Yule–Walker type estimators in periodic bilinear models: strong consistency and asymptotic normality

Abstract

Periodic bilinear processes, Yule–Walker type estimator, Wald statistics, Strong consistency, Asymptotic normality,

Consistent and Asymptotically Normal Estimators for Periodic Bilinear Models

“…The analysis of the simulation results for the MDE in the case n = 100 shows that the asymptotic theory should be used with caution for moderate sample sizes. In addition, even for large samples, the rejection relative frequency of H (2) 0 is far from the theoretical 5% for n = 2000. Regarding the results given into brackets, but for the columns dealing with b M n , it is worth noting that, on the one hand, the effect of the t 5 -distribution is insignificant apart from the RMSE or RMSE * .…”

“…In this subsection, we concentrate on covariance structures for {X 2 t (υ)} t∈Z defined by (2). Such results will be crucially needed for applications to the estimation procedure we will consider in Section 3.…”

“…. , X 2n ) be a sample from a periodically stationary bilinear process generated by (2). This is equivalent to having a sample (X 1 , .…”

“…In Section 2, we give conditions ensuring existence of moments up to the p-th order (p ≥ 2) for the process defined by (2) and the explicit form of the covariance function of the squared process. Further details for the proofs given in this section can be found in Bibi and Aknouche [2]. Section 3 presents results on asymptotic inference.…”

“…Based on Bibi and Gautier [3] and further extensions, the main purpose of this paper is to consider the problem of estimating the model (1) with P = 2 and s = 2, i.e., (2) X t (υ) = b υ X t (υ − 2) t (υ − 1) + t (υ),…”