Statistical inference using higher-order information

Abstract

This paper presents a class of minimum contrast estimators for stochastic processes with possible longrange dependence based on the information on higher-order spectral densities. The results on consistency and asymptotic normality of the proposed estimators are provided.

On the Estimation of Spectral Functionals of the Fourth Order for Stationary Random Fields

“…Note that integrals (1) appear, in particular, in the problems of parameter estimation in the spectral domain within the minimum contrast (or quasi-likelihood) method developed in Anh, Leonenko, and Sakhno (2004), Anh, Leonenko, and Sakhno (2007b), where estimators are based on the Ibragimov functional constructed with the use of information on higher-order spectral densities. The integrals (3) appear in the expressions for covariance matrices of the asymptotic normal law for the Whittle estimators for Gaussian processes and fields, as well as for the Ibragimov estimators based on the second order spectra.…”

Asymptotic properties of parameter estimates for random fields with tapered data

“…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]).…”

Evaluation of bias in higher-order spectral estimation

“…As examples, we mention estimators based on the Whittle functional and the functionals investigated in [20]. Estimators based on the Kullback-Leibler divergence considered in [1]- [3] use information of the spectral densities not only of the second order but also of higher order. To study the properties of minimum contrast estimators based on different contrast functionals (objective functions), one needs to know the limit theorems for the spectral functionals of the form (2) with particular weight functions ϕ k .…”

“…Functionals of the second order have been mostly used for these purposes and, therefore, have been extensively studied by now. But with the minimum contrast techniques based on higher-order spectral densities (as those elaborated in [1]- [3]), one needs the limit theorems for the functionals (2) with k > 2. Besides, we should mention that the expressions for the variance matrices in the limiting normal distribution for minimum contrast estimators contain spectral functionals, and in order to apply asymptotic theory, these functionals have to be estimated.…”