Hotelling’sT2in separable Hilbert spaces

Bivariate Densities in Bayes Spaces: Orthogonal Decomposition and Spline Representation

“…The clr-transformation of the estimated parameters β 0 , β 1 are reported in Figure 6. A permutation test on the global significance of the parameter β 1 -run using a Freedman and Lane scheme [8,30] with test statistics T 2 = β 1 2 B 2 (P) -confirms the statistical significance of this parameter (p-value 0.032), suggesting that the time variation in the interaction between the random variables is indeed relevant. In the light of the shape of β c 1 (Figure 6) one can conclude that, along time, the interaction between weights and heights tends to get more concentrated at medium-high values or low values of the height, whereas it deflates for medium-low values of the height.…”

Functional Two-Sample Test Based on Projection

“…The commonly used eigenfunctions with the largest eigenvalues (Horváth et al, 2013;Pomann et al, 2016) do not necessarily align with the direction of mean difference. Moreover, employing all eigenfunctions with non-zero eigenvalues (Pini et al, 2018) can introduce significant noise into the statistic, thereby disrupting the testing. Additionally, the classical L 2 -norm-based test (Benko et al, 2009) can be interpreted as a test based on the projection function µ µ µ = µ µ µ 1 − µ µ µ 2 , which does not take the covariance function K into consideration, thus making it less sensitive in testing.…”

“…For instance, Górecki and Smaga (2015) proposed a test based on a basis function representation, while Horváth et al (2013) and Pomann et al (2016) projected observations into the linear space spanned by the leading eigenfunctions of the estimated sample covariance function, and the number of eigenfunctions used is determined by the fraction of variation explained, which is usually around 85%. 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.…”

Thresholding mean test for functional data with power enhancement

“…One possible remedy is the F‐type test; see Shen and Faraway (2004); Zhang and Liang (2014). The other is by extending the classical Hotelling's $$$ {T}^2 $$$ test (Pini et al, 2018; Qiu et al, 2021); however, it is difficult to obtain the inverse of covariance function in FDA due to its compactness. One can also construct the simultaneous confidence band as in Degras (2017) and Telschow and Schwartzman (2022) to deal with such a testing problem but the test power may often be conservative.…”