Thresholding mean test for functional data with power enhancement

Abstract

We consider the two‐sample mean testing problem for infinite‐dimensional functional data and present a new testing procedure. The proposed hard‐thresholding test statistic is based on the normalized functional principal component scores and allows the number of components diverging with the sample size. The hard‐thresholding part is introduced for the power improvement. The asymptotic normality of the statistic is derived under both the null hypothesis and some local alternatives. We also design a power enhancement component based on L2‐norm to further alleviate the power loss under certain alternatives. We conduct extensive numerical simulations and analyze a real example to demonstrate the superiority of the proposed test to several competitors.

Functional Two-Sample Test Based on Projection

“…The main difference between our test in this section and the one proposed by Wang et al (2022) is the selection criterion for the eigenfunctions. While Wang et al (2022) defines Q k directly and uses a hard threshold to sum up Q k with two manually specified parameters, we select the eigenfunctions that can extract the most useful information, which is determined by Q k as derived from our analysis. Moreover, to cope with the situation when µ is orthogonal to the linear span of the eigenfunctions, Wang et al (2022) adds a power enhancement component to their test statistic.…”

“…While Wang et al (2022) defines Q k directly and uses a hard threshold to sum up Q k with two manually specified parameters, we select the eigenfunctions that can extract the most useful information, which is determined by Q k as derived from our analysis. Moreover, to cope with the situation when µ is orthogonal to the linear span of the eigenfunctions, Wang et al (2022) adds a power enhancement component to their test statistic.…”

“…We also consider seven other existing testing methods. The first four meth-ods are based on principal components, HT (Wang et al, 2022) based on permutation, GHT (Pini et al, 2018), PC K λ (Horváth et al, 2013) and PC M λ (Pomann et al, 2016). Furthermore, we include the method based on random projection function (RP, Cuesta-Albertos and Febrero-Bande, 2010), as well as the methods based on distance, L 2 (Benko et al, 2009) and MMD (Wynne and Duncan, 2022).…”

“…Additionally, Pini et al (2018) proposed a generalized Hotelling's T 2 test (GHT) in separable Hilbert spaces, which is equivalent to using all eigenfunctions with non-zero eigenvalues as the projection functions. And Wang et al (2022) proposed a testing method based on eigenfunctions chosen via hard thresholding and also introduced a power enhancement component to improve the power. However, the crucial problem of controlling information loss still remains unaddressed.…”

Simultaneous Inference for the Distribution of Functional Principal Component Scores

“…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. See also Liang et al (2022) and Wang et al (2022) more recently. Along this line, the number of FPC scores we focus on also diverges to infinity as the sample size increases in this paper, preserving the infinite-dimensional nature of functional data, and bringing many challenges in our methodological and theoretical developments.…”