Bernstein inequality for the parameter of the $p$th order autoregressive process AR$(p)$

Samir Benaissa¹

¹ Laboratoire de Mathématiques Université Djillali Liabès B.P. 89, Sidi Bel Abbès 22000, Algeria

Applicationes Mathematicae, Tome 33 (2006) no. 3-4, pp. 253-264

Résumé

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.

Détail

DOI : 10.4064/am33-3-1

Keywords: autoregressive process takes important part predicting problems leading decision making practice least squares method estimate parameter widetilde theta first order autoregressive process taking values real separable banach space arb satisfies following relation widetilde widetilde theta widetilde t widetilde varepsilon paper study convergence distribution linear operator widetilde theta widetilde theta widetilde theta t widetilde theta widetilde theta t widetilde theta construct inequalities bernstein type operator