On the rate of convergence for the autocorrelation operator in functional autoregression

TL;DR

This paper develops a general convergence rate result for estimating the autocorrelation operator in functional autoregressive processes using a Tikhonov approach, applicable under various observation modes and without strict spectral decay assumptions.

Contribution

It introduces a Tikhonov-based method to establish convergence rates for the autocorrelation operator estimation in functional autoregression, accommodating diverse data observation modes.

Findings

Convergence rates depend on autocovariance estimation accuracy.

Applicable to complete, discrete, sparse, and noisy observations.

Does not require spectral decay assumptions on autocovariances.

Abstract

We consider the problem of estimating the autocorrelation operator of an autoregressive Hilbertian process. By means of a Tikhonov approach, we establish a general result that yields the convergence rate of the estimated autocorrelation operator as a function of the rate of convergence of the estimated lag zero and lag one autocovariance operators. The result is general in that it can accommodate any consistent estimators of the lagged autocovariances. Consequently it can be applied to processes under any mode of observation: complete, discrete, sparse, and/or with measurement errors. An appealing feature is that the result does not require delicate spectral decay assumptions on the autocovariances but instead rests on natural source conditions. The result is illustrated by application to important special cases.