Estimation of harmonic component in regression with cyclically dependent errors

Abstract

This paper deals with the estimation of hidden periodicities in a non-linear regression model with stationary noise displaying cyclical dependence. Consistency and asymptotic normality are established for the least-squares estimates.

Detecting hidden periodicities for models with cyclical errors

“…Although the model definition included possible LRD in the error term, this property has not been considered to show the asymptotic properties of the LSE. That is, for αm > 1, α = min j=0,...,κ α j , m is Hermite rank of G (see below), the consistency and limiting Gaussian distribution of the LSE for general regression function are proven in [16]. Using limit theorems of [15] it can be seen that these results hold for α > 1/2 and m = 1.…”

“…In papers [13,14] asymptotic distributions of a class of M-estimates and L p -estimates (1 < p < 2) in nonlinear regression model with LRD form were presented. The problem of the estimation of the unknown parameters of the trigonometric regression with cyclical dependent stationary noise is studied in [16]. The authors derived LSE consistency and asymptotic normality of the regression function (2) parameters, and error term ε being a zero-mean stationary process, generated by nonlinear transformation of a stationary Gaussian process ξ displaying cyclical dependence.…”

On the Whittle estimator for linear random noise spectral density parameter in continuous-time nonlinear regression models

“…Some results on consistency of the LSE α T in the observation model of the type (1) with stationary noise ε(t), t ∈ R, were obtained, for example, in Ivanov and Leonenko [34][35][36][37], Ivanov [31,33], Ivanov et al [38] to mention several of the relevant works. In this section we formulate a generalization of Malinvaud theorem [47] on α T consistency for linear stochastic process (4) and consider an example of nonlinear regression function g(t, α) satisfying the conditions of this theorem and conditions C 1 , C 2 .…”

“…Using the theorem, just as in the works cited above (for definiteness, we turn our attention to Ivanov et al [38]), it can be proved that, if a number of additional conditions on the regression function are satisfied, the normalized LSE d T (α 0 ) ( α T − α 0 ) is asymptotically normal N (0, Σ LSE ), with Note that, firstly, our conditions N 3 , 1), 2) are included in the conditions for the LSE asymptotic normality of Ivanov et al [38], and, secondly, the trigonometric regression function (56) satisfies the conditions of Ivanov et al [38]. Moreover, using (70) and (59) we conclude that for the trigonometric model the normalized LSE…”

“…Cumbersome sets of conditions on the behavior of the nonlinear regression function are used in the proofs of the LSE asymptotic normality of the model parameter can be found, for example, in [32,34,38], and it does not make sense to write here all of them. We will comment only on the conditions associated with the proof of the CLT for one weighted integral of the linear process ε in the observation model (1).…”