Interval Estimation for a First-Order Positive Autoregressive Process

We are interested in constructing confidence intervals for the autoregressive (AR) coefficient of a first-order AR model with i.i.d. positive errors via an extreme value estimate (EVE). We assume that the error distribution has a density function f_ε(x) behaving like b_1,0x^{𝛼0−1} as x→0, where b_1,0 and α₀ are unknown positive constants. These specifications imply that the EVE has a limiting distribution depending on b_1,0 and α₀ from which only an infeasible interval estimate can be obtained. To alleviate this difficulty, we introduce a novel procedure to estimate these two constants and establish the desired consistency. This consistency result enables us not only to gain a better understanding of the underlying error distribution, but also to construct a feasible, asymptotically valid confidence interval of the AR coefficient, without resorting to a bootstrap procedure described in Datta and McCormick (1995). The performance of the proposed interval estimate is further illustrated through simulation studies and real data analysis. Copyright © 2018 John Wiley & Sons Ltd