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

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.

💡 Research Summary

The paper addresses the problem of estimating the probability density of the innovation (error) term in functional autoregressive (FAR) models, where observations are functions rather than scalars. A FAR model is written as (X_t = \Phi(X_{t-1}) + \varepsilon_t), with (\Phi) a possibly nonlinear operator acting on a Hilbert space and (\varepsilon_t) an i.i.d. innovation with unknown density (f). Accurate knowledge of (f) is crucial for model diagnostics, construction of prediction intervals, and detection of outliers.

The authors propose a two‑step procedure. First, they estimate the regression operator (\Phi) non‑parametrically from the observed functional series ({X_t}_{t=1}^n). The estimator (\widehat{\Phi}_n) can be built using functional kernel regression, nearest‑neighbors, or a combination of functional principal component analysis (FPCA) followed by multivariate regression on the reduced coordinates. Second, they compute residual functions (\widehat{\varepsilon}_t = X_t - \widehat{\Phi}n(X{t-1})) and apply a standard univariate kernel density estimator to these residuals:

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