Afrika Statistika

Estimation of a stationary multivariate ARFIMA process

Kévin Stanislas Mbeke and Ouagnina Hili

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In this note, we consider an \(m\text{-dimensional}\) stationary multivariate long memory ARFIMA (AutoRegressive Fractionally Integrated Moving Average) process, which is defined as : \(A(L)D(L)\left( y_1(t),\ldots,y_m(t)\right)^{'}= B(L)\left( \varepsilon_1(t),\ldots,\varepsilon_m(t)\right)^{'}\), where \(M^{'}\) denotes the transpose of the matrix \(M\). We determine the minimum Hellinger distance estimator (MHDE) of the parameters of a stationary multivariate long memory ARFIMA. This method is based on the minimization of the Hellinger distance between the random function of \(f_{n}(.)\) and a theoretical probability density \(f_{\theta}(.)\). We establish, under some assumptions, the almost sure convergence of the estimator and its asymptotic normality.


Dans cette note, nous considérons un processus ARFIMA (AutoRegressive Fractionally Integrated Moving Average) stationnaire multivarié à longue mémoire défini par : \(A(L)D(L)\left( y_1(t),\ldots,y_m(t)\right)^{'}= B(L)\left( \varepsilon_1(t),\ldots,\varepsilon_m(t)\right)^{'}\), où \(M^{'}\) représente la transposée de la matrice $M$. Nous déterminons le minimum de distance de Hellinger d’un estimateur (MHDE) de paramètres d’un processus ARFIMA stationnaire multivarié à longue mémoire. Cette méthode consiste à minimiser la distance de Hellinger entre la densité de probabilité théorique \(f_{\theta}(.)\) et une fonction ale´atoire \(f_{n}(.)\). Sous quelques hypothèses, nous établissons la convergence presque sûre de l’estimateur et sa normalité asymptotique.

Article information

Afr. Stat., Volume 13, Number 3 (2018), 1717-1732.

First available in Project Euclid: 12 December 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62H12: Estimation

stationary multivariate ARFIMA process Estimation Long memory Minimum Hellinger distance


Mbeke, Kévin Stanislas; Hili, Ouagnina. Estimation of a stationary multivariate ARFIMA process. Afr. Stat. 13 (2018), no. 3, 1717--1732. doi:10.16929/as/1717.130.

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