Asymptotic properties of parameter estimates for random ﬁelds with tapered data

: In this paper we present novel results on the asymptotic behavior of the so-called Ibragimov minimum contrast estimates. The case of tapered data for various models of Gaussian random ﬁelds is investigated


Introduction
Parameter estimation of stationary stochastic processes and random fields constitutes a central topic in statistical inference from temporally or spatially correlated data.The maximum likelihood estimation method (MLE) is one of the most popular estimation tools.The problem of parameter estimation of fractionally integrated processes with seasonal components was addressed in [37].To estimate the fractional parameters, they propose several log-periodogram regression estimators with different bandwidths selected around or between the seasonal frequencies.The same methodology was used in [34] for fractionally differenced autoregressive-moving average processes in the stationary time series context.Several contributions have also been made for the MLE of long memory spatial processes (see, for example, [8]).For two-dimensional spatial data the paper [12] introduced a spatial unilateral first-order autoregressive moving average (ARMA) model.To implement MLE they provided a proper treatment to border cell values with a substantial effect in the estimation of the parameters.Gaussian maximum likelihood estimation, in the context of ARMA models, was applied in [44].
Linear and non-linear functionals of the periodogram play a key role in the design of Minimum Contrast Estimation (MCE) techniques (see, for example, [42]).Whittle estimation procedure was treated quite extensively in the literature (see [27,28,36], for the case of random fields).Special attention has been paid to minimum contrast parameter estimation of fractional, fractal and long-range dependence stochastic models (see, for example, [18]).For instance, [43] applied the minimum contrast parameter estimation to approximate the drift parameter of the Ornstein-Uhlenbeck process, when the corresponding stochastic differential equation is driven by the fractional Brownian motion.Consistency and asymptotic normality of the Whittle maximum likelihood estimator for stationary seasonal autoregressive fractionally integrated moving-average (SARFIMA) processes was proved in [26].Maximization of the Whittle likelihood has also been considered in the papers [17,16,32].A continuous version of the Whittle contrast functional, supplied with a specific weight function, was formulated in [32], where the problem of parameter estimation of continuous stationary processes was addressed, deriving the consistency and the asymptotic normality of the formulated MCE.Modified Whittle estimation of multilateral models on a lattice is considered in [39].Estimation of the spectral density for aggregated possibly strong-dependent Gaussian random fields, from an expansion in terms of orthogonal Gegenbauer polynomials, was studied in [33].These fields are constructed from accumulation of i.i.d.short memory fields, via an unknown mixing density (to be estimated).
An alternative to Whittle family of linear functionals was proposed in [30] (for generalizations, see also [2,3,4,5,6]).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.Minimum contrast estimators, based on the nonlinear objective functions of the periodogram, have been studied in [35] and [42].
In this paper we study Ibragimov minimum contrast estimators based on tapered data.The case of Gaussian stationary random fields on Z d is considered.
Note that benefits of tapering data have been widely reported in the literature.It is well known that tapering reduces leakage effects, especially when spectral densities contain high peaks.Even more importantly, the use of tapers leads to the bias reduction, which is a key issue in Spatial Statistics.Namely, tapers can help to fight the so-called "edge effects"; see, e.g., [20,28].
The main contributions of the paper are two-fold: 1) We state the theorems on consistency and asymptotic normality for Ibragimov minimum contrast estimators for long-range dependent random fields in such a form which was not presented in the previous papers devoted to Ibragimov functional (see, e.g.[2,3,4]), namely, in the present paper: (i) conditions are given in the form prescribing behavior of the spectral density and its derivatives at the point of singularity, and (ii) exact conditions on tapers are given.This makes the results very convenient to apply for particular models, the form of conditions gives the key to construct the weight function, which is incorporated into the Ibragimov functional in order to compensate singularities of the spectral density.
2) For the case of long-range dependent Gaussian random fields, we state the central limit theorem for spectral functionals (or quadratic forms) based on the tapered data, which is of independent importance and also serves as the main tool to derive the asymptotic properties of estimators in the present paper.
The numerical part contributes to investigations on convergence rates of minimum contrast estimators.It also demonstrates that the obtained theoretical results can be further extended to other classes of fractional models.
For convenience of a reader, the statements of the main results are accompanied by discussion to clarify the conditions used and also some digressions intended to comment the related results existing in the literature.
The paper is organized as follows.
In Section 2, we first explain the general methodology of minimum contrast estimation, and introduce Ibragimov contrast functional based on tapered data.Then, in subsections 2.4-2.5 we state our main results on consistency and asymptotic normality of the corresponding minimum contrast estimators.The conditions cover the case of long-range dependent random fields, with a particular model of spectral densities with singularities which factorize.We also formulate exact conditions on tapers, leading to an asymptotic normality result, with standard normalizing factor, for dimensions d = 1, 2, 3.
In Section 3 we revise the results concerning the bias control in minimum contrast estimation based on the Whittle functional: results in [27,28] and in [36], for the cases of for short-and long-range dependent Gaussian random fields correspondingly.
Section 4 contains the proofs of the results from Section 2 on asymptotic properties of our minimum contrast estimators.
The proofs are based on the central limit theorem for spectral functionals J T (φ) = T φ(λ)I h T (λ) dλ (with I h T (λ) being the periodogram based on tapered data), which we state and prove in Appendix A. This theorem (Theorem A.2) provides an extension of the classical result in [24] to the case of fields and tapered data.Beyond its application here for derivation of asymptotic properties of MCE, it is of separate interest itself.
Since the main part of the paper is devoted to the study of Ibragimov minimum contrast estimators, we placed this central limit theorem with its derivation and related observations in Appendix A, which can be considered, to some extend, as a self-contained part of the paper.With this structure of the paper, the reader can choose to focus firstly on the statistical estimation results and their applications and then consider in detail the central limit theorem for spectral functionals based on tapered data.
Section 5 presents simulation studies to demonstrate performance of the estimations technique developed in the paper.Namely, in the simulation study undertaken, we illustrate consistency and asymptotic normality of the plug-in MCE considered, based on tapered periodogram, in the context of spatial Gaussian fractional autoregressive processes, and spatial Gegenbauer random fields.

General definition of minimum contrast estimators
We introduce the definition of minimum contrast estimators following [28], pp.119-127, where these estimators have been studied for some classes of discretetime random fields.
Let a random field Y (t), t ∈ Z d , be observed on a sequence L T of increasing finite domains.We will suppose that L T is a hypercube: Consider a parametric statistical model with a family of distributions P θ , θ ∈ Θ, where Θ is a compact subset of R q , and the true parameter value θ 0 ∈ int Θ, the interior of Θ. Denote P 0 = P θ0 .
2) a contrast field for a contrast function K(θ 0 ; θ), which is a random field U T (θ), T ∈ Z, θ ∈ Θ, related to observations Y (t), t ∈ L T , and such that the following relation holds: Then the minimum contrast estimator θT is defined as a minimum point of the functional U T (θ), that is, θT = arg min θ∈Θ U T (θ).
Note that the usual way to realize the above definition in practice is to construct, basing on observations, a function U T (θ) which converges in P 0probability to some function U (θ) such that U (θ) − U (θ 0 ) = K(θ 0 ; θ).
In this paper we apply minimum contrast technique for parameter estimation in the spectral domain.We consider a measurable stationary zero-mean realvalued Gaussian random field Y (t), t ∈ Z d .We suppose that the parametric form of its spectral density is known: , being a compact, and we are interested in estimation of θ 0 , the true parameter value.
In the present paper we will study the so-called Ibragimov minimum contrast estimators, where K(θ 0 ; θ) has the form (2.7), the functions U T (θ) and U (θ) are defined by (2.6) and (2.8) correspondingly (see Section 2.3 below).Another example of minimum contrast estimators is provided by the well known Whittle estimators, for which we present the expressions for contrast field and contrast function in Section 3. We refer to [42] for some other examples of minimum contrast estimators and also to [28] where many useful general results for minimum contrast method can be found.
Asymptotic properties of parameter estimates for random fields with tapered data 3337

Data tapers and tapered periodograms
We will base our analysis on tapered data.The use of tapers leads to the bias reduction, which is important when dealing with spatial data: tapers can help to fight the so-called "edge effects".
H1. h (t), t ∈ R, is a positive even function of bounded variation with bounded support: h (t) = 0 for |t| > 1. Denote where hT (t) = h (t/T ), λ = (λ 1 , ..., λ d ), and k is a positive integer number.Note that evaluation of asymptotic behavior of spectral estimates is based on the properties of functions Hk,T (λ), which, in its own turn, is based on properties of functions h (t).For example, the assumption that h (t) is of bounded variation allows to write down useful upper bounds for Hk,T (λ).In what follows we will introduce some further assumptions on tapers.
Define the finite Fourier transform of tapered data {h T (t) Y (t) , t ∈ L T } : and the tapered periodogram of the second order (provided that H 2,T (0) = 0):

Ibragimov contrast function and estimators
We begin with the following assumptions concerning our parametric model to be estimated.
We restrict our study to the fields with d ≤ 3. Note that the results on consistency of estimators will hold for the general d ≥ 1, however, for asymptotic normality we will impose the restriction d ≤ 3, since only for this dimensions we are able to control the bias with the help of tapers.

Remark 2.1.
In what follows, by differentiability with respect to θ, we mean differentiability in the interior of Θ.
To define the Ibragimov functional, we will use a weight function satisfying the next assumption.
B3.There exists a nonnegative function w(λ), λ ∈ T, such that Under this condition, we set and represent the spectral density in the form: For the function ψ(λ, θ) we have and we additionally suppose: B4.The derivatives ∇ θ ψ(λ, θ) exist and it is legitimate to differentiate under the integral sign in equation (2.5), i.e.
We consider the following contrast field based on the tapered periodogram defined above: and the minimum contrast estimator θT = arg min θ∈Θ U T (θ).
We also define the functions

7)
Asymptotic properties of parameter estimates for random fields with tapered data 3339 and The minimum contrast property of the function K(θ 0 , θ) is stated in Theorem 2.1 below, as well as the conditions under which U T (θ) is the contarst field related to K(θ 0 , θ).
Remark 2.2.The weight function w used in the contrast functional U T is aimed to compensate for possible singularities of the spectral density.In some cases, the weight function is not required, for example, when spectral density is bounded.

Assumptions
We now formulate conditions needed to state the results on consistency and asymptotic normality of Ibragimov estimators for Gaussian fields .
We will suppose that there exist functions α i : Θ → (0, 1), i = 1, ..., d, such that the following conditions are satisfied (with the proper choice of a weight function w(λ)).To simplify notations in formulation of the conditions below we will omit the argument θ in functions α i (θ) and write them simply as α i .
Remark 2.3.In conditions B5-B7 the behavior of the spectral density and some of its derivatives is prescribed at the point of singularity.These conditions appear quite naturally and are analogous, for example, to those which are used in [23] for the case of Whittle estimators, with evident modifications since we use another functional to construct estimators.It is clear from assumptions B6-B7 that the weight function w(λ) is constructed to compensate the singularity of the spectral density.
Remark 2.5.One example of a long-range dependent random field for which all the above conditions are satisfied (with the proper choice of the weight function w(λ)), is the solution to the following equation: ) being a two-dimensional white noise and B i being a backward shift operator for i-th coordinate, i = 1, 2. This example of fractional spatial autoregression, in more general form, is considered in detail in Section 5, together with other two exaples, namely, the Gegenbauer random fields.Appropriate for these models weight functions w(λ) are presented.Note that the estimation of fields obeying the fractional spatial authoregressive model have been considered in [16] (based on Whittle functional), however, only the consistency of the estimators was stated therein.Within the approach of the present paper, we are able to construct consistent and asymptotically normal estimates for this model (see Section 5.1 below).

Theorems on consistency and asymptotic normality of the estimators
Theorem 2.1.Let the conditions B1-B3, B5-B6, and H1 to hold.Then, the function K(θ 0 ; θ) defined by (2.7) is the contrast function for the contrast field U T (θ) defined by (2.6).Moreover, the minimum contrast estimator θT θT = arg min is a consistent estimator of the parameter vector θ.That is, there is a convergence in P 0 probability: The estimator σ2 T is an estimator of the parameter σ 2 (θ 0 ) given by The proof will be given in Section 4.
Theorem 2.2.Let the conditions B1-B9 and H1-H2 to hold and d ≤ 3. Then the minimum contrast estimator defined by (2.9) is asymptotically normal, that is, as where the entries of the matrices S(θ) = (s ij (θ)) and A(θ) = (a ij (θ)) are defined in condition B8, e(h) is given by the formula and N q (•, •), denotes the q-dimensional Gaussian law.
The proof will be given in Section 4.
Remark 2.6.In the discrete-time case the domain over which the field is observed is usually taken to be L T = [1, T ] d .Our results remain valid for such a domain as well, we just need to adjust the assumptions on a taper h(t).Namely, assumption H1 must be modified as follows: h(t) is a positive measurable function of bounded variation with support on [0, 1] and h(0

This is just a modification of the well known cosine bell (or the Tukey-Hanning taper)
suitable for the domain Remark 2.8.To establish the consistency of a minimum contrast estimator θT , which corresponds to a functional U T (θ), the following standard reasonings are used: one needs to check that the convergence (2.1) holds in probability, and then, due to Theorem 3.4.1 [28], it is sufficient to prove that the convergence (2.1) holds uniformly with respect to θ.
The standard approach to state the asymptotic normality of the estimator θ T is to consider the relation: and then evaluate the asymptotic behavior of

Therefore, one needs the results on large sample properties of the empirical spectral functionals of the form:
where I h T (λ) is the periodogram based on tapered data.In particular, for derivation of the asymptotic normality results it is important to state the conditions guaranteeing the proper rate of convergence of bias to zero, that is, the following relation to hold: (2.12) In the above Theorem 2.2 the conditions which help to control bias are H1-H2 and B9.Remark 2.9.Note that the previous results on Ibragimov estimators (see, e.g.[2], [3], [4], [7], [11]) were stated under the conditions of integrability of the spectral density f (λ, θ) and function w(λ) log ψ(λ, θ) (and integrability of some derivatives of the latter function), and the condition for bias control (2.12) was just imposed.We mention also that the investigation of bias for estimators of spectral functionals, in non-parametric setting, was presented in [5], [40], [41].
In comparison with previous papers on Ibragimov estimators, in the present paper we give all conditions for consistency and asymptotic normality in a different form, prescribing exactly the behavior of the spectral density f (λ, θ) , the function w(λ) log ψ(λ, θ) and some derivatives at the point of singularity; we also give the corresponding conditions needed for the proper rate of convergence of bias (that is for (2.12) to hold) in the exact form.The form of conditions makes them very operational and easily applicable to concrete models of random fields with singularities (as demonstrated with some examples in Section 5).
We mention that for the case of Whittle estimators for Gaussian random fields with spectral densities possessing singularities of multiplicative form, the results on consistency and asymptotic normality were obtained in [36], therein the weight function w(λ) and taper are both related to the order of singularity of the spectral density.We discuss the results from [36] in Section 3.2, and we also describe in Section 3.1 conditions used in [28] for the case of weakly dependent random fields.Remark 2.10.As follows from the results by Guyon (see, e.g., [28]) outlined in the next section, to control bias for Ibragimov minimum contrast estimators in the case of short-range dependent random fields the following conditions can be used: the taper h(t) is in C 2 [−1, 1]; the spectral density f (λ, θ) and the function ϕ(λ, θ) = w(λ) ∂ ∂θi log ψ(λ, θ) belong to C 2 (T).Remark 2.11.Concerning the condition B8.If instead of condition 3 in B7 we use the following condition 3 : the second order derivatives ∂ 2 ∂θi∂θj log ψ(λ, θ), i, j = 1, . . ., q are continuous in both (λ, θ), λ = 0, then one can use just nondegenerate S(θ) in B8.Indeed, since the point θ 0 is the point of maximum for the functional −U (θ) = T f (λ, θ 0 )w(λ) log ψ(λ, θ) dλ, then under the above 3 the matrix ∇ θ ∇ θ (−U T (θ 0 )) have to be negatively semi-definite, and supposing is negatively definite, and, therefore, S(θ 0 ) = ∇ θ ∇ θ (U T (θ 0 )) is positively definite.(See, for example, [31], comments after condition N8 therein).

On the Whittle estimators with tapered data
In this section we will give a brief review of some results from the literature on the Whittle estimators based on tapered data, with a particular focus on the conditions on tapers which help to control the bias of estimators.
Recall that the Whittle estimators can be defined as minimum contrast estimators with the following contrast field: and corresponding contrast function Note that here again the function w is aimed to compensate for possible singularities of the spectral density and in some cases is not required.

Case of weakly dependent random fields:
Results from [28] Apparently, one of the first publications that addressed the problem of edge effects was [27]: considering the usual parametric Whittle estimates for lattice data, it found that such estimates had bias of order N −1/d , for fields observed on a rectangle We review the results by [27,28].There the domain L T is taken to be of the form L T = d i=1 [1, T i ], we simplify our exposition and consider the cube, that is, all Instead of the observed data {Y (t), t ∈ L T }, one considers the tapered data {h T (t)Y (t), t ∈ L T }, where the tapers are of the following form: and = 1, and it is increasing and belonging to C 2 , and 0 ≤ ρ ≤ 1.(In such a way one obtains the so-called g-taper which tapers 100(1 − ρ)% of the edge values.) We next formulate the results on bias control for tapered estimators stated by Guyon (see, [28]).
Consider J(φ) = T f (λ)φ(λ) dλ and the corresponing estimator based on tapered data J T (φ) = T I h T (λ)φ(λ) dλ.Proposition 1.Let φ ∈ C(T) and assume that the taper h and the spectral density f belong to C 2 .Then as T → ∞, where C is a constant.
Therefore, under the above conditions, if ρ is fixed the bias is of order T −2 , thus smaller than , which is still smaller than T −d/2 .That is, for d = 1, 2, 3, lim Moreover, if φ 1 , φ 2 ∈ C(T), the taper h and the spectral density f belong to C 2 , then for a Gaussian random field we have the following asymptotic behavior of the covariance: lim where the taper factor e(h) is given by (2.11), e(h) ≥ 1 and e(h) = 1 if there is no tapering.It it possible to choose ρ T such that the taper factor will tend to 1.

Asymptotic properties of parameter estimates for random fields with tapered data 3345
We also mention that the asymptotic normality of the Whittle estimators was stated in [28] under some mixing assumptions.[36] In the paper [36] the following two models of spectral densities with singularities were considered:

Case of strongly dependent random fields: results from
where we assume for all δ > 0 that is a positive twice continuously differentiable function of (λ, θ) A2 In this case, we assume there exists a function α : Θ → (0, d), such that the spectral density f (λ, θ) satisfies for all δ > 0 : 1)) where the o( 1) is uniform with respect to λ. (For more details, see [36]).
The authors study the Whittle estimator θT = arg min θ∈Θ U T (θ) for both models A1 and A2 using the tapered data {h T,ε (t)Y (t)}, the functional U T (θ) is given by (3.1), where the function w(λ) (called by the authors smoothing or regularizing function) depends on the shape of singularity.Under A1 w(λ) In the course of derivation of asymptotic normality for the Whittle estimator the following result on bias was obtained for the functional J T (ϕ) = T I h T (λ) ϕ (λ) dλ taken as an estimator for where κ = ν under A1, or κ = 1 under A2, and, therefore, the bias is of order Note that in the derivation of the above result, the form of ϕ (as needed for a particular case of the Whittle functional) was essentially used.With this function, and under conditions of Proposition 2, some regularity properties of convolution T ϕ(λ − μ)f (λ) dλ were obtained, and used for bias evaluation, as well as an interplay between a taper, and the function w(λ) under A1.Multiplicatives structure of the functions defining J(ϕ) was also essential under A1.
We also mention that asymptotic normality of the Whittle estimators was stated in [36] under the assumption A1 using the CLT analogous to that of [24], and, under the assumption A2, the CLT stated in [21] was used.

Proofs
The proofs are based on the results on asymptotic properties of the functionals J T (ϕ) obtained via evaluation of their cumulants.These cumulants can be represented in the form of some integrals involving spectral densities, weight functions and Fejér type kernels.Although these kernels are different for nontapered and tapered cases, all technique for the proofs works similarly in both cases (see more details in Appendix).
The main difference from the previous papers on Ibragimov estimators is that here, using the tapers, we are able to formulate the exact conditions to achieve proper rate of convergence of bias to state asymptotic normality.We present here principal steps for the proofs and state Lemma 4.1, which gives the possibility to control bias as needed for Theorem 2.2.
The main tool to obtain asymptotic normality of the estimators is Theorem A.2 (see Appendix).This theorem gives sufficient conditions for asymptotic normality of J T (ϕ).We will also need conditions for convergence in probability of functionals J T (ϕ) which we discuss in the next remark.

Remark 4.1. To state the convergence in probability
it is sufficient to show that: Therefore, for (4.1) to hold we can use conditions guaranteeing the convergence (i) and convergence of the variance of the normalized functional T d/2 (J T (ϕ) − EJ T (ϕ)) to a finite limit.In particular, convergence (4.1) holds under the conditions on f (λ) and ϕ (λ) imposed in Theorem A.2.
Proof of Theorem 2.1.In view of Remark 4.1, taking into account the expression for U T (θ), we conclude that convergence lim Moreover, under conditions B5 and B6, the above convergence holds uniformly with respect to θ ∈ Θ.Indeed, denoting by η (ε) the modulus of continuity of the function h(λ, θ), we can write: and the integral in the r.h.s. is asymptotically bounded in probability under conditions B5 and B6.Therefore, in view of Theorem 3.4.1 from Guyon [28], we conclude that the estimator θT is consistent: θT P0 −→ θ 0 , as T → ∞.Minimum contrast property for K(θ 0 ; θ) follows from Jensen's inequality: therefore, K(θ 0 ; θ) ≥ 0, and, moreover, K(θ 0 ; θ) > 0 if ψ(λ, θ 0 ) ≡ ψ(λ, θ) for θ 0 = θ almost everywhere with respect to the Lebesgue measure.Convergence can be stated with the use of the same arguments as those for convergence U T (θ) P0 −→ U (θ), which was stated above.Namely, we can apply again our observation form Remark 4.1, that is to state (4.2) we can use Theorem A.2.One can see that under condition B5, w(λ) compensates the singularity of f (λ, θ 0 ), and therefore, the convergence (4.2) follows.
It follows from the definition of minimum contrast estimators that for T sufficiently large So, we need to show that where A(θ) and S(θ) are given in condition B8.Then, by Slutsky's lemma, relation (2.10) is a consequence of (4.3) and (i)-(ii).Note that in view of B4 ..,q , and convergence (ii) will hold if we show that and where ϕ i (λ) = w(λ) ∂ ∂θi log ψ(λ, θ 0 ).Convergence (4.4) holds under conditions B5 and B7(2) in view of Theorem A.2; convergence (i) holds under B5, B6, B7(1), B7(3) in view of Remark 4.1 (and taking into account consistency of θT ).Convergence (4.5) will hold under conditions H1-H2 and B9 in view of Lemma 4.1 stated below.Note that in the previous papers convergence (4.5) was imposed as assumption (see, for example, [7,10]).
Proof.The analog of this lemma for the continuous-parameter fields was obtained in [40], the results on bias evaluation for the spectral functionals of higher-orders in continuous and discrete contexts were stated in [7,10].The present proof uses ideas from these papers.With direct calculations, the bias of J T (ϕ) can be represented as follows: Asymptotic properties of parameter estimates for random fields with tapered data 3349 where 2 is an even function which integrates to 1 (more precisely, this is the kernel of Fejér type, see Appendix), with H 1,T (u) being given in (2.2) for k = 1, and we have denoted g(u) = f (λ)ϕ (λ + u) dλ.Therefore, to evaluate the bias we need to analyze the asymptotic behavior of the expressions (4.7)-(4.9).Here we can apply standard arguments if we impose conditions of regularity on the functions f , ϕ, or, more generally, on their convolution g.
Consider the expression (4.7) and conditions (i).By Taylor's theorem and assumption that f (λ) is twice boundedly differentiable on T, Consider the expression Since H1,T (u i ) 2 • (2π H2,T (0)) −1 is a kernel of Fejér type and integrates to 1 (see, for example, [19]), the inner integrals in (4.10) reduce to the expressions which is equal to zero since H1,T (u i ) 2 is an even function.Therefore, (4.10) is equal to zero and The inner integrals in (4.11) reduce to the expressions We note that H2,T (0) ∼ T h2 (t)dt (4.12) as T → ∞, since H2,T (0) supplied with the factor 1 T gives, in fact, the partial sum for the integral appearing in the r.h.s. of (4.12).
Next, as in Brillinger [15] we can write: where Δ t (u) has the form of a product of (sin(u/2)) −1 with some bounded with respect to T and u function, Δ 0 (u) ≡ 1.For more details we refer, e.g., to [15] (see the proof of Corollary 7.2.1),[19].Now, taking into account that the taper function h(t) is Lipschitz-continuous, we can write the following estimate: Similar arguments are used also in [38] (see the proof of Theorem 3 and formula (3.21) therein).Therefore, each term at the right-hand side of (4.11) is bounded by const•T −2 , which gives the asymptotics (4.6) under the condition (i).Analogously we can deduce (4.6) from the expressions (4.8) or (4.9) applying the conditions (ii) or (iii) respectively.

Numerical examples
This section illustrates the obtained results and numerically investigates properties of the considered minimum contrast estimators.Numerical results fully support the theoretical findings.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.
In the following examples we use random field models with spectral densities of the form where In Example 1 the values of parameters u 1 and u 2 are known and equal u 1 = u 2 = 1.In example 3 α = β = 0 and φ(z 1 , z 2 , 0, 0) ≡ 1. Example 2 considers the general case when singularities are located outside of the origin and we suppose again α = β = 0.
We show that in all cases the minimum contrast approach gives very good convergence rates and asymptotic normality.
We chose the taper of the form h The taper h(•) also satisfies the conditions in Remark 2.6.

Spatial fractional autoregressive processes
First we start with the family of spatial fractional autoregressive processes introduced in [16].Let Y t , t = (t 1 , t 2 ) ∈ Z 2 , be the spatial process satisfying the following fractional autoregressive model: where , with B i denoting the backward-shift operator for the coordinate t i , i = 1, 2, i.e., B 1 Y t1,t2 = Y t1−1,t2 , and The spectral density of the process defined by (5.2) is given by see [16].
Assume that the values of the parameters α and β are known, or they have been estimated before.Then we are interested to estimate the parameter vector By direct calculations it is easy to check that all conditions for consistency and asymptotic normality of the estimator θT are fulfilled.In particular, for condition B6 we can use the function 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.
So, we have consistency of the estimator and also asymptotic normality where for the chosen taper we have e(h) ≈ 3.78.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). 100 realizations of Y t1,t2 were generated, for each T in the increasing sample size sequence T = 10, 30, 50, 70, 90, 110, 130, 150.The following values of parameters were used in simulations: 2) was expanded in a double power series with respect to each B i , i = 1, 2. Realizations of Y were approximated using the truncated sums with powers not exceeding 30 for each B i .The periodogram I h T was computed with h T (t 1 , t 2 ) and ω(λ 1 , λ 2 ) given above.
The minimizing arguments θ T of the functional U T (θ) in (2.6) were found numerically for each simulation.
For each T Table 1 shows estimated values d1 and d2 of the parameters and the mean square errors (MSE) of d1 and d2 .For each value of T Figure 1 produces a boxplot of estimated values d1 and d2 based on 100 realizations.It is clear that the estimated values are centered at the true values d 1 = 0.2 and d 2 = 0.3.Moreover, Figure 1 and Table 1 demonstrate that θ T = ( d1 , d2 ) converges to θ 0 = (0.2, 0.3) as T increases.
Normal Q-Q plots of d1 and d2 were built for large values of T to verify asymptotic normality of the estimators.Figure 2 demonstrates that the empirical distributions of d1 and d2 match the theoretical normal distributions.
Finally, we investigated asymptotic properties of σT .Similarly to the above cases, for each T we built boxplots and plotted sample probabilities for 100 simulations.Figures 4 and 5 support convergence in probability

Spatial Gegenbauer random fields: singularities at non-zero locations
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 , where t1,t2 , (t 1 , t 2 ) ∈ Z 2 , is a zero-mean white noise field with the common variance E[ 2 t1,t2 ] = σ 2 , where, for j = 1, 2, As before B j , j = 1, 2, denotes the backward-shift operator for each spatial coordinate and u = cos ν, i.e. ν = arccos(u).
There exists the following representation of a spatial stationary Gegenbauer random field, see [22], where d i = 0, and n (u) is given by The spectral density f of the spatial Gegenbauer random field is defined as We generated 200 replications of random field given in (5.4), for the increasing sequence of sample sizes T = 10, 30, 50, 70, 90, 110, using the parameter values d 1 = 0.2, d 2 =0.3, u 1 = 0.4, u 2 = 0.3, and σ 2 = 1.Analogously to the example in section 5.1 realizations of Y t1,t2 were approximated by the truncated sums with 100 terms in (5.4).The periodogram The analysis, plots and explanations below are analogous to the example in section 5.1.The minimizing arguments θ T of the functional U T (θ) in (2.6) were found numerically for each simulation.Figure 6 and Table 2 demonstrate that θ T converges to θ 0 as T increases.The plot of the sample probabilities

Spatial Gegenbauer random fields: singularities at the origin
Finally in the third example we consider the case of spectral densities with singularities at the origin and the degenerated φ(B 1 , B 2 , α, β) = I, i.e. α = β = 0.The method shows very good performance in this case as well.
In this example we also used random fields Y satisfying (5.4).Their realisations were simulated for the parameter values The analysis, plots and explanations below are analogous to the example in section 5.1.The minimizing arguments θ T of functional U T (θ) in (2.6) were found numerically for each simulation.Figure 11 and Table 3 show that θT converges to θ T as T increases.
The Normal Q-Q plots of d1 and d2 in Figure 12 match with the theoretical normal distributions.The plot of the sample probabilities P 0 (| θ T − θ 0 | < ε) in Figure 13 also confirms convergence in probability.Table 3 shows MSEs of d1 and d2 , computed for different T values.Figures 14 and 15      Appendix A: Central limit theorems for spectral functionals (or quadratics forms) of Gaussian random fields, from tapered data One of the classical approaches to derive central limit theorem for spectral functionals J T (ϕ) = T I T (λ) ϕ (λ) dλ consists of calculating and evaluating their cumulants.We present here some details of this approach, and the corresponding results for the case of Gaussian fields, and tapered data, as needed for consideration in the present paper.To state CLT for the normalized functional JT (ϕ) = T d/2 (J T (ϕ) − EJ T (ϕ)) it is enough to provide conditions for convergence to the finite limit of the second order cumulant of JT (ϕ), and for convergence to zero of all cumulants of higher orders.
Note Throughout this section we will omit the superscript 'h' and write simply I T (λ) to denote the tapered periodogram and J T (ϕ) to denote the corresponding spectral functional.Only Proposition A.1.here below concerns the non-tapered case.The cumulant of the general k-th order can be represented in the form: λ,t) , and h T (t) = h(t/T ).Details of calculations of the cumulants of spectral functionals can be found, for example, in [2], [10], [14] for the nontapered case, and in [3], [19], [20] for the tapered case.The calculations are based on the so-called product formula for cumulants which gives the expression for cumulants of products of random variables in terms of cumulants of the individual variables, the mentioned formula reduces to a particular simple form in the Gaussian case.
Note that the functions are multidimensional kernels of Fejér type over T k−1 , or approximate identities for convolution.
In the non-tapered case, when h(t) ≡ 1, this was shown in [13,14].In the case under consideration, when the taper factorizes (as defined in Section 2.2), and the domain of observation is a cube L T = [−T, T ] d , this fact follows as a straightforward generalization of the corresponding result by [19] for dimension d = 1.
The kernel property of Φ h k,T (λ 1 , ..., λ k−1 ) implies lim provided that the function G is bounded and continuous at the point (v 1 , ..., v k−1 ).We have, in particular, Asymptotic analysis of expressions for cumulants, based on the property (A.2), allows to state the following asymptotic normality result for the functional J T (ϕ) = J h T (ϕ) = T I h T (λ) ϕ (λ) dλ in the case of tapered data.Theorem A.1.Let X(t), t ∈ Z d , be a zero-mean Gaussian random field with spectral density f (λ) ∈ L p and ϕ(λ) ∈ L q , where 1 p + 1 q ≤ 1 2 .Then where e(h) is defined in (2.11).
The convergence of the variance (A.3) to σ 2 can be obtained by the following arguments: (i) Under the integrability conditions imposed in the theorem on f and ϕ, the inner integral over λ in expression (A.3) is a bounded and continuous function of (u 1 , u 2 , u 3 ), say G(u 1 , u 2 , u 3 ), therefore, the overall integral in (A.3) can be written as G(u 1 , u 2 , u 3 )Φ h 4,T (u 1 , u 2 , u 3 )du 1 du 2 du 3 ; then, due We present here an extension of Proposition A.1.for the case of Gaussian fields and tapered periodogram in the following form.
Theorem A.2. Let X(t), t ∈ Z d , be a zero-mean Gaussian random field with spectral density f (λ) such that for some 0 The sets of discontinuities of functions f (λ) and ϕ(λ) have Lebesgue measure zero, and these functions are bounded for δ ≤ |λ| ≤ π for all δ > 0. Then where σ 2 is the same as in Theorem A.1 From Theorem A.2 and Lemma 4.1 (see Section 4) we can deduce the following corollary.
Corollary 2. Let the conditions of Theorem A.2 and Lemma 4.1 be satisfied.Then where σ 2 is given by (A.5).
Remark A.1.Proposition A.1 is stated for the functions f (λ) and ϕ(λ) which have singularities at the point λ = 0.However, it can be shown (see, [25]) that the choice λ = 0 is not essential and Proposition A.
where ∇ = 1 − B, B is the backward shift operator (BX(t) = X(t − 1)), and where I Y T (λ) = For the functional which corresponds to the process Y (t) we readily have the convergence T E| JY T (ψ)| 2 → σ 2 as T → ∞.We will show that under the conditions of the theorem T E| JX T (ϕ)| 2 and T E JX T (ϕ) JY T (ψ) also tend to σ 2 as T → ∞, and, therefore convergence (A.12) holds.
Using (A.3) we can write: For non-tapered case this was shown in [24].The result holds for the tapered case as well under the conditions on taper introduced.This can be shown by the same arguments as in [24], namely, by considering the Fourier coefficients of measures μ and μ T .Convergence of Fourier coefficients of μ T to those of μ can be deduced from the kernel property (A.2).
Next we note that the following estimate holds for H 1,T : |H 1,T (λ)| ≤ const • l T (λ), where l T (u) denotes 2π-periodic extension of the function l * T (u), which is defined as: l * T (u) = T for |u| ≤ 1 T , and l * T (u) = 1 |u| for 1 T < |u| ≤ π.From this point convergence of (A.14), (A.15), (A.16) to σ 2 defined by (A.11) can be obtained following the corresponding lines of the proof in [24], more precisely, we do not need their arguments in full generality, but just use those arguments for the second order cumulant, and parts a) of their Propositions 6.1 and 6.2 work in our case.
The proof for the case d = 1 can be directly extended for d > 1 in the case under consideration, when the singularities of spectral density factorize as described in the formulation of theorem, and for the taper which factorizes, so that the integrals can be split as d-tuple of integrals, which appear when d = 1, and, therefore, corresponding reasonings can be preserved.Note that the filtered field is introduced as

Appendix B: Expression for the elements of S(θ) and A(θ)
The following expressions are relevant for the example considered in Section 5.1.Specifically, to write down the expression for the elements of the matrices S(θ) and A(θ), we will use the following derivatives: We obtain, see pages 671 and 672 in [22], the indicator function on the d-torus) under A2.The taper function h T,ε (t) = d i=1 hε ti T , where hε : [0, 1] → [0, 1] is of the following form:

Fig 7 .
Fig 7. The Normal Q-Q plots of d1 and d2 .

Table 1
Mean, standard deviation and MSE of d1 and d2 .
Fig 1. Boxplots of sampled values of d1 and d2 .

Table 2
Mean, standard deviation and MSE of d1 and d2 .

Table 3
Mean, standard deviation and MSE of d1 and d2 .