Two sample inference for the second-order property of temporally dependent functional data

TL;DR

This paper introduces new statistical tests based on functional principal component analysis and self-normalization to compare two spatio-temporal datasets, effectively handling dependence and providing robust inference in climate data analysis.

Contribution

The paper develops novel self-normalized tests for comparing covariance structures of functional time series, accommodating weak dependence and dependence between samples.

Findings

SN-based tests outperform existing methods in size accuracy.

The tests demonstrate high power in simulations.

Applied to climate data, they effectively detect differences in spatial dynamics.

Abstract

Motivated by the need to statistically quantify the difference between two spatio-temporal datasets that arise in climate downscaling studies, we propose new tests to detect the differences of the covariance operators and their associated characteristics of two functional time series. Our two sample tests are constructed on the basis of functional principal component analysis and self-normalization, the latter of which is a new studentization technique recently developed for the inference of a univariate time series. Compared to the existing tests, our SN-based tests allow for weak dependence within each sample and it is robust to the dependence between the two samples in the case of equal sample sizes. Asymptotic properties of the SN-based test statistics are derived under both the null and local alternatives. Through extensive simulations, our SN-based tests are shown to outperform existing alternatives in size and their powers are found to be respectable. The tests are then applied to the gridded climate model outputs and interpolated observations to detect the difference in their spatial dynamics.