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ICMRA Seminar Series
ICMRA Seminar Series
Contribution to the Estimation via Projection of the Operator of a First-Order Hilbertian Moving Average
Contribution to the Estimation via Projection of the Operator of a First-Order Hilbertian Moving Average
Organizers
Speaker
Konane Fourtoua Victorien
Time
Friday, April 17, 2026 4:00 PM - 5:00 PM
Venue
A6-101
Online
Zoom 712 322 9571
(BIMSA)
Abstract
We consider the problem of estimating the operator of a Hilbert-valued moving average process of order 1 (MAH(1)). Such processes, of the form $X_n = \varepsilon_n+L(\varepsilon_{n-1})$, arise naturally as infinite-dimensional generalizations of classical MA(1) models and provide a flexible framework for functional time series arising in insurance, finance, and other applied fields.
The central challenge is that, unlike the finite-dimensional setting, the operator $L$ cannot be recovered by simple matrix inversion: the covariance operators $C$ and $D$ are compact and non-invertible on an infinite-dimensional Hilbert space. We address this by projecting the moment equation $L^2D^*-LC+D=0$ onto a finite-dimensional subspace spanned by a common eigenvector basis of $C$ and $D$, yielding a tractable system of equations for the eigenvalues of $L$.
We present the theoretical results, establishing the almost sure convergence of the empirical eigenvalue estimators of $C$ and $D$, the convergence rate of the eigenvalue estimators of $L$, the convergence of the operator estimator $\hat{L}$, and a uniform convergence result for the projection estimator. The method is illustrated on an application to insurance turnover data.
The central challenge is that, unlike the finite-dimensional setting, the operator $L$ cannot be recovered by simple matrix inversion: the covariance operators $C$ and $D$ are compact and non-invertible on an infinite-dimensional Hilbert space. We address this by projecting the moment equation $L^2D^*-LC+D=0$ onto a finite-dimensional subspace spanned by a common eigenvector basis of $C$ and $D$, yielding a tractable system of equations for the eigenvalues of $L$.
We present the theoretical results, establishing the almost sure convergence of the empirical eigenvalue estimators of $C$ and $D$, the convergence rate of the eigenvalue estimators of $L$, the convergence of the operator estimator $\hat{L}$, and a uniform convergence result for the projection estimator. The method is illustrated on an application to insurance turnover data.