The maximum of the periodogram of Hilbert space valued time series

TL;DR

This paper develops a statistical test based on the maximum Hilbert-Schmidt norm of the periodogram operator to detect unknown periodic signals in Hilbert space valued time series, with applications to air quality data.

Contribution

It introduces the asymptotic distribution analysis of the test statistic for Hilbert space time series, extending to functional linear processes and providing implementation details.

Findings

Asymptotic distribution derived for independent noise and functional linear processes.

Test successfully applied to real air quality data from Graz, Austria.

Simulation results confirm finite sample accuracy of the asymptotic theory.

Abstract

We are interested to detect periodic signals in Hilbert space valued time series when the length of the period is unknown. A natural test statistic is the maximum Hilbert-Schmidt norm of the periodogram operator over all fundamental frequencies. In this paper we analyze the asymptotic distribution of this test statistic. We consider the case where the noise variables are independent and then generalize our results to functional linear processes. Details for implementing the test are provided for the class of functional autoregressive processes. We illustrate the usefulness of our approach by examining air quality data from Graz, Austria. The accuracy of the asymptotic theory in finite samples is evaluated in a simulation experiment.