Distributional data analysis: Difference between revisions

Revision as of 14:20, 29 December 2023

Revision as of 19:47, 29 December 2023

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=Hypothesis testing=

==Wasserstein <math>F</math>-test==

Wasserstein <math>F</math>-test has been proposed to test for the effects of the predictors in the Fréchet regression framework with the Wasserstein metric.<ref name="ftest">{{Cite journal|last1=Petersen|first1=A.|last2=Liu|first2=X.|last3=Divani|first3=A.A.|date=2021|title=Wasserstein F-tests and confidence bands for the Fréchet regression of density response curves|journal=Annals of Statistics|volume=49|issue=1|pages=590–611|doi=10.1214/20-AOS1971 |arxiv=1910.13418 |s2cid=204950494 }}</ref> Consider Euclidean predictors <math>X \in \R^p</math> and distributional responses <math>\nu \in \mathcal{W}_2</math>. Denote the Wasserstein mean of <math>\nu</math> as <math>\mu_\oplus^*</math>, and the sample Wasserstein mean as <math>\hat{\mu}_\oplus^*</math>. Consider the global Wasserstein-Fréchet regression model <math>m_\oplus (x)</math> defined in ({{EquationNote|1}}), which is the conditional Wasserstein mean given <math>X=x</math>. The estimator of <math>m_\oplus (x)</math>, <math>\hat{m}_\oplus (x)</math> is obtained by minimizing the empirical version of the criterion.