On a class of minimum contrast estimators for Gegenbauer random fields

Abstract

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

Asymptotic properties of parameter estimates for random fields with tapered data

“…First, we consider the family of spatial fractional autoregressive processes introduced in [16]. The simulation studies also suggest that similar results are valid for Gegenbauer random fields, see [22], and the approach is applicable to more general models.…”

“…The elements of the matrices S(θ) and A(θ) are calculated in the explicit form, quite similarly to the corresponding calculations in [22], see Appendix B. Now we provide numerical results for the spatial fractional autoregressive model (5.2).…”

“…In particular, for condition B6 we can use the function v(λ) = v(λ 1 , λ 2 ) = |λ 1 | 2β |λ 2 | 2β , β ∈ (0, 1/2). To check the positive definiteness of matrices S(θ) and A(θ) (as required in condition B8), we can argue analogously to the very detailed consideration in [22] (see verification of condition A8 therein). As the matter of fact, the character of singularities in both models, here, and in [22] is the same, with the only difference that here we have singularity of the spectral density at the origin (0, 0), and, in [22], singularity can be shifted from the origin to some another point.…”

“…This section illustrates that the proposed minimum contrast estimation technique works even for the case of Gegenbauer random fields, see [22]. Let Y t1,t2 be a discrete random field satisfying…”

“…In particular, [2] derived consistency and asymptotic normality of a class of MCEs based on Ibragimov functional for fractional Riesz-Bessel motion (see [1]). This functional was considered in [22] for MCE of long-range dependence spatial time series, constructed from fractional difference operators associated with Gegenbauer polynomials. Consistency and asymptotic normality results were derived as well.…”