Functional Analysis of Variance for Hilbert-Valued Multivariate Fixed Effect Models

TL;DR

This paper develops a functional analysis of variance framework for fixed effect models with correlated Hilbert-valued Gaussian errors, providing new insights into the geometry of the error space and statistical testing procedures.

Contribution

It introduces a novel approach to analyze variance in Hilbert-valued multivariate models, incorporating the geometry of the RKHS of errors and deriving distributional properties of test statistics.

Findings

Derived distributional characteristics of sum of squares statistics.

Formulated fixed effect hypothesis testing in Hilbert-valued Gaussian models.

Provided new methods for variance analysis in functional data models.

Abstract

This paper presents new results on Functional Analysis of Variance for fixed effect models with correlated Hilbert-valued Gaussian error components. The geometry of the Reproducing Kernel Hilbert Space (RKHS) of the error term is considered in the computation of the total sum of squares, the residual sum of squares, and the sum of squares due to the regression. Under suitable linear transformation of the correlated functional data, the distributional characteristics of these statistics, their moment generating and characteristic functions, are derived. Fixed effect linear hypothesis testing is finally formulated in the Hilbert-valued multivariate Gaussian context considered.