On spectral and bispectral estimator of the parameter of nongaussian data

Abstract

An estimator for a parameter of a non-Gaussian series data obtained as minimum of a particular quadratic form which depends on the second and the third order spectra is considered. This estimate is shown to be consistent and asymptotically normal under certain set of assumptions.

Non-Gaussian Ornstein–Uhlenbeck-based Models and Some of Their Uses in Financial Economics

“…Nevertheless, they do not pay enough attention to estimation in the frequency domain, which has considerable potential for non-Gaussian data if we can use not only the secondorder information (the second-order spectral densities) but also the higher order information (the higher order spectral densities). Some results in this direction for discrete time stochastic processes have been obtained by Leonenko et al (1998).…”

Forecasting bubbles with mixed causal-noncausal autoregressive models

“…In this Section, we present the estimation function that was first proposed by Leonenko et al (1998). However, we use the variation presented by Velasco and Lobato (2018) in terms of the scaling factor in the denominators of Equation 35.…”

“…This normalization by preliminary estimates has no effect on the identification of the parameters, given that they are invariant to any inversion of the polynomial roots. This step significantly simplifies the estimation compared with Leonenko et al (1998). The estimation function is…”

“…Higher-order cumulants, also named higher-order correlations, have been used to estimate noncausal AR models, see Giannakis and Swami (1990); K. Lii and Rosenblatt (1982); Nikias and Chiang (1987); Rosenblatt (1980); Terdik (1999); Tugnait (1986). We use in this paper the procedure proposed by Brillinger (1985); Leonenko, Sikorskii, and Terdik (1998); Velasco and Lobato (2018), constructing a function of the second and third-order spectral densities (the spectrum and the bispectrum, respectively) in a minimum distance estimation with the periodogram and biperiodogram. The estimation of the parameters is performed by minimizing the function, achieving the identification of the model in its global minimum.…”

Statistical inference using higher-order information

“…In Brillinger [17], a criterion involving the second-and third-order spectral densities was considered with the intention to obtain improved estimates for non-Gaussian time series, as well as for testing the hypothesis of non-Gaussianity. A similar approach was followed in Leonenko et al [46] (see also [54]). …”