Bernstein inequality for the parameter of the $p$th order autoregressive process AR$(p)$
Volume 33 / 2006
Applicationes Mathematicae 33 (2006), 253-264
MSC: 60B, 46E15, 60F15, 60G25.
DOI: 10.4064/am33-3-1
Abstract
The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter $\widetilde{\theta}$ of the first-order autoregressive process taking values in a real separable Banach space $B$ $(ARB(1))$, if it satisfies the following relation: $$ \widetilde{X}_t=\widetilde{\theta} \widetilde{X}_{t-1}+ \widetilde{\varepsilon}_t. $$ In this paper we study the convergence in distribution of the linear operator $I(\widetilde{\theta}_T, \widetilde{\theta})= (\widetilde{\theta}_T-\widetilde{\theta})\widetilde{\theta}^{T-2}$ for $\|\widetilde{\theta}\|>1$ and so we construct inequalities of Bernstein type for this operator.
