# Higher-order Accurate Spectral Density Estimation of Functional Time Series

## TL;DR

This paper introduces a new class of spectral density kernel estimators for functional time series using flat-top kernels, achieving higher-order accuracy and bias reduction, with methods to ensure positive semi-definiteness.

## Contribution

The paper proposes a novel flat-top kernel-based spectral density estimator for functional time series, improving bias and accuracy while addressing positive semi-definiteness.

## Key findings

- Higher-order accuracy in spectral density estimation
- Bias reduction achieved by flat-top kernels
- Finite-sample performance confirms theoretical advantages

## Abstract

Under the frequency domain framework for weakly dependent functional time series, a key element is the spectral density kernel which encapsulates the second-order dynamics of the process. We propose a class of spectral density kernel estimators based on the notion of a flat-top kernel. The new class of estimators employs the inverse Fourier transform of a flat-top function as the weight function employed to smooth the periodogram. It is shown that using a flat-top kernel yields a bias reduction and results in a higher-order accuracy in terms of optimizing the integrated mean square error (IMSE). Notably, the higher-order accuracy of flat-top estimation comes at the sacrifice of the positive semi-definite property. Nevertheless, we show how a flat-top estimator can be modified to become positive semi-definite (even strictly positive definite) in finite samples while retaining its favorable asymptotic properties. In addition, we introduce a data-driven bandwidth selection procedure realized by an automatic inspection of the estimated correlation structure. Our asymptotic results are complemented by a finite-sample simulation where the higher-order accuracy of flat-top estimators is manifested in practice.