# The effect of the spatial domain in FANOVA models with ARH(1) error term

## TL;DR

This paper investigates how the spatial domain shape influences FANOVA models with ARH(1) errors, introducing a new statistical test and analyzing boundary effects on dependence and estimation stability.

## Contribution

It extends FANOVA models with ARH(1) errors to rectangular and circular supports under Dirichlet boundary conditions, and develops a new significance test for fixed effects.

## Key findings

- Boundary conditions influence error dependence range.
- Eigenvalue decay affects estimation stability.
- Simulation and fMRI application demonstrate method effectiveness.

## Abstract

Functional Analysis of Variance (FANOVA) from Hilbert-valued correlated data with spatial rectangular or circular supports is analyzed, when Dirichlet conditions are assumed on the boundary. Specifically, a Hilbert-valued fixed effect model with error term defined from an Autoregressive Hilbertian process of order one (ARH(1) process) is considered, extending the formulation given in Ruiz-Medina (2016). A new statistical test is also derived to contrast the significance of the functional fixed effect parameters. The Dirichlet conditions established at the boundary affect the dependence range of the correlated error term. While the rate of convergence to zero of the eigenvalues of the covariance kernels, characterizing the Gaussian functional error components, directly affects the stability of the generalized least-squares parameter estimation problem. A simulation study and a real-data application related to fMRI analysis are undertaken to illustrate the performance of the parameter estimator and statistical test derived.