A two-sample test for mean functions with increasing number of projections

Thresholding mean test for functional data with power enhancement

“…Such a threshold level has already been widely adopted in existing works; see Zhong et al (2013). Thirdly, compared to related works based on the increasing $$$ {k}_n $$$ such as Fremdt et al (2014) and Ghale‐Joogh and Hosseini‐Nasab (2018), the hard‐thresholding technique avoids excessive estimation errors caused by $$$ {\hat{V}}_k $$$.…”

“…The competitors can be categorized into two types, those with increasing $$$ {k}_n $$$ and those with pre‐fixed $$$ {k}_n $$$. The former type includes our hard‐thresholding test abbreviated as HT, the test in Fremdt et al (2014) (abbreviated as FR) and the test in Ghale‐Joogh and Hosseini‐Nasab (2018) (abbreviated as GH), while the latter contains the $$$ {L}_2 $$$‐norm based test (abbreviated as $$$ {L}_2 $$$) proposed in Zhang, Peng, et al (2010) and Horváth and Kokoszka (2012) and the normalized‐projection‐based test in Horváth et al (2013) (abbreviated as HK).…”

“…In the case that the signals only lie in the subspaces corresponding to very small eigenvalues, the pre‐specified $$$ k $$$ FPC scores may fail to encompass sufficient information. As a partial solution to this problem, Fremdt et al (2014) and Ghale‐Joogh and Hosseini‐Nasab (2018) spread the discussion to the case where $$$ k={k}_n $$$ is assumed to tend to infinity rather than fixed; see also Liang et al (2022). Nonetheless, there are two main shortcomings.…”

“…Specifically, we implement spectral decomposition on the pooled sample covariance function then obtain the estimates for the FPC scores. We allow the number of principal components to grow with the sample size at a proper polynomial rate, which is along the line of Fremdt et al (2014) and Ghale-Joogh and Hosseini-Nasab (2018). The hard-thresholding technique is then applied to filter out some FPC scores, and the testing statistic is the sum of the square of the remaining ones.…”

Simultaneous Inference for the Distribution of Functional Principal Component Scores

“…However, in contrast to multivariate data, one major obstacle in the analysis of functional data is the infinite dimensionality of the data, which also explains the intractability of the asymptotic distribution of L 2 -type statistics in functional data analysis. To reflect this, Fremdt et al (2014) and Ghale-Joogh and Hosseini-Nasab (2018) propose procedures by considering the projections on subspaces spanned by an increasing number of FPCs. They point out that this approach allows to derive procedures which are fairly insensitive to the selection of the number of FPC scores used for inference.…”