On $ {\mathbb{L}}_{2}$-structure of bilinear models on $ {\mathbb{Z}}^{d}$

Abdelouahab Bibi; Karima Kimouche

doi:10.1016/j.crma.2012.04.002

Probability Theory/Statistics

On

L_{2}

-structure of bilinear models on

Z^{d}

Presented by: the Editorial Board

Abdelouahab Bibi ¹; Karima Kimouche ¹

¹ Département de mathematiques, université Mentouri-Constantine, 25000 Constantine, Algeria

Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 427-432.

Abstract
Résumé

One-dimensionally indexed bilinear $(BL)$ models are widely used for modeling non-Gaussian dataset. Extending BL models to multidimensionally indexed (spatial) $(SBL)$ one yields a novel class of models which are capable of taking into account the non-Gaussianity character and spatiality behavior. Hence, the main contribution here is to study the $L_{2}$ -structure of some SBL models which play an important role in spatial statistical analysis. So, we establish necessary and sufficient conditions for the existence of regular second order stationary and ergodic solutions in terms of its transfer functions. As a consequence, we observe that the second order structure is similar to a weak ARMA field, and that the variance of the best linear prediction error is always greater than the one obtained from an SBL model.

Les modèles bilinéaires (BL) classiques sont largement utilisés pour la modélisation des données non gaussiennes. Cependant, lʼextension de ces modèles au cas spatial (SBL) donne une nouvelle classe de modèles susceptibles de prendre en considération la non gaussianité et le comportement spatial. Le but principal de cette Note consiste à étudier la structure $L_{2}$ de certains modèles SBL qui jouent un rôle très important dans lʼanalyse statistique spatiale. Nous établissons des conditions nécessaires et suffisantes pour lʼexistence de solutions stationnaires aux seconds ordres, réguliers et ergodiques basées sur les fonctions de transferts. En utilisant la représentation ARMA spatiale, on montre que la variance de lʼerreur de prédiction linéaire est toujours plus grande que celle obtenue par SBL.

Received: 2011-12-20
Accepted: 2012-04-04
Published online: 2012-04-01

DOI: 10.1016/j.crma.2012.04.002

Author's affiliations:

Abdelouahab Bibi ¹; Karima Kimouche ¹

¹ Département de mathematiques, université Mentouri-Constantine, 25000 Constantine, Algeria

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     author = {Abdelouahab Bibi and Karima Kimouche},
     title = {On $ {\mathbb{L}}_{2}$-structure of bilinear models on $ {\mathbb{Z}}^{d}$},
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     pages = {427--432},
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     doi = {10.1016/j.crma.2012.04.002},
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Abdelouahab Bibi; Karima Kimouche. On $ {\mathbb{L}}_{2}$-structure of bilinear models on $ {\mathbb{Z}}^{d}$. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 427-432. doi : 10.1016/j.crma.2012.04.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.04.002/

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