Bootstrap Inference for a First-Order Autoregression with Positive Innovations

Abstract

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 ϕ. 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 ϕ, 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 ϕ 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.

Key Words:

• Autocorrelation coefficient
• Bootstrap
• Extreme value estimator
• Point processes
• Positive AR(1) processes
• Regular variation