Strong uniform consistency and asymptotic normality of a kernel based error density estimator in functional autoregressive models

TL;DR

This paper introduces a kernel-based method for estimating the innovation density in functional autoregressive models, establishing its strong uniform consistency and asymptotic normality under certain conditions.

Contribution

It proposes a residual-based kernel estimator for innovation density in functional autoregressive models and analyzes its asymptotic properties.

Findings

Estimator is strongly uniformly consistent.

Estimator is asymptotically normal.

Conditions for asymptotic properties are identified.

Abstract

Estimating the innovation probability density is an important issue in any regression analysis. This paper focuses on functional autoregressive models. A residual-based kernel estimator is proposed for the innovation density. Asymptotic properties of this estimator depend on the average prediction error of the functional autoregressive function. Sufficient conditions are studied to provide strong uniform consistency and asymptotic normality of the kernel density estimator.