Some Properties of the Normalized Periodogram of a Fractionally Integrated Separable Spatial ARMA (FISSARMA) Model

Abstract

In this article, we study the properties of the normalized periodogram of the Fractionally Integrated Separable Spatial ARMA (FISSARMA) models. In particular, we establish the asymptotic mean of the normalised periodogram and the asymptotic second-order moments of the normalised Fourier coefficients. We also establish the asymptotic distribution of the normalised periodogram. Some numerical results are also provided.

Long memory conditional random fields on regular lattices

“…As a result, the GPH estimator belongs to the so-called class of narrowband estimators because f x (𝝀) is evaluated only in the neighborhood of zero frequency. The extension of GPH to the spatial domain was considered by Shitan (2008) and Ghodsi and Shitan (2016) for the separable fractional processes of Equations ( 32) and ( 39), and by Wang (2009) for the isotropic model of Equation ( 25). Suppose, for example, that X is a separable LM unilateral ARMA model considered by Shitan (2008) and Ghodsi and Shitan (2016), with spectral density function 4sin…”

“…As a result, the GPH estimator belongs to the so‐called class of narrowband estimators because $$$ {f}_x\left(\boldsymbol{\lambda} \right) $$$ is evaluated only in the neighborhood of zero frequency. The extension of GPH to the spatial domain was considered by Shitan (2008) and Ghodsi and Shitan (2016) for the separable fractional processes of Equations (32) and (39), and by Wang (2009) for the isotropic model of Equation (25).…”

“…Suppose, for example, that $$$ X $$$ is a separable LM unilateral ARMA model considered by Shitan (2008) and Ghodsi and Shitan (2016), with spectral density function $$$$ {f}_x\left({\lambda}_1,{\lambda}_2\right)={\left|2\sin \left(\frac{\lambda_1}{2}\right)\right|}^{-2{d}_1}{\left|2\sin \left(\frac{\lambda_2}{2}\right)\right|}^{-2{d}_2}{f}_y\left({\lambda}_1,{\lambda}_2\right),\kern1em 0<{d}_1,\kern1em {d}_2<1/2. $$$$ Then, by taking the log of the spectral density function, it follows that …”

“…We note further that only the lowest $$$ {\kappa}_1 $$$ and $$$ {\kappa}_2 $$$ harmonic frequencies are involved to fit the regression in GPH. An optimal choice of $$$ {\kappa}_r $$$ as positive integers which tend to infinity slower than $$$ {n}_r $$$ for $$$ r=1,2 $$$ was proposed, for example, by Ghodsi and Shitan (2016) with $$$ {\kappa}_r=\sqrt{n_r} $$$ following Geweke and Porter‐Hudak (1983) in time series. In contrast, Wang (2009) chose $$$ {\kappa}_r $$$ equal to $$$ {n}_r^{4/5} $$$ as suggested by Hurvich et al (1998).…”