Time-varying first-order autoregressive processes with irregular innovations

TL;DR

This paper develops an optimal estimation method for a time-varying autoregressive model with irregular innovations, using a quasi-maximum likelihood approach and advanced concentration inequalities, revealing new complexity dependencies.

Contribution

It introduces a novel estimation technique for irregular innovations in time-varying AR models and derives optimal bounds considering irregularity and smoothness.

Findings

Establishes a concentration inequality for weakly dependent processes.

Derives upper and lower minimax bounds showing estimator optimality.

Reveals complexity depends on both smoothness and irregularity parameters.

Abstract

We consider a time-varying first-order autoregressive model with irregular innovations, where we assume that the coefficient function is H\"{o}lder continuous. To estimate this function, we use a quasi-maximum likelihood based approach. A precise control of this method demands a delicate analysis of extremes of certain weakly dependent processes, our main result being a concentration inequality for such quantities. Based on our analysis, upper and matching minimax lower bounds are derived, showing the optimality of our estimators. Unlike the regular case, the information theoretic complexity depends both on the smoothness and an additional shape parameter, characterizing the irregularity of the underlying distribution. The results and ideas for the proofs are very different from classical and more recent methods in connection with statistics and inference for locally stationary processes.