Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence

Abstract

Abstract. This paper provides a quasilikelihood/minimum contrast-type method for parameter estimation of random &elds in the frequency domain based on higher-order information. The estimation technique uses the spectral density of the general k-th order and allows for possible long-range dependence in the random &elds. To avoid bias due to edge effects, data tapering is incorporated in the method. The suggested minimum contrast functional is linear with respect to the periodogram of k-th order, hence kernel estimation for the spectral densities is not needed. Furthermore, discretisation is not required in the estimation of continuously observed random &elds. The consistency and asymptotic normality of the resulting estimators are established. Illustrative application of the method to some problems in mathematical &nance and signal detection will be indicated.

Asymptotic properties of parameter estimates for random fields with tapered data

“…Note that the previous results on Ibragimov estimators (see, e.g. [2], [3], [4], [7], [11]) where 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].…”

“…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.…”

“…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]).…”

“…[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 make 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.…”

On multivariate nonlinear regression models with stationary correlated errors

“…This shows that the next Theorem, due to Robinson (1972), is indeed valid not only for a compact set , but also for more general classes of parameters, for example convex parameter sets as well. The minimum contrast method is also called the quasi-likelihood and it is very efficient in several cases even in non-Gaussian situations, for instance see Anh et al (2004).…”