The article introduces spatial long-range dependent models based on the fractional difference operators associated with the Gegenbauer polynomials. The results on consistency and asymptotic normality of a class of minimum contrast estimators of long-range dependence parameters of the models are obtained. A methodology to verify assumptions for consistency and asymptotic normality of minimum contrast estimators is developed. Numerical results are presented to confirm the theoretical findings.Keywords Gegenbauer random field · long-range dependence · minimum contrast estimator · consistency · asymptotic normality

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This paper addresses the problem of spatial functional extrapolation in the framework of spatial autoregressive Hilbertian processes of order one (SARH(1) processes) introduced in Ruiz-Medina (J Muitivar Anal 102:292-305, 2011a). Moment-based estimators of the operators involved in the state equation of these processes are computed by projection into a suitable orthogonal basis. Specifically, the eigenfunction basis diagonalizing the autocovariance operator is considered. An estimation algorithm is designed for the implementation of the resulting SARH(1)-plug-in projection extrapolator from temporal curves irregularly distributed in space. Its performance is illustrated with a real-data example, where the problem of spatial functional extrapolation of ocean surface temperature profiles is addressed. This problem is crucial in the assessment of climate change anomalies. The data are collected from the public oceanographic biooptical database: The World-wide Ocean Optics Database. Cross Validation (C.V.) procedures are applied for the evaluation of the estimation results derived.

A linear multiple regression model in function spaces is formulated, under temporal correlated errors. This formulation involves kernel regressors. A generalized least-squared regression parameter estimator is derived. Its asymptotic normality and strong consistency is obtained, under suitable conditions. The correlation analysis is based on a componentwise estimator of the residual autocorrelation operator. When the dependence structure of the functional error term is unknown, a plug-in generalized least-squared regression parameter estimator is formulated. Its strong-consistency is proved as well. A simulation study is undertaken to illustrate the performance of the presented approach, under different regularity conditions. An application to financial panel data is also considered.Keywords ARH(1) errors · dynamical functional multiple regression · firm leverage maps · generalized least squared estimator · kernel regressors Mathematics Subject Classification (2010) MSC code1 60G25 · 60G60 and 62J05 · MSC code2 62J10

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This paper introduces spatial long-range dependence time series models, based on the consideration of fractional di erence operators associated with Gegenbauer polynomials. Their structural properties are analyzed. The spatial autoregressive Gegenbauer case is also studied, including the case of factors with multiple singularities. An extension to the Hilbert-valued context is nally formulated leading to the introduction of a new class of spatial functional time series models.

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