.6e5  (2404, 1.58)  0  19.9  (1.58,189)  0  17.1  (0.046,23.1)  0  50.6  0 
[0.80,1]  4.5e3  ( 483, 8.40)  0.347  1.12  ( 8.40,100)  0.358  1.15  (0.114,9.51)  0.375  3.73  0.308 
Figure 1 shows that the raw orthosis curves of the first volunteer are very noisy and so is the Fvalue line of the pointwise Ftest. This indicates that the measurement errors from the individual raw orthosis curves may have some impact on the pointwise Ftest and the F_{max}test as well, possibly resulting in misleading results in some situations. To overcome this problem, we may apply some smoothing technique as described in Section 2.4 to largely remove these measurement errors. For the first volunteer's orthosis data, for simplicity, we applied the regression spline smoothing method to each of the 40 raw orthosis curves, respectively, but with a common set of interior knots equally spaced over [0,1]. The optimal number of interior knots is 20, selected by the GCV rule (Zhang, 2013, ch.3). The regression spline smoothing technique also allows to reconstruct the orthosis curves at any desired resolution. For comparison purpose, the reconstructed orthosis curves of the first volunteer were evaluated at the 256 design time points of the first volunteer's raw orthosis data. Panels (A)–(D) of Figure 2 presents the 40 reconstructed orthosis curves of the first volunteer under the four experimental conditions with each 10 panel curves. It appears that the measurement errors of the raw orthosis curves are largely removed.
We now apply the pointwise Ftest and L^{2}normbased, Ftype, GPF, and F_{max} tests to the reconstructed orthosis data of the first volunteer. The resulting Fvalue line (solid) of the pointwise Ftest, together with the associated 95% pointwise Ftest critical line (dashed) and the 95% F_{max}test critical line (dotdashed), is displayed in Figure 2(E), and the test results of the L^{2}normbased, Ftype, GPF, and F_{max} tests are displayed in Table 3. The main conclusions based on the reconstructed orthosis data of the first volunteer are similar to those based on his raw orthosis data described earlier. However, the impact of the measurement errors and regression spline smoothing is still spotted. The aftersmoothing pvalues (0.333, 0.346, 0.372) of the first three tests over [0.80,1] are comparable with but are generally smaller than their beforesmoothing pvalues ( 0.347,0.358,0.375), probably indicating an improvement of the powers of the three tests. The aftersmoothing pvalue (0.378) of the F_{max}test over [0.80,1] is comparable with the before and aftersmoothing pvalues of the first three tests but is much larger than its beforesmoothing pvalue (0.308), probably indicating a reduction of the impact of the measurement errors on the F_{max}test. It appears that the first three tests are relatively robust against the measurement errors but this is not the case for the F_{max}test. This is probably because the first three tests are the integrals of some pointwise test statistics, whereas the F_{max}test is the maximum value of the pointwise Ftest statistics so that the impact of the measurement errors on the first three tests may have been integrated out partially before smoothing is applied while it is not the case for the F_{max}test. Therefore, smoothing may be more preferred for noisy functional data when the F_{max}test is conducted than when the first three tests are conducted.
Table 3. Oneway anova of the first volunteer's reconstructed orthosis data by the L^{2}normbased, Ftype, GPF, and F_{max} tests[a,b]  L^{2}normbased test  Ftype test  GPF test  F_{max} test 

S_{n}  ()  pvalue  F_{n}  ()  pvalue  T_{n}  ()  pvalue  F_{max}  pvalue 

[0,1]  7.5e5  (2761,11.4)  0  23.8  (11.4,137)  0  19.6  (0.065,16.3)  0  48.7  0 
[0.80,1]  3.2e3  ( 675, 4.08)  0.333  1.15  ( 4.08, 49)  0.346  1.14  (0.232,4.57)  0.372  1.89  0.378 
Concluding remarks
 Top of page
 ABSTRACT
 Introduction
 Main results
 Simulation studies
 An illustrative example
 Concluding remarks
 Acknowledgements
 References
 Appendix: proofs of the main results
In this paper, we proposed and studied the GPF test for the oneway anova problem (2) for functional data under very general conditions. With help of the Welch–Satterthwaite χ^{2}approximation, the GPF test can be easily conducted. Via intensive simulation studies, we found that in terms of size controlling and power, the GPF test is comparable with the L^{2}normbased test (Faraway, 1997; Zhang & Chen, 2007) and the Ftype test (Shen & Faraway, 2004; Zhang, 2011) adopted for the oneway anova problem for functional data.
Notice that the GPF test is widely applicable. Actually, it is rather easy to extend the GPF test for various hypothesis testing problems for functional data because for these hypothesis testing problems, it is very easy to construct the associated pointwise Ftest (Ramsay & Silverman, 2005; Zhang, 2013). These hypothesis testing problems include twoway anova problems for functional data (Zhang, 2013, ch.5) and functional linear models with functional responses (Zhang, 2013, ch.6), among others.
In the introduction section, we briefly introduced the F_{max}test. It is an alternative for globalizing the pointwise Ftest. Further studies about the F_{max}test is interesting and warranted.