Bootstrap inference for a first-order autoregression with positive innovations

In this article we consider statistical inference for the autoregressive parameter of a first-order autoregressive sequence with positive innovations via an extreme value estimator <(phi)over cap>. We show that a bootstrap procedure correctly estimates the sampling distribution of an asymptotically pivotal quantity (whose distribution depends only on the exponent of regular variation of the innovation distribution) based on <(phi)over cap>, provided that the ratio of the bootstrap sample size m and the original sample size n converges to zero. This result enables us to construct a totally nonparametric confidence interval for the autoregressive parameter. We also consider bootstrapping a normalized version of <(phi)over cap> with an application toward bias correction. To obtain the bootstrap validity results, we develop a continuous convergence result for certain associated point processes. We also present results of simulation studies and a numerical example.